Mathematical relations dehipFrom Wikipedia, the free encyclopedia

Contents

1 Binary relation 11.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Demonic composition 112.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Dense order 123.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Dependence relation 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Dependency relation 14

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5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Directed set 166.1 Equivalent definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Equality (mathematics) 197.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 Equipollence (geometry) 238.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Equivalence class 259.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Equivalence relation 30

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10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Euclidean relation 3911.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Relation to transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

12 Exceptional isomorphism 4012.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

12.1.1 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.2 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.3 Alternating groups and symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.4 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.1.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.1.6 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.2 Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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12.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13 Fiber (mathematics) 4513.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.1.1 Fiber in naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.1.2 Fiber in algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.2 Terminological variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

14 Finitary relation 4714.1 Informal introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.2 Relations with a small number of “places” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

15 Foundational relation 5115.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 Hypostatic abstraction 5216.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

17 Idempotence 5417.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

17.1.1 Unary operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.1.2 Idempotent elements and binary operations . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.1.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17.2 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.2.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.2.3 Idempotent ring elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.2.4 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

17.3 Computer science meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

17.4 Applied examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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17.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18 Idempotent relation 5918.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

19 Intransitivity 6019.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

20 Inverse relation 6320.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

20.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21 Inverse trigonometric functions 6521.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

21.1.1 Etymology of the arc- prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

21.2.1 Principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.2.2 Relationships between trigonometric functions and inverse trigonometric functions . . . . . 6621.2.3 Relationships among the inverse trigonometric functions . . . . . . . . . . . . . . . . . . . 6621.2.4 Arctangent addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

21.3 In calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.3.1 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 6721.3.2 Expression as definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.3.3 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.3.4 Indefinite integrals of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . 70

21.4 Extension to complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.4.1 Logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

21.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.5.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.5.2 In computer science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

22 Near sets 7922.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8322.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8322.4 Efremovič proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8322.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8422.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8422.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.8 Descriptive δ -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8822.10Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

22.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.11Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9022.12Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 9122.13Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9122.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.17Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

23 Partial equivalence relation 9823.1 Properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9823.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

23.2.1 Euclidean parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9823.2.2 Kernels of partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.2.3 Functions respecting equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 99

23.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

24 Partial function 10024.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10024.2 Total function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.3 Discussion and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

24.3.1 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.3.2 Subtraction of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.3.3 Bottom element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.3.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.3.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

24.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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24.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

25 Partially ordered set 10325.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10425.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 10525.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10525.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10625.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10725.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10825.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10925.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

26 Preorder 11026.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11026.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11126.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11226.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11326.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

27 Prewellordering 11427.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

27.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

27.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

28 Propositional function 11628.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11628.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 117

28.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11728.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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28.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Chapter 1

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

1.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

1

2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, if G = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

1.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

1.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

4 CHAPTER 1. BINARY RELATION

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

1.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B → C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A → C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]

Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

1.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

1.4. OPERATIONS ON BINARY RELATIONS 5

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

1.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

6 CHAPTER 1. BINARY RELATION

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

1.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

1.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

1.5. SETS VERSUS CLASSES 7

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

1.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

1.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

1.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

8 CHAPTER 1. BINARY RELATION

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

1.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

1.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

1.10. REFERENCES 9

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

1.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and

Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

10 CHAPTER 1. BINARY RELATION

1.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Chapter 2

Demonic composition

In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinarycomposition of relations but is robust to refinement of the relations into (partial) functions or injective relations.Unlike ordinary composition of relations, demonic composition is not associative.

2.1 Definition

Suppose R is a binary relation between X and Y and S is a relation between Y and Z. Their right demonic compositionR ;→ S is a relation between X and Z. Its graph is defined as

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (xR y ⇒ y S z)}.

Conversely, their left demonic composition R ;← S is defined by

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (y S z ⇒ xR y)}.

2.2 References• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, Mathematical

Structures in Computer Science 3 (4): 417–433, doi:10.1017/S096012950000030X, MR 1249420.

11

Chapter 3

Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z inX such that x < z < y.

3.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

3.2 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

3.3 See also• Dense set

• Dense-in-itself

• Kripke semantics

3.4 References• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

12

Chapter 4

Dependence relation

Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.Let X be a set. A (binary) relation ◁ between an element a of X and a subset S of X is called a dependence relation,written a ◁ S , if it satisfies the following properties:

• if a ∈ S , then a ◁ S ;

• if a ◁ S , then there is a finite subset S0 of S , such that a ◁ S0 ;

• if T is a subset of X such that b ∈ S implies b ◁ T , then a ◁ S implies a ◁ T ;

• if a ◁ S but a ̸◁S − {b} for some b ∈ S , then b ◁ (S − {b}) ∪ {a} .

Given a dependence relation ◁ on X , a subset S of X is said to be independent if a ̸◁S−{a} for all a ∈ S. If S ⊆ T, then S is said to span T if t ◁ S for every t ∈ T. S is said to be a basis of X if S is independent and S spans X.

Remark. If X is a non-empty set with a dependence relation ◁ , then X always has a basis with respect to ◁.Furthermore, any two bases of X have the same cardinality.

4.1 Examples• Let V be a vector space over a field F. The relation ◁ , defined by υ ◁ S if υ is in the subspace spanned by S ,

is a dependence relation. This is equivalent to the definition of linear dependence.

• Let K be a field extension of F. Define ◁ by α◁S if α is algebraic over F (S). Then ◁ is a dependence relation.This is equivalent to the definition of algebraic dependence.

4.2 See also• matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

13

Chapter 5

Dependency relation

For other uses, see Dependency (disambiguation).Not to be confused with Dependence relation, which is a generalization of the concept of linear dependence amongmembers of a vector space.

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, andreflexive; i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs D , such that

• If (a, b) ∈ D then (b, a) ∈ D (symmetric)• If a is an element of the set on which the relation is defined, then (a, a) ∈ D (reflexive)

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation bydiscarding transitivity.Let Σ denote the alphabet of all the letters of D . Then the independency induced by D is the binary relation I

I = Σ× Σ \D

That is, the independency is the set of all ordered pairs that are not in D . The independency is symmetric andirreflexive.The pairs (Σ, D) and (Σ, I) , or the triple (Σ, D, I) (with I induced by D ) are sometimes called the concurrentalphabet or the reliance alphabet.The pairs of letters in an independency relation induce an equivalence relation on the free monoid of all possiblestrings of finite length. The elements of the equivalence classes induced by the independency are called traces, andare studied in trace theory.

5.1 Examples

Consider the alphabet Σ = {a, b, c} . A possible dependency relation is

D = {a, b} × {a, b} ∪ {a, c} × {a, c}= {a, b}2 ∪ {a, c}2

= {(a, b), (b, a), (a, c), (c, a), (a, a), (b, b), (c, c)}

The corresponding independency is

ID = {(b, c) , (c, b)}

Therefore, the letters b, c commute, or are independent of one another.

14

5.1. EXAMPLES 15

Aa

b

c

Chapter 6

Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexiveand transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has anupper bound.[1] In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set. A downward directed set is definedanalogously,[2] meaning when every doubleton is bounded below.[3] Some authors (and this article) assume that adirected set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if itis directed both upward and downward.[4]

Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets(contrast partially ordered sets which need not be directed). Join semilattices (which are partially ordered sets) aredirected sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limitused in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

6.1 Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder suchthat every finite subset of A has an upper bound. In this definition, the existence of an upper bound of the emptysubset implies that A is nonempty.

6.2 Examples

Examples of directed sets include:

• The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).

• Let D1 and D2 be directed sets. Then the Cartesian product set D1 × D2 can be made into a directed set bydefining (n1, n2) ≤ (m1, m2) if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is theproduct direction on the Cartesian product.

• It follows from previous example that the set N × N of pairs of natural numbers can be made into a directedset by defining (n0, n1) ≤ (m0, m1) if and only if n0 ≤ m0 and n1 ≤ m1.

• If x0 is a real number, we can turn the set R − {x0} into a directed set by writing a ≤ b if and only if|a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is an example of a directedset that is not ordered (neither totally nor partially).

• A (trivial) example of a partially ordered set that is not directed is the set {a, b}, in which the only orderrelations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the “reals directed towardsx0" but in which the ordering rule only applies to pairs of elements on the same side of x0.

16

6.3. CONTRAST WITH SEMILATTICES 17

• If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed setby writing U ≤ V if and only if U contains V.

• For every U: U ≤ U; since U contains itself.• For every U,V,W : if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U

contains W.• For every U, V: there exists the set U ∩ V such that U ≤ U ∩ V and V ≤ U ∩ V; since both U and V

contain U ∩ V.

• In a poset P, every lower closure of an element, i.e. every subset of the form {a| a in P, a ≤x} where x is afixed element from P, is directed.

6.3 Contrast with semilattices

Witness

Directed sets are a more general concept than (join) semilattices: every join semilattice is a directed set, as the joinor least upper bound of two elements is the desired c. The converse does not hold however, witness the directed set

18 CHAPTER 6. DIRECTED SET

{1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the lastbit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without1111, the set is not directed.)

6.4 Directed subsets

The order relation in a directed sets is not required to be antisymmetric, and therefore directed sets are not alwayspartial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subsetA of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order:in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on theelements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if itsdownward closure is an ideal. While the definition of a directed set is for an “upward-directed” set (every pair ofelements has an upper bound), it is also possible to define a downward-directed set in which every pair of elementshas a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.Directed subsets are used in domain theory, which studies directed complete partial orders.[5] These are posets inwhich every upward-directed set is required to have a least upper bound. In this context, directed subsets againprovide a generalization of convergent sequences.

6.5 See also• Filtered category

• Centered set

• Linked set

6.6 Notes[1] Kelley, p. 65.

[2] Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.

[3] Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.

[4] Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and IntegralEquations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.

[5] Gierz, p. 2.

6.7 References• J. L. Kelley (1955), General Topology.

• Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN0-521-80338-1.

Chapter 7

Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions,asserting that the quantities have the same value or that the expressions represent the same mathematical object. Theequality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign".

7.1 Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”,“level”, “fair”, “just”).

7.2 Types of equalities

7.2.1 Identities

Main article: Identity (mathematics)

When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function.Such an equality of functions is sometimes called an identity. An example is (x + 1)2 = x2 + 2x + 1.

7.2.2 Equalities as predicates

When A and B are not fully specified or depend on some variables, equality is a proposition, which may be truefor some values and false for some other values. Equality is a binary relation, or, in other words, a two-argumentspredicate, which may produce a truth value (false or true) from its arguments. In computer programming, its com-putation from two expressions is known as comparison.

7.2.3 Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties thatare considered. This is, in particular the case in geometry, where two geometric shapes are said equal when one maybe moved to coincide with the other. The word congruence is also used for this kind of equality.

7.2.4 Equations

An equation is the problem of finding values of some variables, called unknowns, for which the specified equalityis true. Equation may also refer to an equality relation that is satisfied only for the values of the variables that oneis interested on. For example x2 + y2 = 1 is the equation of the unit circle. There is no standard notation that

19

20 CHAPTER 7. EQUALITY (MATHEMATICS)

distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriateinterpretation from the semantic of expressions and the context.

7.2.5 Equivalence relations

Main article: Equivalence relation

Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set:those binary relations that are reflexive, symmetric, and transitive. The identity relation is an equivalence relation.Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of allelements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equalityis the smallest equivalence relation on any set S, in the sense that it is the relation that has the smallest equivalenceclasses (every class is reduced to a single element).

7.3 Logical formalizations of equality

There are several formalizations of the notion of equality in mathematical logic, usually by means of axioms, such asthe first few Peano axioms, or the axiom of extensionality in ZF set theory.For example, Azriel Lévy gives as the five axioms for equality, first the three properties of an equivalence relation,and these two:

x = y ∧ x ∈ z ⇒ y ∈ z, andx = y ∧ z ∈ x ⇒ z ∈ y.[1]

These extra two conditions allow substitution of equal quantities into complex expressions.There are also some logic systems that do not have any notion of equality. This reflects the undecidability of theequality of two real numbers defined by formulas involving the integers, the basic arithmetic operations, the logarithmand the exponential function. In other words, there cannot exist any algorithm for deciding such an equality.

7.4 Logical formulations

Equality is always defined such that things that are equal have all and only the same properties. Some people defineequality as congruence. Often equality is just defined as identity.A stronger sense of equality is obtained if some form of Leibniz’s law is added as an axiom; the assertion of this axiomrules out “bare particulars”—things that have all and only the same properties but are not equal to each other—whichare possible in some logical formalisms. The axiom states that two things are equal if they have all and only the sameproperties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective “if and only if” can be weakened to “if"; the modified law is equivalent to the original.Instead of considering Leibniz’s law as an axiom, it can also be taken as the definition of equality. The property ofbeing an equivalence relation, as well as the properties given below, can then be proved: they become theorems. Ifa=b, then a can replace b and b can replace a.

7.5 Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e.are well-formed).

7.6. RELATION WITH EQUIVALENCE AND ISOMORPHISM 21

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functionalpredicate).Some specific examples of this are:

• For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);

• For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);

• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);

• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined,is not transitive (it may seem so at first sight, but many small differences can add up to something big). However,equality almost everywhere is transitive.Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitutionand reflexive properties are assumed instead.

7.6 Relation with equivalence and isomorphism

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism.[2] For example, one may distin-guish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 aredistinct as fractions, as different strings of symbols, but they “represent” the same rational number, the same pointon a number line. This distinction gives rise to the notion of a quotient set.Similarly, the sets

{A,B,C} and {1, 2, 3}

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of threeelements, and thus isomorphic, meaning that there is a bijection between them, for example

A 7→ 1,B 7→ 2,C 7→ 3.

However, there are other choices of isomorphism, such as

A 7→ 3,B 7→ 2,C 7→ 1,

and these sets cannot be identified without making such a choice – any statement that identifies them “dependson choice of identification”. This distinction, between equality and isomorphism, is of fundamental importance incategory theory, and is one motivation for the development of category theory.

22 CHAPTER 7. EQUALITY (MATHEMATICS)

7.7 See also• Equals sign

• Inequality

• Logical equality

• Extensionality

7.8 References[1] Azriel Lévy (1979) Basic Set Theory, page 358, Springer-Verlag

[2] (Mazur 2007)

• Mazur, Barry (12 June 2007), When is one thing equal to some other thing? (PDF)

• Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.

Chapter 8

Equipollence (geometry)

In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB frompoint A to point B has the opposite direction to line segment BA. Two directed line segments are equipollent whenthey have the same length and direction.The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently the term vectorwas adopted for a class of equipollent line segments. Bellavitis’s use of the idea of a relation to compare differentbut similar objects has become a common mathematical technique, particularly in the use of equivalence relations.Bellavitis used a special notation for the equipollence of segments AB and CD:

AB ≏ CD.

The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:

Equipollences continue to hold when one substitutes for the lines in them, other lines which are respec-tively equipollent to them, however they may be situated in space. From this it can be understood howany number and any kind of lines may be summed, and that in whatever order these lines are taken, thesame equipollent-sum will be obtained...

In equipollences, just as in equations, a line may be transferred from one side to the other, provided thatthe sign is changed...

Thus oppositely directed segments are negatives of each other: AB +BA ≏ 0.

The equipollence AB ≏ n.CD, where n stands for a positive number, indicates that AB is both parallelto and has the same direction as CD, and that their lengths have the relation expressed by AB = n.CD .

8.1 References

• Giusto Bellavitis (1835) “Saggio di applicasioni di un nuovo metodo di Geometria Analitica (Calculo delleequipollenze)", Annali delle Scienze del Regno Lombardo-Veneto, Padova 5: 244–59.

• Giusto Bellavitis (1854) Sposizione del Metodo della Equipollenze, link from Google Books.

• Michael J. Crowe (1967) A History of Vector Analysis, “Giusto Bellavitis and His Calculus of Equipollences”,pp 52–4, University of Notre Dame Press.

• Lena L. Severance (1930) The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis,link from HathiTrust.

23

24 CHAPTER 8. EQUIPOLLENCE (GEOMETRY)

8.2 External links• Axiomatic definition of equipollence

Chapter 9

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the first two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

9.1 Notation and formal definition

An equivalence relation is a binary relation ~ satisfying three properties:[1]

• For every element a in X, a ~ a (reflexivity),

• For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

• For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

25

26 CHAPTER 9. EQUIVALENCE CLASS

The equivalence class of an element a is denoted [a] and is defined as the set

[a] = {x ∈ X | a ∼ x}

of elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7→ [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this defines an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more “natural” than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdefined by a ~ b if a − b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identified, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

9.2 Examples• If X is the set of all cars, and ~ is the equivalence relation “has the same color as.” then one particular equivalence

class consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

• Let X be the set of all rectangles in a plane, and ~ the equivalence relation “has the same area as”. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

• Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

• Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentified with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal definition of the set of rational numbers.[5] The same construction can be generalized to the fieldof fractions of any integral domain.

• If X consists of all the lines in, say the Euclidean plane, and L ~ M means that L and M are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at infinity.

9.3 Properties

Every element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

9.4. GRAPHICAL REPRESENTATION 27

• x ∼ y

• [x] = [y]

• [x] ∩ [y] ̸= ∅.

9.4 Graphical representation

Any binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

9.5 Invariants

If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. Some authors use “compatible with ~" or just “respects ~" instead of “invariantunder ~".Any function f : X → Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

9.6 Quotient space in topology

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original space’s topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relation’s set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a set X either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the definition of invariants of equivalence relations given above.

9.7 See also• Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible

28 CHAPTER 9. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs

• Homogeneous space, the quotient space of Lie groups.

• Transversal (combinatorics)

9.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 77–78

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

9.9 References• Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8

• Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

• Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

• Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X

• Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematician’s Toolbox, Freeman, ISBN 978-0-7167-3050-7

9.10 Further reading

This material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

• Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall

• Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole)

• Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

• O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall

• Lay (2001), Analysis with an introduction to proof, Prentice Hall

• Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall

• Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent

• Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan

• D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

9.10. FURTHER READING 29

• Cupillari, The Nuts and Bolts of Proofs, Wadsworth

• Bond, Introduction to Abstract Mathematics, Brooks/Cole

• Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall

• Ash, A Primer of Abstract Mathematics, MAA

Chapter 10

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two different cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

10.1 Notation

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a ≡R b", or "aRb" otherwise.

10.2 Definition

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

• a ~ a. (Reflexivity)

• if a ~ b then b ~ a. (Symmetry)

• if a ~ b and b ~ c then a ~ c. (Transitivity)

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as[a] = {b ∈ X | a ∼ b} .

10.3 Examples

10.3.1 Simple example

Let the set {a, b, c} have the equivalence relation {(a, a), (b, b), (c, c), (b, c), (c, b)} . The following sets are equivalenceclasses of this relation:[a] = {a}, [b] = [c] = {b, c} .The set of all equivalence classes for this relation is {{a}, {b, c}} .

30

10.4. CONNECTIONS TO OTHER RELATIONS 31

10.3.2 Equivalence relations

The following are all equivalence relations:

• “Has the same birthday as” on the set of all people.

• “Is similar to” on the set of all triangles.

• “Is congruent to” on the set of all triangles.

• “Is congruent to, modulo n" on the integers.

• “Has the same image under a function" on the elements of the domain of the function.

• “Has the same absolute value” on the set of real numbers

• “Has the same cosine” on the set of all angles.

10.3.3 Relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 doesnot imply that 5 ≥ 7. It is, however, a partial order.

• The relation “has a common factor greater than 1 with” between natural numbers greater than 1, is reflexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

• The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reflexive. (If X is also empty then R is reflexive.)

• The relation “is approximately equal to” between real numbers, even if more precisely defined, is not an equiv-alence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is defined asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point,then this defines an equivalence relation.

10.4 Connections to other relations

• A partial order is a relation that is reflexive, antisymmetric, and transitive.

• Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set thatis reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence relation can substitute for one another, but not individuals within a class.

• A strict partial order is irreflexive, transitive, and asymmetric.

• A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and onlyif for all a ∈ X, there exists a b ∈ X such that a ~ b.

• A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.

• A preorder is reflexive and transitive.

• A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare defined. E.g. the congruence relations on groups correspond to the normal subgroups.

32 CHAPTER 10. EQUIVALENCE RELATION

10.5 Well-definedness under an equivalence relation

If ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use “compatible with ~" or just “respects ~" instead of “invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

10.6 Equivalence class, quotient set, partition

Let a, b ∈ X . Some definitions:

10.6.1 Equivalence class

Main article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := {x ∈ X | a ∼ x} denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

10.6.2 Quotient set

Main article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/∼ := {[x] | x ∈ X} , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

10.6.3 Projection

Main article: Projection (relational algebra)

The projection of ~ is the function π : X → X/∼ defined by π(x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X → B be such that a ~ b → f(a) = f(b). Then there is aunique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is abijection.

10.6.4 Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by x ∼ y ⇐⇒ f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

10.6.5 Partition

Main article: Partition of a set

10.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 33

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

∞∑k=0

kn

k!,

where the above is one of the ways to write the nth Bell number.

10.7 Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions:[2][3][4]

• An equivalence relation ~ on a set X partitions X.

• Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

10.8 Comparing equivalence relations

If ~ and ≈ are two equivalence relations on the same set S, and a~b implies a≈b for all a,b ∈ S, then ≈ is said to be acoarser relation than ~, and ~ is a finer relation than ≈. Equivalently,

• ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalenceclass of ≈ is a union of equivalence classes of ~.

• ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial orderrelation.

10.9 Generating equivalence relations

• Given any set X, there is an equivalence relation over the set [X→X] of all possible functions X→X. Two suchfunctions are deemed equivalent when their respective sets of fixpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[X→X], and these equivalence classes partition [X→X].

34 CHAPTER 10. EQUIVALENCE RELATION

• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X → X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

• The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X × X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi₊ ₁)∈R or (xi₊₁,xi)∈R, i = 1, ..., n−1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalencerelation ~ generated by:

• • Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y;• Any subset of the identity relation on X has equivalence classes that are the singletons of X.

• Equivalence relations can construct new spaces by “gluing things together.” Let X be the unit Cartesian square[0,1] × [0,1], and let ~ be the equivalence relation on X defined by ∀a, b ∈ [0,1] ((a, 0) ~ (a, 1) ∧ (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identified (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

10.10 Algebraic structure

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

10.10.1 Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Then thefollowing three connected theorems hold:[6]

• ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations, mentionedabove);

• Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion‡;

• Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations differs fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

10.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 35

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A → A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b ↔ (ab−1 ∈ H). The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosetsof H in G. Interchanging a and b yields the left cosets.‡Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that ∀x ∈ A ∀g ∈ G ([g(x)] = [x]), because G satisfies the following fourconditions:

• G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

• Existence of identity function. The identity function, I(x)=x, is an obvious element of G;• Existence of inverse function. Every bijective function g has an inverse g−1, such that gg−1 = I;• Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

Let f and g be any two elements of G. By virtue of the definition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. □Related thinking can be found in Rosen (2008: chpt. 10).

10.10.2 Categories and groupoids

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

• Whereas the notion of “free equivalence relation” does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a “presentation of an equivalence relation,” i.e., a presentation of thecorresponding groupoid;

• Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

• In many contexts “quotienting,” and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

10.10.3 Lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X → Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: X→X toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

10.11 Equivalence relations and mathematical logic

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties defining a relation can be proved independent of each other(and hence necessary parts of the definition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three defining properties ofequivalence relations can be proved mutually independent by the following three examples:

36 CHAPTER 10. EQUIVALENCE RELATION

• Reflexive and transitive: The relation ≤ on N. Or any preorder;

• Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation;

• Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3.”Or any dependency relation.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

• The number of equivalence classes is finite or infinite;

• The number of equivalence classes equals the (finite) natural number n;

• All equivalence classes have infinite cardinality;

• The number of elements in each equivalence class is the natural number n.

10.12 Euclidean relations

Euclid's The Elements includes the following “Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing “equal” by “are in relationwith”). By “relation” is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc ∧ bRc) → aRb (Left-Euclidean relation)(cRa ∧ cRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(aRc ∧ bRc) → aRb [a/c] = (aRa ∧ bRa) → aRb [reflexive; erase T∧] = bRa → aRb. Hence R is symmetric.

(aRc ∧ bRc) → aRb [symmetry] = (aRc ∧ cRb) → aRb. Hence R is transitive. □

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed thereflexivity of equality too obvious to warrant explicit mention.

10.13 See also

• Partition of a set

• Equivalence class

• Up to

• Conjugacy class

• Topological conjugacy

10.14. NOTES 37

10.14 Notes[1] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

[3] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker

[5] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.

[10] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.

[11] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.

[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

10.15 References• Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.

• Castellani, E., 2003, “Symmetry and equivalence” in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.

• Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow equivalence relations arise in lattice theory.

• Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.

• John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.

• Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

• Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdefining equivalence, pp 48–50, John Wiley & Sons.

10.16 External links• Hazewinkel, Michiel, ed. (2001), “Equivalence relation”, Encyclopedia of Mathematics, Springer, ISBN 978-

1-55608-010-4

• Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009

• Equivalence relation at PlanetMath

• Binary matrices representing equivalence relations at OEIS.

38 CHAPTER 10. EQUIVALENCE RELATION

Logical matrices of the 52 equivalence relations on a 5-element set (Colored fields, including those in light gray, stand for ones; whitefields for zeros.)

Chapter 11

Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity thatformalizes Euclid's “Common Notion 1” in The Elements: things which equal the same thing also equal one another.

11.1 Definition

A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for everya, b, c in X, if a is related to b and c, then b is related to c.[1]

To write this in predicate logic:

∀a, b, c ∈ X (aR b ∧ aR c → bR c).

Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b isrelated to c:

∀a, b, c ∈ X (bR a ∧ cR a → bR c).

11.2 Relation to transitivity

The property of being Euclidean is different from transitivity: both the Euclidean property and transitivity infer arelation between b and c from relations between a and b and between a and c, but with different argument orderingsin the relations. However, if a relation is symmetric, then the argument orders do not matter; thus a symmetric relationwith any one of these three properties (transitive, right Euclidean, left Euclidean) must have all three.[1]

If a relation is Euclidean and reflexive, then it must also be symmetric and hence transitive (following the previousparagraph), and so it must be an equivalence relation. Consequently, equivalence relations are exactly the reflexiveEuclidean relations.[1]

11.3 References[1] Fagin, Ronald (2003), Reasoning About Knowledge, MIT Press, p. 60, ISBN 978-0-262-56200-3.

39

Chapter 12

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism betweenmembers ai and bj of two families (usually infinite) of mathematical objects, that is not an example of a pattern ofsuch isomorphisms.[note 1] These coincidences are at times considered a matter of trivia,[1] but in other respects theycan give rise to other phenomena, notably exceptional objects.[1] In the below, coincidences are listed in all placesthey occur.

12.1 Groups

12.1.1 Finite simple groups

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special lineargroups and alternating groups, and are:[1]

• L2(4) ∼= L2(5) ∼= A5, the smallest non-abelian simple group (order 60);

• L2(7) ∼= L3(2), the second-smallest non-abelian simple group (order 168) – PSL(2,7);

• L2(9) ∼= A6,

• L4(2) ∼= A8,

• PSU4(2) ∼= PSp4(3), between a projective special orthogonal group and a projective symplectic group.

12.1.2 Groups of Lie type

In addition to the aforementioned, there are some isomorphisms involving SL, PSL, GL, PGL, and the natural mapsbetween these. For example, the groups over F5 have a number of exceptional isomorphisms:

• PSL(2, 5) ∼= A5∼= I, the alternating group on five elements, or equivalently the icosahedral group;

• PGL(2, 5) ∼= S5, the symmetric group on five elements;

• SL(2, 5) ∼= 2 · A5∼= 2I, the double cover of the alternating group A5, or equivalently the binary icosahedral

group.

12.1.3 Alternating groups and symmetric groups

There are coincidences between alternating groups and small groups of Lie type:

• L2(4) ∼= L2(5) ∼= A5,

40

12.1. GROUPS 41

The compound of five tetrahedra expresses the exceptional isomorphism between the icosahedral group and the alternating group onfive letters.

• L2(9) ∼= Sp4(2)′ ∼= A6,

• Sp4(2) ∼= S6,

• L4(2) ∼= O6(+, 2)′ ∼= A8,

• O6(+, 2) ∼= S8.

These can all be explained in a systematic way by using linear algebra (and the action of Sn on affine n -space) todefine the isomorphism going from the right side to the left side. (The above isomorphisms for A8 and S8 are linkedvia the exceptional isomorphism SL4/µ2

∼= SO6 .) There are also some coincidences with symmetries of regularpolyhedra: the alternating group A5 agrees with the icosahedral group (itself an exceptional object), and the doublecover of the alternating group A5 is the binary icosahedral group.

12.1.4 Cyclic groups

Cyclic groups of small order especially arise in various ways, for instance:

• C2∼= {±1} ∼= O(1) ∼= Spin(1) ∼= Z∗ , the last being the group of units of the integers

42 CHAPTER 12. EXCEPTIONAL ISOMORPHISM

12.1.5 Spheres

The spheres S0, S1, and S3 admit group structures, which arise in various ways:

• S0 ∼= O(1) ,

• S1 ∼= SO(2) ∼= U(1) ∼= Spin(2) ,

• S3 ∼= Spin(3) ∼= SU(2) ∼= Sp(1) .

12.1.6 Coxeter groups

B2 C2≅ ≅

D3A3 ≅

≅

≅

E4A4 ≅

E5D5 ≅

≅

The exceptional isomorphisms of connected Dynkin diagrams.

12.2. LIE THEORY 43

There are some exceptional isomorphisms of Coxeter diagrams, yielding isomorphisms of the corresponding Coxetergroups and of polytopes realizing the symmetries. These are:

• A2 = I2(2) (2-simplex is regular 3-gon/triangle);

• BC2 = I2(4) (2-cube (square) = 2-cross-polytope (diamond) = regular 4-gon)

• A3 = D3 (3-simplex (tetrahedron) is 3-demihypercube (demicube), as per diagram)

• A1 = B1 = C1 (= D1?)

• D2 = A1 × A1

• A4 = E4

• D5 = E5

Closely related ones occur in Lie theory for Dynkin diagrams.

12.2 Lie theory

In low dimensions, there are isomorphisms among the classical Lie algebras and classical Lie groups called acciden-tal isomorphisms. For instance, there are isomorphisms between low-dimensional spin groups and certain classicalLie groups, due to low-dimensional isomorphisms between the root systems of the different families of simple Liealgebras, visible as isomorphisms of the corresponding Dynkin diagrams:

• Trivially, A0 = B0 = C0 = D0

• A1 = B1 = C1 , or sl2 ∼= so3 ∼= sp1

• B2 = C2, or so5 ∼= sp2

• D2 = A1 × A1, or so4 ∼= sl2 ⊕ sl2 ; note that these are disconnected, but part of the D-series

• A3 = D3 sl4 ∼= so6

• A4 = E4; the E-series usually starts at 6, but can be started at 4, yielding isomorphisms

• D5 = E5

Spin(1) = O(1)Spin(2) = U(1) = SO(2)Spin(3) = Sp(1) = SU(2)Spin(4) = Sp(1) × Sp(1)Spin(5) = Sp(2)Spin(6) = SU(4)

12.3 See also• Exceptional object

• Mathematical coincidence, for numerical coincidences

12.4 References[1] Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions),

but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).

44 CHAPTER 12. EXCEPTIONAL ISOMORPHISM

12.5 References[1] Wilson, Robert A. (2009), “Chapter 1: Introduction”, The finite simple groups, Graduate Texts in Mathematics 251 251,

Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792, 2007preprint; Chapter doi:10.1007/978-1-84800-988-2_1.

Chapter 13

Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

1. In naive set theory, the fiber of the element y in the set Y under a map f : X → Y is the inverse image of thesingleton {y} under f.

2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because,in general, not every point is closed.

13.1 Definitions

13.1.1 Fiber in naive set theory

Let f : X → Y be a map. The fiber of an element y ∈ Y , commonly denoted by f−1(y) , is defined as

f−1({y}) = {x ∈ X | f(x) = y}.

In various applications, this is also called:

• the inverse image of {y} under the map f

• the preimage of {y} under the map f

• the level set of the function f at the point y.

The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuousfunction and if y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and moregenerally a hypersurface of dimension d-1.

13.1.2 Fiber in algebraic geometry

In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered productX ×Y Spec k(p) where k(p) is the residue field at p.

13.2 Terminological variance

The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

• the fiber of the element y under the map f

45

46 CHAPTER 13. FIBER (MATHEMATICS)

• the inverse image of the set {y} under the map f

• the preimage of the set {y} under the map f

• the level set of the function f at the point y.

By abuse of language, the following terminology is sometimes used but should be avoided:

• the fiber of the map f at the element y• the inverse image of the map f at the element y• the preimage of the map f at the element y• the level set of the point y under the map f.

13.3 See also• Fibration

• Fiber bundle

• Fiber product

• Image (category theory)

• Image (mathematics)

• Inverse relation

• Kernel (mathematics)

• Level set

• Preimage

• Relation

• Zero set

Chapter 14

Finitary relation

This article is about the set-theoretic notion of relation. For the common case, see binary relation.For other uses, see Relation (disambiguation).

In mathematics, a finitary relation has a finite number of “places”. In set theory and logic, a relation is a propertythat assigns truth values to k -tuples of individuals. Typically, the property describes a possible connection betweenthe components of a k -tuple. For a given set of k -tuples, a truth value is assigned to each k -tuple according towhether the property does or does not hold.An example of a ternary relation (i.e., between three individuals) is: "X was introduced to Y byZ ", where (X,Y, Z)is a 3-tuple of persons; for example, "Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp" istrue, while "Karl Marx was introduced to Friedrich Engels by Queen Victoria" is false.

14.1 Informal introduction

Relation is formally defined in the next section. In this section we introduce the concept of a relation with a familiareveryday example. Consider the relation involving three roles that people might play, expressed in a statement of theform "X thinks that Y likes Z ". The facts of a concrete situation could be organized in a table like the following:Each row of the table records a fact or makes an assertion of the form "X thinks that Y likes Z ". For instance, thefirst row says, in effect, “Alice thinks that Bob likes Denise”. The table represents a relation S over the set P of peopleunder discussion:

P = {Alice, Bob, Charles, Denise}.

The data of the table are equivalent to the following set of ordered triples:

S = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

By a slight abuse of notation, it is usual to write S(Alice, Bob, Denise) to say the same thing as the first row ofthe table. The relation S is a ternary relation, since there are three items involved in each row. The relation itselfis a mathematical object defined in terms of concepts from set theory (i.e., the relation is a subset of the Cartesianproduct on {Person X, Person Y, Person Z}), that carries all of the information from the table in one neat package.Mathematically, then, a relation is simply an “ordered set”.The table for relation S is an extremely simple example of a relational database. The theoretical aspects of databasesare the specialty of one branch of computer science, while their practical impacts have become all too familiar in oureveryday lives. Computer scientists, logicians, and mathematicians, however, tend to see different things when theylook at these concrete examples and samples of the more general concept of a relation.For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematicsat the very least concerns itself with potential infinity. This difference in perspective brings up a number of ideas thatmay be usefully introduced at this point, if by no means covered in depth.

47

48 CHAPTER 14. FINITARY RELATION

14.2 Relations with a small number of “places”

The variable k giving the number of "places" in the relation, 3 for the above example, is a non-negative integer,called the relation’s arity, adicity, or dimension. A relation with k places is variously called a k -ary, a k -adic, ora k -dimensional relation. Relations with a finite number of places are called finite-place or finitary relations. Itis possible to generalize the concept to include infinitary relations between infinitudes of individuals, for exampleinfinite sequences; however, in this article only finitary relations are discussed, which will from now on simply becalled relations.Since there is only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations: the one thatalways holds, and the one that never holds. They are sometimes useful for constructing the base case of an inductionargument. One-place relations are called unary relations. For instance, any set (such as the collection of Nobellaureates) can be viewed as a collection of individuals having some property (such as that of having been awardedthe Nobel prize). Two-place relations are called binary relations or, in the past, dyadic relations. Binary relations arevery common, given the ubiquity of relations such as:

• Equality and inequality, denoted by signs such as ' = ' and ' < ' in statements like ' 5 < 12 ';

• Being a divisor of, denoted by the sign ' | ' in statements like ' 13 | 143 ';

• Set membership, denoted by the sign ' ∈ ' in statements like ' 1 ∈ N '.

A k -ary relation is a straightforward generalization of a binary relation.

14.3 Formal definitionsWhen two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some

connexion, that connexion is called a relation.—Augustus De Morgan[1]

The simpler of the two definitions of k-place relations encountered in mathematics is:Definition 1. A relation L over the sets X1, …, Xk is a subset of their Cartesian product, written L ⊆ X1 × … × Xk.Relations are classified according to the number of sets in the defining Cartesian product, in other words, accordingto the number of terms following L. Hence:

• Lu denotes a unary relation or property;• Luv or uLv denote a binary relation;• Luvw denotes a ternary relation;• Luvwx denotes a quaternary relation.

Relations with more than four terms are usually referred to as k-ary or n-ary, for example, “a 5-ary relation”. A k-aryrelation is simply a set of k-tuples.The second definition makes use of an idiom that is common in mathematics, stipulating that “such and such is ann-tuple” in order to ensure that such and such a mathematical object is determined by the specification of n componentmathematical objects. In the case of a relation L over k sets, there are k + 1 things to specify, namely, the k sets plusa subset of their Cartesian product. In the idiom, this is expressed by saying that L is a (k + 1)-tuple.Definition 2. A relation L over the sets X1, …, Xk is a (k + 1)-tuple L = (X1, …, Xk, G(L)), where G(L) is a subsetof the Cartesian product X1 × … × Xk. G(L) is called the graph of L.Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element a =(a1, …, ak) or the variable element x = (x1, …, xk).A statement of the form "a is in the relation L " or "a satisfies L " is taken to mean that a is in L under the firstdefinition and that a is in G(L) under the second definition.The following considerations apply under either definition:

14.4. HISTORY 49

• The sets Xj for j = 1 to k are called the domains of the relation. Under the first definition, the relation does notuniquely determine a given sequence of domains.

• If all of the domains Xj are the same set X, then it is simpler to refer to L as a k-ary relation over X.

• If any of the domains Xj is empty, then the defining Cartesian product is empty, and the only relation over sucha sequence of domains is the empty relation L = ∅ . Hence it is commonly stipulated that all of the domainsbe nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that fallsunder it will be called a relation for the duration of that discussion. If it becomes necessary to distinguish the twodefinitions, an entity satisfying the second definition may be called an embedded or included relation.If L is a relation over the domains X1, …, Xk, it is conventional to consider a sequence of terms called variables, x1,…, xk, that are said to range over the respective domains.Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values,typically 0 = false and 1 = true. The characteristic function of the relation L, written ƒL or χ(L), is the Boolean-valuedfunction ƒL : X1 × … × Xk → B, defined in such a way that ƒL( x ) = 1 just in case the k-tuple x is in the relation L.Such a function can also be called an indicator function, particularly in probability and statistics, to avoid confusionwith the notion of a characteristic function in probability theory.It is conventional in applied mathematics, computer science, and statistics to refer to a Boolean-valued function like ƒLas a k-place predicate. From the more abstract viewpoint of formal logic and model theory, the relation L constitutesa logical model or a relational structure that serves as one of many possible interpretations of some k-place predicatesymbol.Because relations arise in many scientific disciplines as well as in many branches of mathematics and logic, thereis considerable variation in terminology. This article treats a relation as the set-theoretic extension of a relationalconcept or term. A variant usage reserves the term “relation” to the corresponding logical entity, either the logicalcomprehension, which is the totality of intensions or abstract properties that all of the elements of the relation inextension have in common, or else the symbols that are taken to denote these elements and intensions. Further, somewriters of the latter persuasion introduce terms with more concrete connotations, like “relational structure”, for theset-theoretic extension of a given relational concept.

14.4 History

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relationin anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan andrelations, see Merrill 1990). Charles Sanders Peirce restated and extended De Morgan’s results. Bertrand Russell(1938; 1st ed. 1903) was historically important, in that it brought together in one place many 19th century results onrelations, especially orders, by Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind, and others. Russell and A.N. Whitehead made free use of these results in their Principia Mathematica.

14.5 Notes[1] De Morgan, A. (1858) “On the syllogism, part 3” in Heath, P., ed. (1966) On the syllogism and other logical writings.

Routledge. P. 119,

14.6 See also• Correspondence (mathematics)

• Functional relation

• Incidence structure

• Hypergraph

50 CHAPTER 14. FINITARY RELATION

• Logic of relatives

• Logical matrix

• Partial order

• Projection (set theory)

• Reflexive relation

• Relation algebra

• Sign relation

• Transitive relation

• Relational algebra

• Relational model

14.7 References• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification

of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9,317–78, 1870. Reprinted, Collected Papers CP 3.45–149, Chronological Edition CE 2, 359–429.

• Ulam, S.M. and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for ParallelComputation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), Analogies Between Analogies: TheMathematical Reports of S.M. Ulam andHis Los Alamos Collaborators, University of California Press, Berkeley,CA.

14.8 Bibliography• Bourbaki, N. (1994) Elements of the History of Mathematics, John Meldrum, trans. Springer-Verlag.

• Carnap, Rudolf (1958) Introduction to Symbolic Logic with Applications. Dover Publications.

• Halmos, P.R. (1960) Naive Set Theory. Princeton NJ: D. Van Nostrand Company.

• Lawvere, F.W., and R. Rosebrugh (2003) Sets for Mathematics, Cambridge Univ. Press.

• Lucas, J. R. (1999) Conceptual Roots of Mathematics. Routledge.

• Maddux, R.D. (2006) Relation Algebras, vol. 150 in 'Studies in Logic and the Foundations of Mathematics’.Elsevier Science.

• Merrill, Dan D. (1990) Augustus De Morgan and the logic of relations. Kluwer.

• Peirce, C.S. (1984) Writings of Charles S. Peirce: A Chronological Edition, Volume 2, 1867-1871. PeirceEdition Project, eds. Indiana University Press.

• Russell, Bertrand (1903/1938) The Principles of Mathematics, 2nd ed. Cambridge Univ. Press.

• Suppes, Patrick (1960/1972) Axiomatic Set Theory. Dover Publications.

• Tarski, A. (1956/1983) Logic, Semantics, Metamathematics, Papers from 1923 to 1938, J.H. Woodger, trans.1st edition, Oxford University Press. 2nd edition, J. Corcoran, ed. Indianapolis IN: Hackett Publishing.

• Ulam, S.M. (1990) Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los AlamosCollaborators in A.R. Bednarek and Françoise Ulam, eds., University of California Press.

• R. Fraïssé, Theory of Relations (North Holland; 2000).

Chapter 15

Foundational relation

In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimalelement.Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relationdefined on A. Then (A, R) is a foundational relation if and only if any nonempty subset in A has a R-minimal element.In predicate logic,

(∀S)(S ⊆ A ∧ S ̸= ∅ ⇒ (∃x ∈ S)(S ∩R−1{x} = ∅)

), [1]

in which ∅ denotes the empty set, and R−1{x} denotes the class of the elements that precede x in the relation R. Thatis,

R−1{x} = {y|yRx}. [2]

Here x is an R-minimal element in the subset S, since none of its R-predecessors is in S.

15.1 See also• Binary relation

• Well-order

15.2 References[1] See Definition 6.21 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York:

Springer-Verlag. ISBN 0387900241.

[2] See Theorem 6.19 and Definition 6.20 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev.ed.). New York: Springer-Verlag. ISBN 0387900241.

51

Chapter 16

Hypostatic abstraction

Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal op-eration that transforms a predicate into a relation; for example “Honey is sweet” is transformed into “Honey possessessweetness”. The relation is created between the original subject and a new term that represents the property expressedby the original predicate.Hypostasis changes a propositional formula of the form X is Y to another one of the form X has the property of beingY or X has Y-ness. The logical functioning of the second object Y-ness consists solely in the truth-values of thosepropositions that have the corresponding concrete term Y as the predicate. The object of thought introduced in thisway may be called a hypostatic object and in some senses an abstract object and a formal object.The above definition is adapted from the one given by Charles Sanders Peirce (CP 4.235, “The Simplest Mathematics”(1902), in Collected Papers, CP 4.227–323). As Peirce describes it, the main point about the formal operation ofhypostatic abstraction, insofar as it operates on formal linguistic expressions, is that it converts an adjective or predicateinto an extra subject, thus increasing by one the number of “subject” slots -- called the arity or adicity -- of the mainpredicate.The transformation of “honey is sweet” into “honey possesses sweetness” can be viewed in several ways:

The grammatical trace of this hypostatic transformation is a process that extracts the adjective “sweet” from thepredicate “is sweet”, replacing it by a new, increased-arity predicate “possesses”, and as a by-product of the reaction,as it were, precipitating out the substantive “sweetness” as a second subject of the new predicate.The abstraction of hypostasis takes the concrete physical sense of “taste” found in “honey is sweet” and gives it formalmetaphysical characteristics in “honey has sweetness”.

52

16.1. SEE ALSO 53

16.1 See also

16.2 References• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6 (1931–1935), Charles Hartshorne and Paul

Weiss, eds., vols. 7–8 (1958), Arthur W. Burks, ed., Harvard University Press, Cambridge, MA.

16.3 External links• J. Jay Zeman, Peirce on Abstraction

Chapter 17

Idempotence

For the concept in matrix algebra, see Idempotent matrix.

Idempotence (/ˌaɪdɨmˈpoʊtəns/EYE-dəm-POH-təns) is the property of certain operations in mathematics and computerscience, that can be applied multiple times without changing the result beyond the initial application. The conceptof idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closureoperators) and functional programming (in which it is connected to the property of referential transparency).The term was introduced by Benjamin Peirce[1] in the context of elements of algebras that remain invariant whenraised to a positive integer power, and literally means "(the quality of having) the same power”, from idem + potence(same + power).There are several meanings of idempotence, depending on what the concept is applied to:

• A unary operation (or function) is idempotent if, whenever it is applied twice to any value, it gives the sameresult as if it were applied once; i.e., ƒ(ƒ(x)) ≡ ƒ(x). For example, the absolute value function, where abs(abs(x))≡ abs(x).

• A binary operation is idempotent if, whenever it is applied to two equal values, it gives that value as the result.For example, the function giving the maximum value of two equal values is idempotent: max (x, x) ≡ x.

• Given a binary operation, an idempotent element (or simply an “idempotent”) for the operation is a value forwhich the operation, when given that value for both of its operands, gives that value as the result. For example,the number 1 is an idempotent of multiplication: 1 × 1 = 1.

17.1 Definitions

17.1.1 Unary operation

A unary operation f , that is, a map from some set S into itself, is called idempotent if, for all x in S ,

f(f(x)) = f(x)

In particular, the identity function idS , defined by idS (x) = x , is idempotent, as is the constant function Kc , wherec is an element of S , defined by Kc (x) = c .An important class of idempotent functions is given by projections in a vector space. An example of a projection isthe function πxy defined by πxy (x, y, z) = (x, y, 0) , which projects an arbitrary point in 3D space to a point on thexy -plane, where the third coordinate ( z ) is equal to 0.A unary operation f : S → S is idempotent if it maps each element of S to a fixed point of f . We can partition aset with n elements into k chosen fixed points and n− k non-fixed points, and then kn−k is the number of differentidempotent functions. Hence, taking into account all possible partitions,

54

17.2. COMMON EXAMPLES 55

n∑k=0

(n

k

)kn−k

is the total number of possible idempotent functions on the set. The integer sequence of the number of idempotentfunctions as given by the sum above for n = {0, 1, 2, . . . } starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, . . . .(sequence A000248 in OEIS)Neither the property of being idempotent nor that of being not is preserved under composition of unary functions.[2]

As an example for the former, f(x) = x mod 3 and g(x) = max(x,5) are both idempotent, but f∘g is not,[3] althoughg∘f happens to be.[4] As an example for the latter, the negation function ¬ on truth values isn't idempotent, but ¬∘¬is.

17.1.2 Idempotent elements and binary operations

Main article: Idempotent element

Given a binary operation ⋆ on a set S , an element x is said to be idempotent (with respect to ⋆ ) if:

x⋆x = x

In particular an identity element of ⋆ , if it exists, is idempotent with respect to the operation ⋆ . The binaryoperation itself is called idempotent if every element of S is idempotent. That is, for all x ∈ S when ∈ denotes setmembership:

x⋆x = x

For example, the operations of set union and set intersection are both idempotent, as are logical conjunction andlogical disjunction, and, in general, the meet and join operations of a lattice.

17.1.3 Connections

The connections between the three notions are as follows.

• The statement that the binary operation ★ on a set S is idempotent, is equivalent to the statement that everyelement of S is idempotent for ★.

• The defining property of unary idempotence, f(f(x)) = f(x) for x in the domain of f, can equivalently berewritten as f ∘ f = f, using the binary operation of function composition denoted by ∘. Thus, the statementthat f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element withrespect to the function composition operation ∘ on functions from S to S.

17.2 Common examples

17.2.1 Functions

As mentioned above, the identity map and the constant maps are always idempotent maps. The absolute value functionof a real or complex argument, and the floor function of a real argument are idempotent. The function that assignsto every subset U of some topological space X the closure of U is idempotent on the power set P (X) of X .It is an example of a closure operator; all closure operators are idempotent functions. The operation of subtractingthe average of a list of numbers from every number in the list is idempotent. For example, consider the numbers3, 6, 8, 8, and10 . The average

∑n1 xn

n ∀xn is 3+6+8+8+105 = 35

5 = 7 . Subtracting 7 from every number in the listyields (−4) , (−1) , 1, 1, 3 . The average

∑n1 xn

n ∀xn of that list is (−4)+(−1)+1+1+35 = 0

5 = 0 . Subtracting 0 fromevery number in that list yields the same list.

56 CHAPTER 17. IDEMPOTENCE

17.2.2 Formal languages

The Kleene star and Kleene plus operators used to express repetition in formal languages are idempotent.

17.2.3 Idempotent ring elements

Main article: Idempotent element

An idempotent element of a ring is, by definition, an element that is idempotent for the ring’s multiplication.[5] Thatis, an element a is idempotent precisely when a2 = a.Idempotent elements of rings yield direct decompositions of modules, and play a role in describing other homologicalproperties of the ring. While “idempotent” usually refers to the multiplication operation of a ring, there are rings inwhich both operations are idempotent: Boolean algebras are such an example.

17.2.4 Other examples

In Boolean algebra, both the logical and and the logical or operations are idempotent. This implies that every elementof Boolean algebra is idempotent with respect to both of these operations. Specifically, x ∧ x = x and x ∨ x = xfor all x . In linear algebra, projections are idempotent. In fact, the projections of a vector space are exactly theidempotent elements of the ring of linear transformations of the vector space. After fixing a basis, it can be shownthat the matrix of a projection with respect to this basis is an idempotent matrix. An idempotent semiring (alsosometimes called a dioid) is a semiring whose addition (not multiplication) is idempotent. If both operations of thesemiring are idempotent, then the semiring is called doubly idempotent.[6]

17.3 Computer science meaning

See also: Referential transparency (computer science), Reentrant (subroutine) and Stable sort

In computer science, the term idempotent is used more comprehensively to describe an operation that will producethe same results if executed once or multiple times.[7] This may have a different meaning depending on the contextin which it is applied. In the case of methods or subroutine calls with side effects, for instance, it means that themodified state remains the same after the first call. In functional programming, though, an idempotent function isone that has the property f(f(x)) = f(x) for any value x.[8]

This is a very useful property in many situations, as it means that an operation can be repeated or retried as oftenas necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keeptrack of whether the operation was already performed or not.

17.3.1 Examples

Looking up some customer’s name and address in a database are typically idempotent (in fact nullipotent), since thiswill not cause the database to change. Similarly, changing a customer’s address is typically idempotent, because thefinal address will be the same no matter how many times it is submitted. However, placing an order for a car for thecustomer is typically not idempotent, since running the method/call several times will lead to several orders beingplaced. Canceling an order is idempotent, because the order remains canceled no matter how many requests aremade.A composition of idempotent methods or subroutines, however, is not necessarily idempotent if a later method inthe sequence changes a value that an earlier method depends on – idempotence is not closed under composition. Forexample, suppose the initial value of a variable is 3 and there is a sequence that reads the variable, then changes it to 5,and then reads it again. Each step in the sequence is idempotent: both steps reading the variable have no side effectsand changing a variable to 5 will always have the same effect no matter how many times it is executed. Nonetheless,executing the entire sequence once produces the output (3, 5), but executing it a second time produces the output (5,5), so the sequence is not idempotent.[9]

17.4. APPLIED EXAMPLES 57

In the HyperText Transfer Protocol (HTTP), idempotence and safety are the major attributes that separate HTTPverbs. Of the major HTTP verbs, GET, PUT, and DELETE are idempotent (if implemented according to the stan-dard), but POST is not.[9] These verbs represent very abstract operations in computer science: GET retrieves aresource; PUT stores content at a resource; and DELETE eliminates a resource. As in the example above, readingdata usually has no side effects, so it is idempotent (in fact nullipotent). Storing a given set of content is usuallyidempotent, as the final value stored remains the same after each execution. And deleting something is generallyidempotent, as the end result is always the absence of the thing deleted.In Event Stream Processing, idempotence refers to the ability of a system to produce the same outcome, even if anevent or message is received more than once.In a load-store architecture, instructions that might possibly cause a page fault are idempotent. So if a page faultoccurs, the OS can load the page from disk and then simply re-execute the faulted instruction. In a processor wheresuch instructions are not idempotent, dealing with page faults is much more complex.

17.4 Applied examples

Applied examples that many people could encounter in their day-to-day lives include elevator call buttons and cross-walk buttons.[10] The initial activation of the button moves the system into a requesting state, until the request issatisfied. Subsequent activations of the button between the initial activation and the request being satisfied have noeffect.

17.5 See also

• Closure operator

• Fixed point (mathematics)

• Idempotent of a code

• Nilpotent

• Idempotent matrix

• Idempotent relation — a generalization of idempotence to binary relations

• List of matrices

• Pure function

• Referential transparency (computer science)

• Iterated function

• Biordered set

• Involution (mathematics)

17.6 References[1] Polcino & Sehgal (2002), p. 127.

[2] If f and g commute, i.e. if f∘g = g∘f, then idempotency of both f and g implies that of f∘g, since f∘g ∘ f∘g = f∘f ∘ g∘g =f ∘ g, using the associativity of composition.

[3] e.g. f(g(7)) = f(7) = 1, but f(g(1)) = f(5) = 2 ≠ 1

[4] also showing that commutation of f and g is not a necessary condition for idempotency preservation

[5] See Hazewinkel et al. (2004), p. 2.

58 CHAPTER 17. IDEMPOTENCE

[6] Gondran & Minoux. Graphs, dioids and semirings. Springer, 2008, p. 34

[7] Rodriguez, Alex. “RESTful Web services: The basics”. IBM developerWorks. IBM. Retrieved 24 April 2013.

[8] http://foldoc.org/idempotent

[9] IETF, Hypertext Transfer Protocol (HTTP/1.1): Semantics and Content. See also HyperText Transfer Protocol.

[10] http://web.archive.org/web/20110523081716/http://www.nclabor.com/elevator/geartrac.pdf For example, this design spec-ification includes detailed algorithm for when elevator cars will respond to subsequent calls for service

17.7 Further reading• “idempotent” at FOLDOC

• Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co.Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975 (93m:16006)

• Gunawardena, Jeremy (1998), “An introduction to idempotency”, in Gunawardena, Jeremy, Idempotency.Based on a workshop, Bristol, UK, October 3–7, 1994 (PDF), Cambridge: Cambridge University Press, pp.1–49, Zbl 0898.16032

• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004), Algebras, rings and modules. vol. 1,Mathematics and its Applications 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380, ISBN 1-4020-2690-0, MR 2106764 (2006a:16001)

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 (2002c:16001)

• Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001 p. 443

• Peirce, Benjamin. Linear Associative Algebra 1870.

• Polcino Milies, César; Sehgal, Sudarshan K. (2002), An introduction to group rings, Algebras and Applications1, Dordrecht: Kluwer Academic Publishers, pp. xii+371, ISBN 1-4020-0238-6, MR 1896125 (2003b:16026)

17.8 External links• Hazewinkel, Michiel, ed. (2001), “Idempotent”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

Chapter 18

Idempotent relation

In mathematics, an idempotent binary relation R ⊆ X × X is one for which R ∘ R = R.[1][2] This notion generalizesthat of an idempotent function to relations. Each idempotent relation is necessarily transitive, as the latter means R ∘R ⊆ R.For example, the relation < on ℚ is idempotent. In contrast, < on ℤ is not, since (<) ∘ (<) ⊇ (<) does not hold: e.g. 1< 2, but 1 < x < 2 is false for every x ∈ ℤ.Idempotent relations have been used as an example to illustrate the application of Mechanized Formalisation of math-ematics using the interactive theorem prover Isabelle/HOL. Besides checking the mathematical properties of finiteidempotent relations, an algorithm for counting the number of idempotent relations has been derived in Isabelle/HOL.[3][4]

18.1 References[1] Florian Kammüller, J. W. Sanders (2004). Idempotent Relation in Isabelle/HOL (PDF) (Technical report). TU Berlin. p.

27. 2004-04. Here:p.3

[2] Florian Kammüller (2011). “Mechanical Analysis of Finite Idempotent Relations”. Fundamenta Informaticae 107. pp.43–65. doi:10.3233/FI-2011-392.

[3] Florian Kammüller (2006). “Number of idempotent relations on n labeled elements”. The On-Line Ecyclopedea of IntegerSequences (A12137).

[4] Florian Kammüller (2008). Counting Idempotent Relations (PDF) (Technical report). TU Berlin. p. 27. 2008-15.

• Berstel, Jean; Perrin, Dominique; Reutenauer, Christophe (2010). Codes and automata. Encyclopedia ofMathematics and its Applications 129. Cambridge: Cambridge University Press. p. 330. ISBN 978-0-521-88831-8. Zbl 1187.94001.

59

Chapter 19

Intransitivity

This article is about intransitivity in mathematics. For the linguistics sense, see Intransitive verb.

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are nottransitive relations. This may include any relation that is not transitive, or the stronger property of antitransitiv-ity, which describes a relation that is never transitive.

19.1 Intransitivity

A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.Some authors call a relation intransitive if it is not transitive, i.e. (if the relation in question is named R )

¬ (∀a, b, c : aRb ∧ bRc ⇒ aRc) .

This statement is equivalent to

∃a, b, c : aRb ∧ bRc ∧ ¬(aRc)

For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.[1] Thus,the feed on relation among life forms is intransitive, in this sense.Another example that does not involve preference loops arises in freemasonry: it may be the case that lodge Arecognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognitionrelation among Masonic lodges is intransitive.

19.2 Antitransitivity

Often the term intransitive is used to refer to the stronger property of antitransitivity.We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feedon rabbits, rabbits feed on carrots, and humans also feed on carrots.A relation is antitransitive if this never occurs at all, i.e.,

∀a, b, c : aRb ∧ bRc ⇒ ¬aRc

Many authors use the term intransitivity to mean antitransitivity.[2][3]

An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated playerB and player B defeated player C, A can have never played C, and therefore, A has not defeated C.

60

19.3. CYCLES 61

19.3 Cycles

The term intransitivity is often used when speaking of scenarios in which a relation describes the relative preferencesbetween pairs of options, and weighing several options produces a “loop” of preference:

• A is preferred to B

• B is preferred to C

• C is preferred to A

Rock, paper, scissors; Nontransitive dice; and Penney’s game are examples.Assuming no option is preferred to itself i.e. the relation is irreflexive, a preference relation with a loop is not transitive.For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this exampleof a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred toC, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.Therefore such a preference loop (or "cycle") is known as an intransitivity.Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, anequivalence relation possesses cycles but is transitive. Now, consider the relation “is an enemy of” and suppose thatthe relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country isnot itself an enemy of the country. This is an example of an antitransitive relation that does not have any cycles. Inparticular, by virtue of being antitransitive the relation is not transitive.Finally, let us work with the example of rock, paper, scissors, calling the three options A, B, and C. Now, the relationover A, B, and C is “defeats” and the standard rules of the game are such that A defeats B, B defeats C, and C defeatsA. Furthermore, it is also true that B does not defeat A, C does not defeat B, and A does not defeat C. Finally, it isalso true that no option defeats itself. This information can be depicted in a table:The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zeroindicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn(with replacement) from the set {A, B, C}: If x defeats y, and y defeats z, then x does not defeat z. Hence the relationis antitransitive.Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.

19.4 Occurrences in preferences• Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet

voting method in which ranking several candidates can produce a loop of preference when the weights arecompared (see voting paradox). Intransitive dice demonstrate that probabilities are not necessarily transitive.

• In psychology, intransitivity often occurs in a person’s system of values (or preferences, or tastes), potentiallyleading to unresolvable conflicts.

• Analogously, in economics intransitivity can occur in a consumer’s preferences. This may lead to consumerbehaviour that does not conform to perfect economic rationality. In recent years, economists and philosophershave questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand(1993)).

19.5 Likelihood

It has been suggested that Condorcet voting tends to eliminate “intransitive loops” when large numbers of votersparticipate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidateson several different units of measure such as by order of social consciousness or by order of most fiscally conservative.In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units ofmeasure in assessing candidates.Such as:

62 CHAPTER 19. INTRANSITIVITY

• 30% favor 60/40 weighting between social consciousness and fiscal conservatism

• 50% favor 50/50 weighting between social consciousness and fiscal conservatism

• 20% favor a 40/60 weighting between social consciousness and fiscal conservatism

While each voter may not assess the units of measure identically, the trend then becomes a single vector on whichthe consensus agrees is a preferred balance of candidate criteria.

19.6 References[1] Wolves do eat grass - see Engel, Cindy (2003). Wild Health: Lessons in Natural Wellness from the Animal Kingdom

(paperback ed.). Houghton Mifflin. p. 141. ISBN 0-618-34068-8..

[2] Guide to Logic, Relations II

[3] IntransitiveRelation

19.7 Further reading• Anand, P (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press..

Chapter 20

Inverse relation

For inverse relationships in statistics, see negative relationship.

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements isswitched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms,if X and Y are sets and L ⊆ X × Y is a relation from X to Y then L−1 is the relation defined so that y L−1 x if andonly if xLy . In set-builder notation, L−1 = {(y, x) ∈ Y ×X | (x, y) ∈ L} .The notation comes by analogy with that for an inverse function. Although many functions do not have an inverse;every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inversein the sense of group inverse; the unary operation that maps a relation to the inverse relation is however an involution,so it induces the structure of a semigroup with involution on the binary relations on a set, or more generally induces adagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimescalled inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes withunion, intersection, complement etc.The inverse relation is also called the converse relation or transpose relation— the latter in view of its similaritywith the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notationsfor the inverse relation include LC , LT , L~ or L̆ or L° or L∨.

20.1 Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected “opposite” order, e.g. ≤−1=≥, <−1= > , etc.

20.1.1 Inverse relation of a function

A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inversefunction.The inverse relation of a function f : X → Y is the relation f−1 : Y → X defined by graph f−1 = {(y, x) | y =f(x)} .This is not necessarily a function: One necessary condition is that f be injective, since else f−1 is multi-valued. Thiscondition is sufficient for f−1 being a partial function, and it is clear that f−1 then is a (total) function if and only iff is surjective. In that case, i.e. if f is bijective, f−1 may be called the inverse function of f.

20.2 Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition ofrelations), the inverse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary

63

64 CHAPTER 20. INVERSE RELATION

relation on X, then L ◦L−1 does not equal the identity relation on X in general. The inverse relation does satisfy the(weaker) axioms of a semigroup with involution: (L−1)−1 = L and (L ◦R)−1 = R−1 ◦ L−1 .[3]

Since one may generally consider relations between different sets (which form a category rather than a monoid,namely the category of relations Rel), in this context the inverse relation conforms to the axioms of a dagger category(aka category with involution).[3] A relation equal to its inverse is a symmetric relation; in the language of daggercategories, it is self-adjoint.Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relationsas sets), and actually an involutive quantale. Similarly, the category of heterogenous relations, Rel is also an orderedcategory.[3]

In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (theoperation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversionalso commutes with unary operation of complementation as well as with taking suprema and infima. Inversion is alsocompatible with the ordering of relations by inclusion.[1]

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

20.3 See also• Bijection

• Function (mathematics)

• Inverse function

• Relation (mathematics)

• Transpose graph

20.4 References[1] Gunther Schmidt; Thomas Ströhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer

Berlin Heidelberg. pp. 9–10. ISBN 978-3-642-77970-1.

[2] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups.Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.

[3] Joachim Lambek (2001). “Relations Old and New”. In Ewa Orlowska, Andrzej Szalas. Relational Methods for ComputerScience Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.

• Halmos, Paul R. (1974), Naive Set Theory, p. 40, ISBN 978-0-387-90092-6

Chapter 21

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions[1]) are the inversefunctions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of thesine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of theangle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, andgeometry.

21.1 Notation

There are many notations used for the inverse trigonometric functions. The notations sin−1 (x), cos−1 (x), tan−1

(x), etc. are often used, but this convention logically conflicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion betweenmultiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of thereciprocal trigonometric functions has its own name—for example, (cos(x))−1=sec(x). Another convention used bysome authors[2] is to use a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1 (x), Cos−1

(x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin−1 (x), cos−1

(x), etc. Yet another convention is to use an arc- prefix, so that the confusion with the −1 superscript is resolvedcompletely, e.g., arcsin (x), arccos (x), etc. This convention is used throughout the article. In computer programminglanguages (also MS Office Excel) the inverse trigonometric functions are usually called asin, acos, atan.According to Cajori,[3] the notation sin−1 (x) was introduced by John Herschel in 1813.[4]

21.1.1 Etymology of the arc- prefix

When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radiusof the circle. Thus, in the unit circle, “the arc whose cosine is x” is the same as “the angle whose cosine is x”, becausethe length of the arc of the circle in radii is the same as the measurement of the angle in radians.[5]

21.2 Basic properties

21.2.1 Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions.Therefore the ranges of the inverse functions are proper subsets of the domains of the original functionsFor example, using function in the sense of multivalued functions, just as the square root function y = √x could bedefined from y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such thatsin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. When only one value is desired, the functionmay be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x)

65

66 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometricfunctions.The principal inverses are listed in the following table.(Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π/2 or π ≤ y < 3π/2 ), because the tangentfunction is nonnegative on this domain. This makes some computations more consistent. For example using thisrange, tan(arcsec(x))=√x2−1, whereas with the range ( 0 ≤ y < π/2 or π/2 < y ≤ π ), we would have to writetan(arcsec(x))=±√x2−1, since tangent is nonnegative on 0 ≤ y < π/2 but nonpositive on π/2 < y ≤ π. For a similarreason, the same authors define the range of arccosecant to be ( -π < y ≤ -π/2 or 0 < y ≤ π/2 ).)If x is allowed to be a complex number, then the range of y applies only to its real part.

21.2.2 Relationships between trigonometric functions and inverse trigonometric func-tions

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is byconsidering the geometry of a right-angled triangle, with one side of length 1, and another side of length x (any realnumber between 0 and 1), then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purelyalgebraic derivations are longer.

21.2.3 Relationships among the inverse trigonometric functions

Complementary angles:

arccosx =π

2− arcsinx

arccotx =π

2− arctanx

arccscx =π

2− arcsecx

Negative arguments:

arcsin(−x) = − arcsinxarccos(−x) = π − arccosxarctan(−x) = − arctanxarccot(−x) = π − arccotxarcsec(−x) = π − arcsecxarccsc(−x) = − arccscx

Reciprocal arguments:

21.3. IN CALCULUS 67

arccos(1/x) = arcsecxarcsin(1/x) = arccscx

arctan(1/x) = π

2− arctanx = arccotx , if x > 0

arctan(1/x) = −π

2− arctanx = arccotx− π , if x < 0

arccot(1/x) = π

2− arccotx = arctanx , if x > 0

arccot(1/x) = 3π

2− arccotx = π + arctanx , if x < 0

arcsec(1/x) = arccosxarccsc(1/x) = arcsinx

If you only have a fragment of a sine table:

arccosx = arcsin√1− x2 , if 0 ≤ x ≤ 1

arctanx = arcsin x√x2 + 1

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positiveimaginary part if the square was negative real).From the half-angle formula, tan θ

2 = sin θ1+cos θ , we get:

arcsinx = 2 arctan x

1 +√1− x2

arccosx = 2 arctan√1− x2

1 + x, if − 1 < x ≤ +1

arctanx = 2 arctan x

1 +√1 + x2

21.2.4 Arctangent addition formula

arctanu+ arctan v = arctan(

u+ v

1− uv

)(mod π) , uv ̸= 1 .

This is derived from the tangent addition formula

tan(α+ β) =tanα+ tanβ1− tanα tanβ ,

by letting

α = arctanu , β = arctan v .

21.3 In calculus

21.3.1 Derivatives of inverse trigonometric functions

Main article: Differentiation of trigonometric functions

68 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

The derivatives for complex values of z are as follows:

ddz arcsin z =

1√1− z2

; z ̸= −1,+1

ddz arccos z = − 1√

1− z2; z ̸= −1,+1

ddz arctan z =

1

1 + z2; z ̸= −i,+i

ddz arccot z = − 1

1 + z2; z ̸= −i,+i

ddz arcsec z =

1

z2√1− 1/z2

; z ̸= −1, 0,+1

ddz arccsc z = − 1

z2√1− 1/z2

; z ̸= −1, 0,+1

Only for real values of x:

ddx arcsecx =

1

|x|√x2 − 1

; |x| > 1

ddx arccscx = − 1

|x|√x2 − 1

; |x| > 1

For a sample derivation: if θ = arcsinx , we get:

d arcsinxdx =

dθd sin θ =

dθcos θdθ =

1

cos θ =1√

1− sin2 θ=

1√1− x2

21.3.2 Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric functionas a definite integral:

arcsinx =

∫ x

0

1√1− z2

dz , |x| ≤ 1

arccosx =

∫ 1

x

1√1− z2

dz , |x| ≤ 1

arctanx =

∫ x

0

1

z2 + 1dz ,

arccotx =

∫ ∞

x

1

z2 + 1dz ,

arcsecx =

∫ x

1

1

z√z2 − 1

dz = π +

∫ −1

x

1

z√z2 − 1

dz , x ≥ 1

arccscx =

∫ ∞

x

1

z√z2 − 1

dz =

∫ x

−∞

1

z√z2 − 1

dz , x ≥ 1

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

21.3.3 Infinite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows.For arcsine, the series can be derived by expanding its derivative, 1√

1−z2, as a binomial series, and integrating term

21.3. IN CALCULUS 69

by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding itsderivative 1

1+z2 in a geometric series and applying the integral definition above (see Leibniz series).

arcsin z = z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+

(1 · 3 · 52 · 4 · 6

)z7

7+ · · · =

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arccos z =π

2− arcsin z =

π

2−(z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+ · · ·

)=

π

2−

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arctan z = z − z3

3+

z5

5− z7

7+ · · · =

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

arccot z =π

2−arctan z =

π

2−(z − z3

3+

z5

5− z7

7+ · · ·

)=

π

2−

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

arcsec z = arccos(1/z) = π

2−(z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · ·

)=

π

2−

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

arccsc z = arcsin(1/z) = z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · · =

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

Leonhard Euler found a more efficient series for the arctangent, which is:

arctan z =z

1 + z2

∞∑n=0

n∏k=1

2kz2

(2k + 1)(1 + z2).

(Notice that the term in the sum for n = 0 is the empty product which is 1.)Alternatively, this can be expressed:

arctan z =∞∑

n=0

22n(n!)2

(2n+ 1)!

z2n+1

(1 + z2)n+1

Variant: Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

arctan z =z

1 +(1z)2

3− 1z2 +(3z)2

5− 3z2 +(5z)2

7− 5z2 +(7z)2

9− 7z2 +. . .

=z

1 +(1z)2

3 +(2z)2

5 +(3z)2

7 +(4z)2

9 +. . .

The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going downthe imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers runningfrom −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) arejust (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by CarlFriedrich Gauss utilizing the Gaussian hypergeometric series.

70 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

21.3.4 Indefinite integrals of inverse trigonometric functions

For real and complex values of z:

∫arcsin z dz = z arcsin z +

√1− z2 + C∫

arccos z dz = z arccos z −√

1− z2 + C∫arctan z dz = z arctan z − 1

2ln(1 + z2

)+ C∫

arccot z dz = z arccot z + 1

2ln(1 + z2

)+ C∫

arcsec z dz = z arcsec z − ln[z

(1 +

√z2 − 1

z2

)]+ C

∫arccsc z dz = z arccsc z + ln

[z

(1 +

√z2 − 1

z2

)]+ C

For real x ≥ 1:

∫arcsecx dx = x arcsecx− ln

(x+

√x2 − 1

)+ C∫

arccscx dx = x arccscx+ ln(x+

√x2 − 1

)+ C

For all real x not between −1 and 1:

∫arcsecx dx = x arcsecx− sgn(x) ln

∣∣∣x+√

x2 − 1∣∣∣+ C∫

arccscx dx = x arccscx+ sgn(x) ln∣∣∣x+

√x2 − 1

∣∣∣+ C

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecantfunctions. The signum function is also necessary due to the absolute values in the derivatives of the two functions,which create two different solutions for positive and negative values of x. These can be further simplified using thelogarithmic definitions of the inverse hyperbolic functions:

∫arcsecx dx = x arcsecx− arcosh |x|+ C∫arccscx dx = x arccscx+ arcosh |x|+ C

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical tothe signum logarithmic function shown above.All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example

Using∫u dv = uv −

∫v du , set

u = arcsinx dv = dx

du =dx√1− x2

v = x

21.4. EXTENSION TO COMPLEX PLANE 71

Then

∫arcsin(x) dx = x arcsinx−

∫x√

1− x2dx

Substitute

w = 1− x2 .

Then

dw = −2x dx

and

∫x√

1− x2dx = −1

2

∫ dw√w

= −√w

Back-substitute for x to yield

∫arcsin(x) dx = x arcsinx+

√1− x2 + C

21.4 Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complexplane. This results in functions with multiple sheets and branch points. One possible way of defining the extensionsis:

arctan z =

∫ z

0

dx1 + x2

z ̸= −i,+i

where the part of the imaginary axis which does not lie strictly between −i and +i is the cut between the principalsheet and other sheets;

arcsin z = arctan z√1− z2

z ̸= −1,+1

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does notlie strictly between −1 and +1 is the cut between the principal sheet of arcsin and other sheets;

arccos z =π

2− arcsin z z ̸= −1,+1

which has the same cut as arcsin;

arccot z =π

2− arctan z z ̸= −i,+i

which has the same cut as arctan;

72 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

arcsec z = arccos 1z

z ̸= −1, 0,+1

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and othersheets;

arccsc z = arcsin 1

zz ̸= −1, 0,+1

which has the same cut as arcsec.

21.4.1 Logarithmic forms

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain tothe complex plane.

arcsin z = −i ln(iz +

√1− z2

)= arccsc 1

z

arccos z = −i ln(z +

√z2 − 1

)=

π

2+ i ln

(iz +

√1− z2

)=

π

2− arcsin z = arcsec 1

z

arctan z = 12 i [ln (1− iz)− ln (1 + iz)] = arccot 1

z

arccot z = 12 i

[ln(1− i

z

)− ln

(1 +

i

z

)]= arctan 1

z

arcsec z = −i ln(√

1

z2− 1 +

1

z

)= i ln

(√1− 1

z2+

i

z

)+

π

2=

π

2− arccsc z = arccos 1

z

arccsc z = −i ln(√

1− 1

z2+

i

z

)= arcsin 1

z

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof

sin(ϕ) = z

ϕ = arcsin z

Using the exponential definition of sine, one obtains

z =eiϕ − e−iϕ

2i

Let

ξ = eiϕ

Solving for ϕ

21.5. APPLICATIONS 73

z =ξ − 1/ξ

2i

2iz = ξ − 1/ξ

ξ − 2iz − 1/ξ = 0

ξ2 − 2iξz − 1 = 0

ξ = iz ±√1− z2 = eiϕ

iϕ = ln(iz ±

√1− z2

)ϕ = −i ln

(iz ±

√1− z2

)(the positive branch is chosen)

ϕ = arcsin z = −i ln(iz +

√1− z2

)Example proof (variant 2)

ϕ = arcsin zeiϕ = cos(ϕ) + i sin(ϕ)Apply the natural logarithm, multiply by -i and substitute phi.arcsin z = −i ln[cos(arcsin z) + i sin(arcsin z)]arcsin z = −i ln(

√1− z2 + iz)

21.5 Applications

21.5.1 General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice ineach interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2,and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishesit at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.This periodicity is reflected in the general inverses where k is some integer:

sin(y) = x ⇔ y = arcsin(x) + 2πk or y = π − arcsin(x) + 2πk

sin(y) = x ⇔ y = (−1)k arcsin(x) + πk

cos(y) = x ⇔ y = arccos(x) + 2πk or y = 2π − arccos(x) + 2πk

cos(y) = x ⇔ y = ± arccos(x) + 2πk

tan(y) = x ⇔ y = arctan(x) + πk

cot(y) = x ⇔ y = arccot(x) + πk

sec(y) = x ⇔ y = arcsec(x) + 2πk or y = 2π − arcsec(x) + 2πk

csc(y) = x ⇔ y = arccsc(x) + 2πk or y = π − arccsc(x) + 2πk

74 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

Application: finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle whenthe lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, itfollows that

θ = arcsin( opposite

hypotenuse

).

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using thePythagorean Theorem: a2 + b2 = h2 where h is the length of the hypotenuse. Arctangent comes in handy inthis situation, as the length of the hypotenuse is not needed.

θ = arctan(opposite

adjacent

).

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, whereθ may be computed as follows:

θ = arctan(opposite

adjacent

)= arctan

( riserun

)= arctan

(8

20

)≈ 21.8◦ .

21.5.2 In computer science and engineering

Two-argument variant of arctangent

Main article: atan2

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. Inother words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positivesign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane,y < 0). It was first introduced in many computer programming languages, but it is now also common in other fieldsof science and engineering.In terms of the standard arctan function, that is with range of (−π/2, π/2), it can be expressed as follows:

atan2(y, x) =

arctan( yx ) x > 0

arctan( yx ) + π y ≥ 0 , x < 0

arctan( yx )− π y < 0 , x < 0π2 y > 0 , x = 0

−π2 y < 0 , x = 0

undefined y = 0 , x = 0

It also equals the principal value of the argument of the complex number x + iy.This function may also be defined using the tangent half-angle formulae as follows:

atan2(y, x) = 2 arctan y√x2 + y2 + x

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable forcomputational use.The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such asthe C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted.These variations are detailed at atan2.

21.6. SEE ALSO 75

Arctangent function with location parameter

In many applications the solution y of the equation x = tan y is to come as close as possible to a given value−∞ < η < ∞ . The adequate solution is produced by the parameter modified arctangent function

y = arctanη x := arctanx+ π · rni η − arctanxπ

.

The function rni rounds to the nearest integer.

Practical considerations

For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in acomputer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −π/2and π/2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

21.6 See also• Argument (complex analysis)

• Complex logarithm

• Gauss’s continued fraction

• Inverse hyperbolic function

• List of integrals of inverse trigonometric functions

• List of trigonometric identities

• Square root

• Tangent half-angle formula

• Trigonometric function

21.7 References[1] For example Dörrie, Heinrich (1965). Triumph der Mathematik. Trans. David Antin. Dover. p. 69. ISBN 0-486-61348-8.

[2] Prof. Sanaullah Bhatti; Ch. Nawab-ud-Din; Ch. Bashir Ahmed; Dr. S. M. Yousuf; Dr. Allah Bukhsh Taheem (1999).“Differentiation of Tigonometric, Logarithmic and Exponential Functions”. In Prof. Mohammad Maqbool Ellahi, Dr.Karamat Hussain Dar, Faheem Hussain. Calculus and Analytic Geometry (in Pakistani English) (First ed.). Lahore: PunjabTextbook Board. p. 140.

[3] Cajori, Florian (1919). A History of Mathematics (2nd ed.). The Macmillan Company, New York. p. 272., at GoogleBooks

[4] Herschel, John F. W. (1813). “On a remarkable Application of Cotes’s Theorem”. Philosophical Transactions (RoyalSociety, London) 103 (1): 10., at Google Books

[5] “Inverse trigonometric functions” in The Americana: a universal reference library, Vol.21, Ed. Frederick Converse Beach,George Edwin Rines, (1912).

21.8 External links• Weisstein, Eric W., “Inverse Trigonometric Functions”, MathWorld.

• Weisstein, Eric W., “Inverse Tangent”, MathWorld.

76 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.

21.8. EXTERNAL LINKS 77

The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.

Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

78 CHAPTER 21. INVERSE TRIGONOMETRIC FUNCTIONS

A C

B

b

ah

(adjacent)

(opposite)(hypotenuse)

A right triangle.

Chapter 22

Near sets

Figure 1. Descriptively, very near sets

In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection.In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. De-scriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjointsets. Spatially near sets are also descriptively near sets.The underlying assumption with descriptively close sets is that such sets contain elements that have location andmeasurable features such as colour and frequency of occurrence. The description of the element of a set is definedby a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near

79

80 CHAPTER 22. NEAR SETS

Figure 2. Descriptively, minimally near sets

sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in setsbased on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based onhuman perception that arise in areas such as image processing, computer vision as well as engineering and scienceproblems.Near sets have a variety of applications in areas such as topology[37], pattern detection and classification[50], abstract al-gebra[51], mathematics in computer science[38], and solving a variety of problems based on human perception[42][82][47][52][56]

that arise in areas such as image analysis[54][14][46][17][18], image processing[40], face recognition[13], ethology[64], aswell as engineering and science problems[55][64][42][19][17][18]. From the beginning, descriptively near sets have provedto be useful in applications of topology[37], and visual pattern recognition [50], spanning a broad spectrum of applica-tions that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analy-sis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, andtopological psychology.As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colourmodel for varying degrees of nearness between sets of picture elements in pictures (see, e.g.,[17] §4.3). The two pairsof ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalenceclass where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals inFig.1 are closer to each other descriptively than the ovals in Fig. 2.

22.1. HISTORY 81

22.1 History

It has been observed that the simple concept of nearness unifies various concepts of topological structures[20] inas-much as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetrictopological spaces and continuous maps[3]),Prox (proximity spaces and δ -maps[8][67]),Unif (uniform spaces and uni-formly continuous maps[81][77]) andCont (contiguity spaces and contiguity maps[24]) as embedded full subcategories[20][59].The categories εANear and εAMer are shown to be full supercategories of various well-known categories, in-cluding the category sTop of symmetric topological spaces and continuous maps, and the category Met∞ ofextended metric spaces and nonexpansive maps. The notation A ↪→ B reads category A is embedded in category B. The categories εAMer and εANear are supercategories for a variety of familiar categories[76] shown in Fig. 3.Let εANear denote the category of all ε -approach nearness spaces and contractions, and let εAMer denote thecategory of all ε -approach merotopic spaces and contractions.

Figure 3. Supercats

Among these familiar categories is sTop , the symmetric form of Top (see category of topological spaces), the cat-egory with objects that are topological spaces and morphisms that are continuous maps between them[1][32]. Met∞

with objects that are extended metric spaces is a subcategory of εAP (having objects ε -approach spaces and con-tractions) (see also[57][75]). Let ρX , ρY be extended pseudometrics on nonempty sets X,Y , respectively. The mapf : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f : (X, νDρX

) −→ (Y, νDρY) is a contraction. For

nonempty subsets A,B ∈ 2X , the distance function Dρ : 2X × 2X −→ [0,∞] is defined by

Dρ(A,B) =

{inf {ρ(a, b) : a ∈ A, b ∈ B}, ifA and Bempty not are ,

∞, ifA or Bempty is .

Thus ε AP is embedded as a full subcategory in εANear by the functor F : εAP −→ εANear defined byF ((X, ρ)) = (X, νDρ) and F (f) = f . Then f : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f :

82 CHAPTER 22. NEAR SETS

(X, νDρX) −→ (Y, νDρY

) is a contraction. Thus εAP is embedded as a full subcategory in εANear by thefunctor F : εAP −→ εANear defined by F ((X, ρ)) = (X, νDρ) and F (f) = f. Since the category Met∞

of extended metric spaces and nonexpansive maps is a full subcategory of εAP , therefore, εANear is also a fullsupercategory of Met∞ . The category εANear is a topological construct[76].

Figure 4. Frigyes Riesz, 1880-1956

The notions of near and far[A] in mathematics can be traced back to works by Johann Benedict Listing and FelixHausdorff. The related notions of resemblance and similarity can be traced back to J.H. Poincaré, who introducedsets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechner’s sensation sensitivityexperiments[10] and a framework for the study of resemblance in representative spaces as models of what he termedphysical continua[63][60][61]. The elements of a physical continuum (pc) are sets of sensations. The notion of a pcand various representative spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article onthe mathematical continuum[63], an 1895 article on space and geometry[60] and a compendious 1902 book on science

22.2. NEARNESS OF SETS 83

and hypothesis[61] followed by a number of elaborations, e.g.,[62]. The 1893 and 1895 articles on continua (Pt. 1,ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in[61]. Later, F. Rieszintroduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians(ICM) in 1908[65].During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception[83]. A.B. Sossinskyobserved in 1986[71] that the main idea underlying tolerance space theory comes from Poincaré, especially[60]. In2002, Z. Pawlak and J. Peters[B] considered an informal approach to the perception of the nearness of physical objectssuch as snowflakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness ofobjects was considered by J. Peters, A. Skowron and J. Stepaniuk[C] in the context of proximity spaces[39][33][35][21].In 2007, descriptively near sets were introduced by J. Peters[D][E] followed by the introduction of tolerance nearsets[41][45]. Recently, the study of descriptively near sets has led to algebraic[22][51], topological and proximity space[37]

foundations of such sets.

22.2 Nearness of sets

The adjective near in the context of near sets is used to denote the fact that observed feature value differences ofdistinct objects are small enough to be considered indistinguishable, i.e., within some tolerance.The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally,in almost any mathematical setting (see, e.g.,[66]). It is especially natural in mathematical applications: practicalproblems, more often than not, deal with approximate input data and only require viable results with a tolerable levelof error[71].The words near and far are used in daily life and it was an incisive suggestion of F. Riesz[65] that these intuitiveconcepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. Thisconcept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitivedefinition of continuity of a function at a point to its rigorous epsilon-delta definition is sometime difficult for teachersto explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., afunction f : R → R is continuous at a point c , provided points {x} near c go into points {f(x)} near f(c) . UsingRiesz’s idea, this definition can be made more precise and its contrapositive is the familiar definition[4][36].

22.3 Generalization of set intersection

From a spatial point of view, nearness (aka proximity) is considered a generalization of set intersection. For disjointsets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that havesimilar features within some tolerance (see, e.g., §3 in[80]). For example, the ovals in Fig. 1 are considered near eachother, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.

22.4 Efremovič proximity space

Let X denote a metric topological space that is endowed with one or more proximity relations and let 2X denote thecollection of all subsets of X . The collection 2X is called the power set of X .There are many ways to define Efremovič proximities on topological spaces (discrete proximity, standard proximity,metric proximity, Čech proximity, Alexandroff proximity, and Freudenthal proximity), For details, see § 2, pp. 93–94 in[6]. The focus here is on standard proximity on a topological space. For A,B ⊂ X , A is near B (denoted byA δ B ), provided their closures share a common point.The closure of a subset A ∈ 2X (denoted by cl(A) ) is the usual Kuratowski closure of a set[F], introduced in § 4, p.20[27], is defined by

cl(A) = {x ∈ X : D(x,A) = 0} , whereD(x,A) = inf {d(x, a) : a ∈ A} .

i.e. cl(A) is the set of all points x in X that are close to A ( D(x,A) is the Hausdorff distance (see § 22, p. 128,

84 CHAPTER 22. NEAR SETS

in[15]) between x and the set A and d(x, a) = |x− a| (standard distance)). A standard proximity relation is definedby

δ ={(A,B) ∈ 2X × 2X : cl(A) ∩ cl(B) ̸= ∅

}.

Whenever sets A and B have no points in common, the sets are farfrom each other (denoted A δ B ).The following EF-proximity[G] space axioms are given by Jurij Michailov Smirnov[67] based on what Vadim Arsenye-vič Efremovič introduced during the first half of the 1930s[8]. Let A,B,E ∈ 2X .

EF.1 If the set A is close to B , then B is close to A .

EF.2 A ∪B is close to E , if and only if, at least one of the sets A or B is close to E .

EF.3 Two points are close, if and only if, they are the same point.

EF.4 All sets are far from the empty set ∅ .

EF.5 For any two sets A and B which are far from each other, there exists C,D ∈ 2X , C ∪D = X , such that Ais far from C and B is far from D (Efremovič-axiom).

The pair (X, δ) is called an EF-proximity space. In this context, a space is a set with some added structure. Witha proximity space X , the structure of X is induced by the EF-proximity relation δ . In a proximity space X , theclosure of A in X coincides with the intersection of all closed sets that contain A .

Theorem 1[67] The closure of any set A in the proximity space X is the set of points x ∈ X that are close to A .

22.5 Visualization of EF-axiom

Let the set X be represented by the points inside the rectangular region in Fig. 5. Also, let A,B be any two non-intersection subsets (i.e. subsets spatially far from each other) inX , as shown in Fig. 5. LetCc = X\C (complementof the set C ). Then from the EF-axiom, observe the following:

A δ B,

B ⊂ C,

D = Cc,

X = D ∪ C,

A ⊂ D, hence, we can writeA δ B ⇒ A δ C and B δ D, for some C,D in X so that C ∪D = X. ■

22.6 Descriptive proximity space

Descriptively near sets were introduced as a means of solving classification and pattern recognition problems arisingfrom disjoint sets that resemble each other[44][43]. Recently, the connections between near sets in EF-spaces and nearsets in descriptive EF-proximity spaces have been explored in[53][48].Again, let X be a metric topological space and let Φ = {ϕ1, . . . , ϕn} a set of probe functions that represent featuresof each x ∈ X . The assumption made here is X contains non-abstract points that have measurable features such asgradient orientation. A non-abstract point has a location and features that can be measured (see § 3 in [26]).A probe function ϕ : X → R represents a feature of a sample point in X . The mapping Φ : X −→ Rn is definedby Φ(x) = (ϕ1(x), . . . , ϕn(x)) , where Rn is an n-dimensional real Euclidean vector space. Φ(x) is a feature vectorfor x , which provides a description of x ∈ X . For example, this leads to a proximal view of sets of picture pointsin digital images[48].

22.6. DESCRIPTIVE PROXIMITY SPACE 85

X

AB

CC c

Figure 5. Example of a descriptive EF-proximity relation between sets A,B , and Cc

To obtain a descriptive proximity relation (denoted by δΦ ), one first chooses a set of probe functions. LetQ : 2X −→2R

n be a mapping on a subset of 2X into a subset of 2Rn . For example, let A,B ∈ 2X and Q(A),Q(B) denotesets of descriptions of points in A,B , respectively. That is,

Q(A) = {Φ(a) : a ∈ A} ,Q(B) = {Φ(b) : b ∈ B} .

The expression A δΦ B reads A is descriptively near B . Similarly, A δΦ B reads A is descriptively far from B . Thedescriptive proximity of A and B is defined by

A δΦ B ⇔ Q(cl(A)) δ Q(cl(B)) ̸= ∅.

The descriptive intersection ∩Φ of A and B is defined by

A ∩Φ B = {x ∈ A ∪B : Q(A) δ Q(B)} .

That is, x ∈ A ∪B is in A ∩Φ B , provided Φ(x) = Φ(a) = Φ(b) for some a ∈ A, b ∈ B . Observe that A and Bcan be disjoint and yet A ∩Φ B can be nonempty. The descriptive proximity relation δΦ is defined by

δΦ ={(A,B) ∈ 2X × 2X : cl(A) ∩Φ cl(B) ̸= ∅

}.

Whenever sets A and B have no points with matching descriptions, the sets are descriptively far from each other(denoted by A δΦ B ).The binary relation δΦ is a descriptive EF-proximity, provided the following axioms are satisfied for A,B,E ⊂ X .

86 CHAPTER 22. NEAR SETS

dEF.1 If the set A is descriptively close to B , then B is descriptively close to A .

dEF.2 A ∪B is descriptively close to E , if and only if, at least one of the sets A or B is descriptively close to E .

dEF.3 Two points x, y ∈ X are descriptively close, if and only if, the description of x matches the description ofy .

dEF.4 All nonempty sets are descriptively far from the empty set ∅ .

dEF.5 For any two sets A and B which are descriptively far from each other, there exists C,D ∈ 2X , C ∪D = X, such that A is descriptively far from C and B is descriptively far from D (Descriptive Efremovič axiom).

The pair (X, δΦ) is called a descriptive proximity space.

22.7 Proximal relator spaces

A relator is a nonvoid family of relations R on a nonempty set X [72]. The pair (X,R) (also denoted X(R) ) iscalled a relator space. Relator spaces are natural generalizations of ordered sets and uniform spaces[73][74]}. With theintroduction of a family of proximity relations Rδ on X , we obtain a proximal relator space (X,Rδ) . For simplicity,we consider only two proximity relations, namely, the Efremovič proximity δ [8] and the descriptive proximity δΦ indefining the descriptive relator RδΦ

[53][48]. The pair (X,RδΦ) is called a proximal relator space [49]. In this work, Xdenotes a metric topological space that is endowed with the relations in a proximal relator. With the introduction of(X,RδΦ) , the traditional closure of a subset (e.g., [9][7]) can be compared with the more recent descriptive closureof a subset.In a proximal relator space X , the descriptive closure of a set A (denoted by clΦ(A) ) is defined by

clΦ(A) = {x ∈ X : Φ(x)δQ(cl(A))} .

That is, x ∈ X is in the descriptive closure of A , provided the closure of Φ(x) and the closure of Q(cl(A)) have atleast one element in common.

Theorem 2 [50] The descriptive closure of any set A in the descriptive EF-proximity space (X,RδΦ) is the set ofpoints x ∈ X that are descriptively close to A .

Theorem 3 [50] Kuratowski closure of a setA is a subset of the descriptive closure ofA in a descriptive EF-proximityspace.

Theorem 4 [49] Let (X,RδΦ) be a proximal relator space, A ⊂ X . Then cl(A) ⊆ clΦ(A) .

Proof Let Φ(x) ∈ Q(X \ cl(A)) such that Φ(x) = Φ(a) for some a ∈ clA . Consequently, Φ(x) ∈ Q(clΦ(A)) .Hence, cl(A) ⊆ clΦ(A)

In a proximal relator space, EF-proximity δ leads to the following results for descriptive proximity δΦ .

Theorem 5 [49] Let (X,RδΦ) be a proximal relator space, A,B,C ⊂ X . Then

1 ◦

A δ B implies A δΦ B .

2 ◦

(A ∪B) δ C implies (A ∪B) δΦ C .

3 ◦

clA δ clB implies clA δΦ clB .

Proof

22.8. DESCRIPTIVE δ -NEIGHBOURHOODS 87

1 ◦

A δ B ⇔ A ∩B ̸= ∅ . For x ∈ A ∩B,Φ(x) ∈ Q(A) and Φ(x) ∈ Q(B) . Consequently, A δΦ B .

1◦ ⇒ 2◦

3 ◦

clA δ clB implies that clA and clA have at least one point in common. Hence, 1 o ⇒ 3o .

■

22.8 Descriptive δ -neighbourhoods

X

BE 1

E 2

A

X \E 2

Figure 6. Example depicting δ -neighbourhoods

In a pseudometric proximal relator space X , the neighbourhood of a point x ∈ X (denoted by Nx,ε ), for ε > 0 , isdefined by

Nx,ε = {y ∈ X : d(x, y) < ε} .

The interior of a set A (denoted by int(A) ) and boundary of A (denoted by bdy(A) ) in a proximal relator space Xare defined by

int(A) = {x ∈ X : Nx,ε ⊆ A} .

bdy(A) = cl(A) \ int(A).

88 CHAPTER 22. NEAR SETS

A set A has a natural strong inclusion in a set B associated with δ [5][6]} (denoted by A ≪δ B ), provided A ⊂ intB, i.e., A δ X \ intB ( A is far from the complement of intB ). Correspondingly, a set A has a descriptive stronginclusion in a set B associated with δΦ (denoted by A ≪Φ B ), provided Q(A) ⊂ Q(intB) , i.e., A δΦ X \ intB (Q(A) is far from the complement of intB ).Let ≪Φ be a descriptive δ -neighbourhood relation defined by

≪Φ ={(A,B) ∈ 2X × 2X : Q(A) ⊂ Q(intB)

}.

That is, A≪Φ B , provided the description of each a ∈ A is contained in the set of descriptions of the points b ∈ intB. Now observe that any A,B in the proximal relator space X such that A δΦ B have disjoint δΦ -neighbourhoods,i.e.,

A δΦ B ⇔ A≪Φ E1, B ≪Φ E2, for some E1, E2 ⊂ X (See Fig. 6).

Theorem 6 [50] Any two sets descriptively far from each other belong to disjoint descriptive δΦ -neighbourhoods ina descriptive proximity space X .

A consideration of strong containment of a nonempty set in another set leads to the study of hit-and-miss topologiesand the Wijsman topology[2].

22.9 Tolerance near sets

Let ε be a real number greater than zero. In the study of sets that are proximally near within some tolerance, the setof proximity relations RδΦ is augmented with a pseudometric tolerance proximity relation (denoted by δΦ,ε ) definedby

DΦ(A,B) = inf {d(Φ(a),Φ(a)) : Φ(a) ∈ Q(A),Φ(a) ∈ Q(B)} ,

d(Φ(a),Φ(a)) =∑n

i=1|ϕi(a)− ϕi(b)|,

δΦ,ε ={(A,B) ∈ 2X × 2X : |D(cl(A), cl(B))| < ε

}.

Let RδΦ,ε = RδΦ ∪{δΦ,ε} . In other words, a nonempty set equipped with the proximal relator RδΦ,ε has underlyingstructure provided by the proximal relator RδΦ and provides a basis for the study of tolerance near sets in X that arenear within some tolerance. Sets A,B in a descriptive pseudometric proximal relator space (X,RδΦ,ε) are tolerancenear sets (i.e., A δΦ,ε B ), provided

DΦ(A,B) < ε.

22.10 Tolerance classes and preclasses

Relations with the same formal properties as similarity relations of sensations considered by Poincaré[62] are nowadays,after Zeeman[83], called tolerance relations. A tolerance τ on a set O is a relation τ ⊆ O × O that is reflexive andsymmetric. In algebra, the term tolerance relation is also used in a narrow sense to denote reflexive and symmetricrelations defined on universes of algebras that are also compatible with operations of a given algebra, i.e., they aregeneralizations of congruence relations (see e.g.,[12]). In referring to such relations, the term algebraic tolerance orthe term algebraic tolerance relation is used. Transitive tolerance relations are equivalence relations. A set O togetherwith a tolerance τ is called a tolerance space (denoted (O, τ) ). A set A ⊆ O is a τ -preclass (or briefly preclasswhen τ is understood) if and only if for any x, y ∈ A , (x, y) ∈ τ .The family of all preclasses of a tolerance space is naturally ordered by set inclusion and preclasses that are maximalwith respect to set inclusion are called τ -classes or just classes, when τ is understood. The family of all classes ofthe space (O, τ) is particularly interesting and is denoted by Hτ (O) . The family Hτ (O) is a covering of O [58].

22.10. TOLERANCE CLASSES AND PRECLASSES 89

The work on similarity by Poincaré and Zeeman presage the introduction of near sets[44][43] and research on similarityrelations, e.g.,[79]. In science and engineering, tolerance near sets are a practical application of the study of sets thatare near within some tolerance. A tolerance ε ∈ (0,∞] is directly related to the idea of closeness or resemblance(i.e., being within some tolerance) in comparing objects. By way of application of Poincaré's approach in definingvisual spaces and Zeeman’s approach to tolerance relations, the basic idea is to compare objects such as image patchesin the interior of digital images.

22.10.1 Examples

Simple ExampleThe following simple example demonstrates the construction of tolerance classes from real data. Consider the 20objects in the table below with |Φ| = 1 .

Let a tolerance relation be defined as

∼=ε= {(x, y) ∈ O ×O : ∥ Φ(x)− Φ(y) ∥2≤ ε}

Then, setting ε = 0.1 gives the following tolerance classes:

H∼=ε(O) ={{x1, x8, x10, x11}, {x1, x9, x10, x11, x14},{x2, x7, x18, x19},{x3, x12, x17},{x4, x13, x20}, {x4, x18},{x5, x6, x15, x16}, {x5, x6, x15, x20},{x6, x13, x20}}.

Observe that each object in a tolerance class satisfies the condition ∥ Φ(x)−Φ(y) ∥2≤ ε , and that almost all of theobjects appear in more than one class. Moreover, there would be twenty classes if the indiscernibility relation wasused since there are no two objects with matching descriptions.Image Processing Example

Figure 7. Example of images that are near each other. (a) and (b) Images from the freely available LeavesDataset (see, e.g.,www.vision.caltech.edu/archive.html).

The following example provides an example based on digital images. Let a subimage be defined as a small subset ofpixels belonging to a digital image such that the pixels contained in the subimage form a square. Then, let the sets Xand Y respectively represent the subimages obtained from two different images, and let O = {X ∪ Y } . Finally, letthe description of an object be given by the Green component in the RGB color model. The next step is to find all

90 CHAPTER 22. NEAR SETS

the tolerance classes using the tolerance relation defined in the previous example. Using this information, toleranceclasses can be formed containing objects that have similar (within some small ε ) values for the Green componentin the RGB colour model. Furthermore, images that are near (similar) to each other should have tolerance classesdivided among both images (instead of a tolerance classes contained solely in one of the images). For example, thefigure accompanying this example shows a subset of the tolerance classes obtained from two leaf images. In thisfigure, each tolerance class is assigned a separate colour. As can be seen, the two leaves share similar toleranceclasses. This example highlights a need to measure the degree of nearness of two sets.

22.11 Nearness measure

Let (U,RδΦ,ε) denote a particular descriptive pseudometric EF-proximal relator space equipped with the proximityrelation δΦ,ε and with nonempty subsets X,Y ∈ 2U and with the tolerance relation ∼=Φ,ε defined in terms of a set ofprobes Φ and with ε ∈ (0,∞] , where

Figure 8. Examples of degree of nearness between two sets: (a) High degree of nearness, and (b) Low degree of nearness.

≃Φ,ε= {(x, y) ∈ U × U | |Φ(x)− Φ(y)| ≤ ε}.

Further, assume Z = X ∪ Y and let HτΦ,ε(Z) denote the family of all classes in the space (Z,≃Φ,ε) .Let A ⊆ X,B ⊆ Y . The distance D

tNM: 2U × 2U :−→ [0,∞] is defined by

DtNM

(X,Y ) =

{1− tNM(A,B), if X and Y are not empty,∞, if X or Y is empty,

where

tNM(A,B) =

( ∑C∈HτΦ,ε

(Z)

|C|

)−1

·∑

C∈HτΦ,ε(Z)

|C|min(|C ∩A|, |[C ∩B|)max(|C ∩A|, |C ∩B|)

.

The details concerning tNM are given in[14][16][17]. The idea behind tNM is that sets that are similar should havea similar number of objects in each tolerance class. Thus, for each tolerance class obtained from the covering ofZ = X ∪Y , tNM counts the number of objects that belong to X and Y and takes the ratio (as a proper fraction) oftheir cardinalities. Furthermore, each ratio is weighted by the total size of the tolerance class (thus giving importanceto the larger classes) and the final result is normalized by dividing by the sum of all the cardinalities. The range oftNM is in the interval [0,1], where a value of 1 is obtained if the sets are equivalent (based on object descriptions)and a value of 0 is obtained if they have no descriptions in common.As an example of the degree of nearness between two sets, consider figure below in which each image consists of twosets of objects, X and Y . Each colour in the figures corresponds to a set where all the objects in the class share the

22.12. NEAR SET EVALUATION AND RECOGNITION (NEAR) SYSTEM 91

same description. The idea behind tNM is that the nearness of sets in a perceptual system is based on the cardinalityof tolerance classes that they share. Thus, the sets in left side of the figure are closer (more near) to each other interms of their descriptions than the sets in right side of the figure.

22.12 Near set evaluation and recognition (NEAR) system

Figure 9. NEAR system GUI.

The Near set Evaluation and Recognition (NEAR) system, is a system developed to demonstrate practical applicationsof near set theory to the problems of image segmentation evaluation and image correspondence. It was motivatedby a need for a freely available software tool that can provide results for research and to generate interest in nearset theory. The system implements a Multiple Document Interface (MDI) where each separate processing task isperformed in its own child frame. The objects (in the near set sense) in this system are subimages of the images beingprocessed and the probe functions (features) are image processing functions defined on the subimages. The systemwas written in C++ and was designed to facilitate the addition of new processing tasks and probe functions. Currently,the system performs six major tasks, namely, displaying equivalence and tolerance classes for an image, performingsegmentation evaluation, measuring the nearness of two images, performing Content Based Image Retrieval (CBIR),and displaying the output of processing an image using a specific probe function.

22.13 Proximity System

The Proximity System is an application developed to demonstrate descriptive-based topological approaches to near-ness and proximity within the context of digital image analysis. The Proximity System grew out of the work of S.Naimpally and J. Peters on Topological Spaces. The Proximity System was written in Java and is intended to run intwo different operating environments, namely on Android smartphones and tablets, as well as desktop platforms run-

92 CHAPTER 22. NEAR SETS

Figure 10. The Proximity System.

ning the Java Virtual Machine. With respect to the desktop environment, the Proximity System is a cross-platformJava application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linuxusing the Sun Java 6 Runtime. In terms of the implementation of the theoretical approaches, both the Android andthe desktop based applications use the same back-end libraries to perform the description-based calculations, wherethe only differences are the user interface and the Android version has less available features due to restrictions onsystem resources.

22.14 See also• Alternative set theory

• Category:Mathematical relations

• Category:Topology

• Feature vector

• Proximity space

• Rough set

• Topology

22.15 Notes1. ^ J.R. Isbell observed that the notions near and far are important in a uniform space. Sets A,B are far

(uniformaly distal), provided the {A,B} is a discrete collection. A nonempty set U is a uniform neighbour-hood of a set A , provided the complement of U is far from U . See, §33 in [23]

2. ^ The intuition that led to the discovery of descriptively near sets is given in Pawlak, Z.;Peters, J.F. (2002,2007) “Jak blisko (How Near)". Systemy Wspomagania Decyzji I 57 (109)

3. ^ Descriptively near sets are introduced in[48]. The connections between traditional EF-proximity and descrip-tive EF-proximity are explored in [37].

22.16. REFERENCES 93

4. ^ Reminiscent of M. Pavel’s approach, descriptions of members of sets objects are defined relative to vectorsof values obtained from real-valued functions called probes. See, Pavel, M. (1993). Fundamentals of patternrecognition. 2nd ed. New York: Marcel Dekker, for the introduction of probe functions considered in thecontext of image registration.

5. ^ A non-spatial view of near sets appears in, C.J. Mozzochi, M.S. Gagrat, and S.A. Naimpally, Symmetricgeneralized topological structures, Exposition Press, Hicksville, NY, 1976., and, more recently, nearness ofdisjoint sets X and Y based on resemblance between pairs of elements x ∈ X, y ∈ Y (i.e. x and y havesimilar feature vectors ϕ(x),ϕ(y) and the norm ∥ ϕ(x)− ϕ(y) ∥p< ε ) See, e.g.,[43][42][53].

6. ^ The basic facts about closure of a set were first pointed out by M. Fréchet in[11], and elaborated by B. Knasterand C. Kuratowski in[25].

7. ^Observe that up to the 1970s, proximity meant EF-proximity, since this is the one that was studied intensively.The pre-1970 work on proximity spaces is exemplified by the series of papers by J. M. Smirnov during thefirst half of the 1950s[68][67][69][70], culminating in the compendious collection of results by S.A. Naimpally andB.D. Warrack[34]. But in view of later developments, there is a need to distinguish between various proximities.A basic proximity or Čech-proximity was introduced by E. Čech during the late 1930s (see §25 A.1, pp. 439-440 in [78]). The conditions for the non-symmetric case for a proximity were introduced by S. Leader[28] andfor the symmetric case by M.W. Lodato[29][30][31].

22.16 References1. ^ Adámek, J.; Herrlich, H.; Strecker, G. E. (1990). Abstract and concrete categories. London: Wiley-

Interscience. pp. ix+482.

2. ^ Beer, G. (1993), “Topologies on closed and closed convex sets”, London, UK: Kluwer Academic Pub., pp.xi + 340pp. Missing or empty |title= (help)

3. ^ Bentley, H. L.; Colebunders, E.; Vandermissen, E. (2009), “A convenient setting for completions and func-tion spaces”, in Mynard, F.; Pearl, E., Contemporary Mathematics, Providence, RI: American MathematicalSociety, pp. 37–88 Missing or empty |title= (help)

4. ^ Cameron, P.; Hockingand, J. G.; Naimpally, S. A. (1974). “Nearness–a better approach to continuity andlimits”. American Mathematical Monthly 81 (7): 739–745. doi:10.2307/2319561.

5. ^ Di Concilio, A. (2008), “Action, uniformity and proximity”, in Naimpally, S. A.; Di Maio, G., Theory andApplications of Proximity, Nearness and Uniformity, Seconda Università di Napoli, Napoli: Prentice-Hall, pp.71–88 Missing or empty |title= (help)

6. ^ a b Di Concilio, A. (2009). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces,boolean algebras and point-free geometry”. ContemporaryMathematics 486: 89–114. doi:10.1090/conm/486/09508.

7. ^ Devi, R.; Selvakumar, A.; Vigneshwaran, M. (2010). " (I, γ) -generalized semi-closed sets in topologicalspaces”. FILOMAT 24 (1): 97–100. doi:10.2298/fil1001097d.

8. ^ a b c Efremovič, V. A. (1952). “The geometry of proximity I (in Russian)". Mat. Sb. (N.S.) 31 (73): 189–200.

9. ^ Peters, J. F. (2008). “A note on a-open sets and e ∗ -sets”. FILOMAT 22 (1): 89–96.

10. ^ Fechner, G. T. (1966). Elements of Psychophysics, vol. I. London, UK: Hold, Rinehart & Winston. pp. H.E. Adler’s trans. of Elemente der Psychophysik, 1860.

11. ^ Fréchet, M. (1906). “Sur quelques points du calcul fonctionnel”. Rend. Circ. Mat. Palermo 22: 1–74.doi:10.1007/bf03018603.

12. ^ Grätzer, G.; Wenzel, G. H. (1989). “Tolerances, covering systems, and the axiom of choice”. ArchivumMathematicum 25 (1-2): 27–34.

13. ^ Gupta, S.; Patnaik, K. (2008). “Enhancing performance of face recognition systems by using near set ap-proach for selecting facial features”. Journal of Theoretical and Applied Information Technology 4 (5): 433–441.

94 CHAPTER 22. NEAR SETS

14. ^ a b Hassanien, A. E.; Abraham, A.; Peters, J. F.; Schaefer, G.; Henry, C. (2009). “Rough sets and near sets inmedical imaging: A review, IEEE”. Transactions on Information Technology in Biomedicine 13 (6): 955–968.doi:10.1109/TITB.2009.2017017.

15. ^ Hausdorff, F. (1914). Grundz¨uge der mengenlehre. Leipzig: Veit and Company. pp. viii + 476.

16. ^ Henry, C.; Peters, J. F. (2010). “Perception-based image classification, International”. Journal of IntelligentComputing and Cybernetics 3 (3): 410–430. doi:10.1108/17563781011066701.

17. ^ a b c d Henry, C. J. (2010), “Near sets: Theory and applications”, Ph.D. thesis, Dept. Elec. Comp. Eng., Uni.of MB, supervisor: J.F. Peters

18. ^ a b Henry, C.; Peters, J. F. (2011). “Arthritic hand-finger movement similarity measurements: Tolerance nearset approach”. Computational and Mathematical Methods in Medicine 2011: 1–14. doi:10.1155/2011/569898.

19. ^ Henry, C. J.; Ramanna, S. (2011). “Parallel Computation in Finding Near Neighbourhoods”. Lecture Notesin Computer Science,: 523–532.

20. ^ a b Herrlich, H. (1974). “A concept of nearness”. General Topology and its Applications 4: 191–212.doi:10.1016/0016-660x(74)90021-x.

21. ^ Hocking, J. G.; Naimpally, S. A. (2009), “Nearness—a better approach to continuity and limits”, AllahabadMathematical Society Lecture Note Series 3, Allahabad: The Allahabad Mathematical Society, pp. iv+66,ISBN 978-81-908159-1-8 Missing or empty |title= (help)

22. ^ Ïnan, E.; Öztürk, M. A. (2012). “Near groups on nearness approximation spaces”. Hacettepe Journal ofMathematics and Statistics 41 (4): 545–558.

23. ^ Isbell, J. R. (1964). Uniform spaces. Providence, Rhode Island: American Mathematical Society. pp. xi +175.

24. ^ Ivanova, V. M.; Ivanov, A. A. (1959). “Contiguity spaces and bicompact extensions of topological spaces(russian)". Dokl. Akad. Nauk SSSR 127: 20–22.

25. ^ Knaster, B.; Kuratowski, C. (1921). “Sur les ensembles connexes”. Fundamenta Mathematicae 2: 206–255.

26. ^Kovár, M. M. (2011). “A new causal topology and why the universe is co-compact”. arXive:1112.0817{[}math-ph]:1–15.

27. ^ Kuratowski, C. (1958), “Topologie i”, Warsaw: Panstwowe Wydawnictwo Naukowe, pp. XIII + 494pp.Missing or empty |title= (help)

28. ^ Leader, S. (1967). “Metrization of proximity spaces”. Proceedings of the American Mathematical Society18: 1084–1088. doi:10.2307/2035803.

29. ^ Lodato, M. W. (1962), “On topologically induced generalized proximity relations”, Ph.D. thesis, RutgersUniversity

30. ^ Lodato, M. W. (1964). “On topologically induced generalized proximity relations I”. Proceedings of theAmerican Mathematical Society 15: 417–422. doi:10.2307/2034517.

31. ^ Lodato, M. W. (1966). “On topologically induced generalized proximity relations II”. Pacific Journal ofMathematics 17: 131–135.

32. ^ MacLane, S. (1971). Categories for the working mathematician. Berlin: Springer. pp. v+262pp.

33. ^ Mozzochi, C. J.; Naimpally, S. A. (2009), “Uniformity and proximity”, Allahabad Mathematical SocietyLecture Note Series 2, Allahabad: The Allahabad Mathematical Society, pp. xii+153, ISBN 978-81-908159-1-8 Missing or empty |title= (help)

34. ^ Naimpally, S. A. (1970). Proximity spaces. Cambridge, UK: Cambridge University Press. pp. x+128. ISBN978-0-521-09183-1.

35. ^ Naimpally, S. A. (2009). Proximity approach to problems in topology and analysis. Munich, Germany:Oldenbourg Verlag. pp. ix + 204. ISBN 978-3-486-58917-7.

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36. ^ Naimpally, S. A.; Peters, J. F. (2013). “Preservation of continuity”. Scientiae Mathematicae Japonicae 76(2): 1–7.

37. ^ a b c d Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near andFar. Singapore: World Scientific.

38. ^ Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013). Near set theory and applications. Special Issue inMathematics in Computer Science 7. Berlin: Springer. p. 136.

39. ^ Naimpally, S. A.; Warrack, B. D. (1970), “Proximity spaces”, Cambridge Tract in Mathematics 59, Cam-bridge, UK: Cambridge University Press, pp. x+128 Missing or empty |title= (help)

40. ^ Pal, S. K.; Peters, J. F. (2010). Rough fuzzy image analysis. Foundations and methodologies. London, UK,:CRC Press, Taylor & Francis Group. ISBN 9781439803295.

41. ^ Peters, J. F. (2009). “Tolerance near sets and image correspondence”. International Journal of Bio-InspiredComputation 1 (4): 239–245. doi:10.1504/ijbic.2009.024722.

42. ^ a b c Peters, J. F.; Wasilewski, P. (2009). “Foundations of near sets”. Information Sciences 179 (18): 3091–3109. doi:10.1016/j.ins.2009.04.018.

43. ^ a b c Peters, J. F. (2007). “Near sets. General theory about nearness of objects”. Applied MathematicalSciences 1 (53): 2609–2629.

44. ^ a b Peters, J. F. (2007). “Near sets. Special theory about nearness of objects”. Fundamenta Informaticae 75(1-4): 407–433.

45. ^ Peters, J. F. (2010). “Corrigenda and addenda: Tolerance near sets and image correspondence”. InternationalJournal of Bio-Inspired Computation 2 (5): 310–318. doi:10.1504/ijbic.2010.036157.

46. ^ Peters, J. F. (2011), “How near are Zdzisław Pawlak’s paintings? Merotopic distance between regions ofinterest”, in Skowron, A.; Suraj, S., Intelligent Systems Reference Library volume dedicated to Prof. ZdzisławPawlak, Berlin: Springer, pp. 1–19 Missing or empty |title= (help)

47. ^ Peters, J. F. (2011), “Sufficiently near sets of neighbourhoods”, in Yao, J. T.; Ramanna, S.; Wang, G. et al.,Lecture Notes in Artificial Intelligence 6954, Berlin: Springer, pp. 17–24 Missing or empty |title= (help);

48. ^ a b c d Peters, J. F. (2013). “Near sets: An introduction”. Mathematics in Computer Science 7 (1): 3–9.doi:10.1007/s11786-013-0149-6.

49. ^ a b c Peters, J. F. (2014). “Proximal relator spaces”. FILOMAT : 1–5 (in press).

50. ^ a b c d e Peters, J. F. (2014). Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces 63.Springer. p. 342. ISBN 978-3-642-53844-5.

51. ^ a b Peters, J. F.; İnan, E.; Öztürk, M. A. (2014). “Spatial and descriptive isometries in proximity spaces”.General Mathematics Notes 21 (2): 125–134.

52. ^ Peters, J. F.; Naimpally, S. A. (2011). “Approach spaces for near families”. General Mathematics Notes 2(1): 159–164.

53. ^ a b c Peters, J. F.; Naimpally, S. A. (2011). General Mathematics Notes 2 (1): 159–164. Missing or empty|title= (help)

54. ^ Peters, J. F.; Puzio, L. (2009). “Image analysis with anisotropic wavelet-based nearness measures”. Interna-tional Journal of Computational Intelligence Systems 2 (3): 168–183. doi:10.1016/j.ins.2009.04.018.

55. ^ Peters, J. F.; Shahfar, S.; Ramanna, S.; Szturm, T. (2007), “Biologically-inspired adaptive learning: A nearset approach”, Frontiers in the Convergence of Bioscience and Information Technologies, Korea Missing orempty |title= (help)

56. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.

57. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.

96 CHAPTER 22. NEAR SETS

58. ^ Peters, J. F.; Wasilewski, P. (2012). “Tolerance spaces: Origins, theoretical aspects and applications”. In-formation Sciences 195: 211–225. doi:10.1016/j.ins.2012.01.023.

59. ^ Picado, J. “Weil nearness spaces”. Portugaliae Mathematica 55 (2): 233–254.

60. ^ a b c Poincaré, J. H. (1895). “L'espace et la géomètrie”. Revue de m'etaphysique et de morale 3: 631–646.

61. ^ a b c Poincaré, J. H. (1902). “Sur certaines surfaces algébriques; troisième complément 'a l'analysis situs”.Bulletin de la Société de France 30: 49–70.

62. ^ a b Poincaré, J. H. (1913 & 2009). Dernières pensées, trans. by J.W. Bolduc as Mathematics and science: Lastessays. Paris & NY: Flammarion & Kessinger. Check date values in: |date= (help)

63. ^ a b Poincaré, J. H. (1894). “Sur la nature du raisonnement mathématique”. Revue de méaphysique et demorale 2: 371–384.

64. ^ a b Ramanna, S.; Meghdadi, A. H. (2009). “Measuring resemblances between swarm behaviours: A percep-tual tolerance near set approach”. Fundamenta Informaticae 95 (4): 533–552. doi:10.3233/FI-2009-163.

65. ^ a b Riesz, F. (1908). “Stetigkeitsbegriff und abstrakte mengenlehre”. Atti del IV Congresso Internazionale deiMatematici II: 182–109.

66. ^ Shreider, J. A. (1975). Equality, resemblance, and order. Russia: Mir Publishers. p. 279.

67. ^ a b c d Smirnov, J. M. (1952). “On proximity spaces”. Mat. Sb. (N.S.) 31 (73): 543–574 (English translation:Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 5–35).

68. ^ Smirnov, J. M. (1952). “On proximity spaces in the sense of V.A. Efremovič". Math. Sb. (N.S.) 84:895–898, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 1–4.

69. ^ Smirnov, J. M. (1954). “On the completeness of proximity spaces. I.”. Trudy Moskov. Mat. Obšč 3:271–306, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 37–74.

70. ^ Smirnov, J. M. (1955). “On the completeness of proximity spaces. II.”. Trudy Moskov. Mat. Obšč 4:421–438, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 75–94.

71. ^ a b Sossinsky, A. B. (1986). “Tolerance space theory and some applications”. Acta Applicandae Mathemati-cae: An International Survey Journal on Applying Mathematics and Mathematical Applications 5 (2): 137–167.doi:10.1007/bf00046585.

72. ^ Száz, Á. (1997). “Uniformly, proximally and topologically compact relators”. Mathematica Pannonica 8 (1):103–116.

73. ^ Száz, Á. (1987). “Basic tools and mild continuities in relator spaces”. Acta Mathematica Hungarica 50:177–201. doi:10.1007/bf01903935.

74. ^ Száz, Á (2000). “An extension of Kelley’s closed relation theorem to relator spaces”. FILOMAT 14: 49–71.

75. ^ Tiwari, S. (2010), “Some aspects of general topology and applications. Approach merotopic structures andapplications”, Ph.D. thesis, Dept. of Math., Allahabad (U.P.), India, supervisor: M. khare

76. ^ a b Tiwari, S.; Peters, J. F. (2013). “A new approach to the study of extended metric spaces”. MathematicaAeterna 3 (7): 565–577.

77. ^ Tukey, J. W. (1940), “Convergence and uniformity in topology”, Annals of Mathematics Studies AM–2,Princeton, NJ: Princeton Univ. Press, p. 90 Missing or empty |title= (help)

78. ^ Čech, E. (1966). Topological spaces, revised ed. by Z. Frolik and M. Katětov. London: John Wiley & Sons.p. 893.

79. ^ Wasilewski, P. (2004), “On selected similarity relations and their applications into cognitive science”, Ph.D.thesis, Dept. Logic

80. ^ Wasilewski, P.; Peters, J. F.; Ramanna, S. (2011). “Perceptual tolerance intersection”. Transactions onRough Sets XIII: 159–174.

22.17. FURTHER READING 97

81. ^ Weil, A. (1938), “Sur les espaces à structure uniforme et sur la topologie générale”, Actualités scientifiqueet industrielles, Paris: Harmann & cie Missing or empty |title= (help)

82. ^ Wolski, M. (2010). “Perception and classification. A note on near sets and rough sets”. Fundamenta Infor-maticae 101: 143–155.

83. ^ a b Zeeman, E. C. (1962), “The topology of the brain and visual perception”, in Fort, Jr., M. K., Topologyof 3-Manifolds and Related Topics, University of Georgia Institute Conference Proceedings (1962): Prentice-Hall, pp. 240–256 Missing or empty |title= (help)

22.17 Further reading• Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near and Far.

World Scientific Publishing . Co. Pte. Ltd. ISBN 978-981-4407-65-6.

• Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013), "Near Set Theory and Applications", Mathematics inComputer Science 7 (1), Berlin: Springer Missing or empty |title= (help)

• Peters, J. F. (2014), "Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces", IntelligentSystems Reference Library 63, Berlin: Springer Missing or empty |title= (help)

• Henry, C. J.; Peters, J. F. (2012), "Near set evaluation and recognition (NEAR) system V3.0", UM CI Labo-ratory Technical Report No. TR-2009-015, Computational Intelligence Laboratory, University of ManitobaMissing or empty |title= (help)

• Concilio, A. Di (2014). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces, booleanalgebras and point-free geometry”. Computational Intelligence Laboratory, University of Manitoba. UM CILaboratory Technical Report No. TR-2009-021.

• Peters, J. F.; Naimpally, S. A. (2012). “Applications of near sets” (PDF). Notices of the AmericanMathematicalSociety 59 (4): 536–542. CiteSeerX: 10 .1 .1 .371 .7903.

Chapter 23

Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restrictedequivalence relation) R on a set X is a relation that is symmetric and transitive. In other words, it holds for alla, b, c ∈ X that:

1. if aRb , then bRa (symmetry)

2. if aRb and bRc , then aRc (transitivity)

If R is also reflexive, then R is an equivalence relation.

23.1 Properties and applications

In a set-theoretic context, there is a simple structure to the general PER R on X : it is an equivalence relation on thesubset Y = {x ∈ X|xRx} ⊆ X . ( Y is the subset of X such that in the complement of Y ( X \ Y ) no element isrelated by R to any other.) By construction, R is reflexive on Y and therefore an equivalence relation on Y . Noticethat R is actually only true on elements of Y : if xRy , then yRx by symmetry, so xRx and yRy by transitivity.Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to definesetoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to theoperations of subset and quotient in classical set-theoretic mathematics.Every partial equivalence relation is a difunctional relation, but the converse does not hold.The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence,i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive.[1]

23.2 Examples

A simple example of a PER that is not an equivalence relation is the empty relation R = ∅ (unless X = ∅ , in whichcase the empty relation is an equivalence relation (and is the only relation on X )).

23.2.1 Euclidean parallelism

In the Euclidean plane, two lines m and n are parallel lines when m ∩ n = ∅. The symmetry of this relation is obviousand the transitivity can be proven in the Euclidean plane, thus Euclidean parallelism is a partial equivalence relation.Nevertheless, mathematicians developing affine geometry prefer the facility of an equivalence relation and thereforesometimes revise the definition of parallelism to allow a line to be parallel to itself, making the new relation of “affineparallelism” that is a reflexive relation.

98

23.3. REFERENCES 99

23.2.2 Kernels of partial functions

For another example of a PER, consider a set A and a partial function f that is defined on some elements of A butnot all. Then the relation ≈ defined by

x ≈ y if and only if f is defined at x , f is defined at y , and f(x) = f(y)

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties,but it is not reflexive since if f(x) is not defined then x ̸≈ x — in fact, for such an x there is no y ∈ A such thatx ≈ y . (It follows immediately that the subset of A for which ≈ is an equivalence relation is precisely the subset onwhich f is defined.)

23.2.3 Functions respecting equivalence relations

Let X and Y be sets equipped with equivalence relations (or PERs) ≈X ,≈Y . For f, g : X → Y , define f ≈ g tomean:

∀x0 x1, x0 ≈X x1 ⇒ f(x0) ≈Y g(x1)

then f ≈ f means that f induces a well-defined function of the quotients X/ ≈X → Y / ≈Y . Thus, the PER≈ captures both the idea of definedness on the quotients and of two functions inducing the same function on thequotient.

23.3 References[1] J. Lambek (1996). “The Butterfly and the Serpent”. In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp.

161–180. ISBN 978-0-8247-9606-8.

• Mitchell, John C. Foundations of programming languages. MIT Press, 1996.

• D.S. Scott. “Data types as lattices”. SIAM Journ. Comput., 3:523-587, 1976.

23.4 See also• Equivalence relation

• Binary relation

Chapter 24

Partial function

Not to be confused with partial function of a multilinear map or the mathematical concept of a piecewise function.

In mathematics, a partial function from X to Y (written as f: X ↛ Y) is a function f: X′ → Y, for some subset X′of X. It generalizes the concept of a function f: X → Y by not forcing f to map every element of X to an elementof Y (only some subset X′ of X). If X′ = X, then f is called a total function and is equivalent to a function. Partialfunctions are often used when the exact domain, X′, is not known (e.g. many functions in computability theory).Specifically, we will say that for any x ∈ X, either:

• f(x) = y ∈ Y (it is defined as a single element in Y) or

• f(x) is undefined.

For example we can consider the square root function restricted to the integers

g : Z → Z

g(n) =√n.

Thus g(n) is only defined for n that are perfect squares (i.e., 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined.

24.1 Basic concepts

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function.Most mathematicians, including recursion theorists, use the term “domain of f" for the set of all values x such thatf(x) is defined (X' above). But some, particularly category theorists, consider the domain of a partial function f:X →Y to be X, and refer to X' as the domain of definition. Similarly, the term range can refer to either the codomain orthe image of a function.Occasionally, a partial function with domain X and codomain Y is written as f: X ⇸ Y, using an arrow with verticalstroke.A partial function is said to be injective or surjective when the total function given by the restriction of the partialfunction to its domain of definition is. A partial function may be both injective and surjective.Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partialfunction which is injective.[1]

An injective partial function may be inverted to an injective partial function, and a partial function which is bothinjective and surjective has an injective function as inverse. Furthermore, a total function which is injective may beinverted to an injective partial function.The notion of transformation can be generalized to partial functions as well. A partial transformation is a functionf: A → B, where both A and B are subsets of some set X.[2]

100

24.2. TOTAL FUNCTION 101

24.2 Total function

Total function is a synonym for function. The use of the prefix “total” is to suggest that it is a special case over a largerset X of a partial function over a subset of X. For example, when considering the operation of morphism compositionin Concrete Categories, the composition operation ◦ : Hom(C) × Hom(C) → Hom(C) is a total function if andonly if Ob(C) has one element. The reason for this is that two morphisms f : X → Y and g : U → V can only becomposed as g ◦ f if Y = U , that is, the codomain of f must equal the domain of g .

24.3 Discussion and examples

The first diagram above represents a partial function that is not a total function since the element 1 in the left-hand setis not associated with anything in the right-hand set. Whereas, the second diagram represents a total function sinceevery element on the left-hand set is associated with exactly one element in the right hand set.

24.3.1 Natural logarithm

Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positivereal is not a real number, so the natural logarithm function doesn't associate any real number in the codomain withany non-positive real number in the domain. Therefore, the natural logarithm function is not a total function whenviewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only includethe positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals),then the natural logarithm is a total function.

24.3.2 Subtraction of natural numbers

Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:

f : N× N → N

f(x, y) = x− y.

It is defined only when x ≥ y .

24.3.3 Bottom element

In denotational semantics a partial function is considered as returning the bottom element when it is undefined.In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEEfloating point standard defines a not-a-number value which is returned when a floating point operation is undefinedand exceptions are suppressed, e.g. when the square root of a negative number is requested.In a programming language where function parameters are statically typed, a function may be defined as a partialfunction because the language’s type system cannot express the exact domain of the function, so the programmerinstead gives it the smallest domain which is expressible as a type and contains the true domain.

24.3.4 In category theory

The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets andpoint-preserving maps.[3] One textbook notes that “This formal completion of sets and partial maps by adding “im-proper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) andin theoretical computer science.”[4]

The category of sets and partial bijections is equivalent to its dual.[5] It is the prototypical inverse category.[6]

102 CHAPTER 24. PARTIAL FUNCTION

24.3.5 In abstract algebra

Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in whichthe multiplicative inversion is the only proper partial operation (because division by zero is not defined).[7]

The set of all partial functions (partial transformations) on a given base X set forms a regular semigroup called thesemigroup of all partial transformations (or the partial transformation semigroup on X), typically denoted by PT X

.[8][9][10] The set of all partial bijections on X forms the symmetric inverse semigroup.[8][9]

24.4 See also• Bijection

• Injective function

• Surjective function

• Multivalued function

• Densely defined operator

24.5 References[1] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.

American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[2] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Lutz Schröder (2001). “Categories: a free tour”. In Jürgen Koslowski and Austin Melton. Categorical Perspectives. SpringerScience & Business Media. p. 10. ISBN 978-0-8176-4186-3.

[4] Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science &Business Media. p. 290. ISBN 978-1-4419-0615-1.

[5] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[6] Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semi-groups. World Scientific. p. 55. ISBN 978-981-4407-06-9.

[7] Peter Burmeister (1993). “Partial algebras – an introductory survey”. In Ivo G. Rosenberg and Gert Sabidussi. Algebrasand Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9.

[8] Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American MathematicalSoc. p. xii. ISBN 978-0-8218-0272-4.

[9] Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5.

[10] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction.Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4.

• Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Re-published by Dover in 1982. ISBN 0-486-61471-9.

• Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam,Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.

• Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Com-pany, New York.

Chapter 25

Partially ordered set

{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

Ø

The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

103

104 CHAPTER 25. PARTIALLY ORDERED SET

25.1 Formal definition

A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive,i.e., which satisfies for all a, b, and c in P:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as “ordered sets”, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they areincomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-rightfigure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no thirdelement c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. Amore concise definition will be given below using the strict order corresponding to "≤". For example, {x} is coveredby {x,z} in the top-right figure, but not by {x,y,z}.

25.2 Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).

• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, theset of sequences ordered by subsequence, and the set of strings ordered by substring.

• The set of natural numbers equipped with the relation of divisibility.

• The vertex set of a directed acyclic graph ordered by reachability.

• The set of subspaces of a vector space ordered by inclusion.

• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕif and only if a ≤ b for all n in ℕ, i.e. a componentwise order.

• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ gif and only if f(x) ≤ g(x) for all x in X.

• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...

25.3 Extrema

There are several notions of “greatest” and “least” element in a poset P, notably:

25.4. ORDERS ON THE CARTESIAN PRODUCT OF PARTIALLY ORDERED SETS 105

• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g.An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest orleast element.

• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a <m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be morethan one maximal element, and similarly for least elements and minimal elements.

• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for eachelement a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is alower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, anda least element is a lower bound of P.

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements;on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, whichis a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not evenhave any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If thenumber 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting posetdoes not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound(though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in theposet); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

25.4 Orders on the Cartesian product of partially ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesianproduct of two partially ordered sets are (see figures):

• the lexicographical order: (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);

• the product order: (a,b) ≤ (c,d) if a ≤ c and b ≤ d;

• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d)or (a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets.Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.See also orders on the Cartesian product of totally ordered sets.

25.5 Sums of partially ordered sets

Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of theunderlying sets X and Y by the order a ≤Z b if and only if:

• a, b ∈ X with a ≤X b, or

• a, b ∈ Y with a ≤Y b, or

• a ∈ X and b ∈ Y.

If two posets are well-ordered, then so is their ordinal sum.[5]

106 CHAPTER 25. PARTIALLY ORDERED SET

25.6 Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In thesecontexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric,i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity),

• if a < b and b < c then a < c (transitivity), and

• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if a ≤ b and a ≠ b

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closuregiven by:

a ≤ b if a < b or a = b.

This is the reason for using the notation "≤".Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b forany c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): everystrict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

25.7 Inverse and order dual

The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partialorder relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of apartially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > isto ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to eachother: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is onethat rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Thenatural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitudewhereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; wecould for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is notordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitudeyields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolutemagnitude but are not equal, violating antisymmetry.

25.8 Mappings between partially ordered sets

Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone,or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S→ T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f:S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving andorder-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective,since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that Scan be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the

25.9. NUMBER OF PARTIAL ORDERS 107

partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams(cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘gyields the identity function on S and T, respectively, then S and T are order-isomorphic. [7]

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set ofnatural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Itis order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neitherinjective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Takinginstead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number tothe set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and itsisomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to awide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".

25.9 Number of partial orders

Partially ordered set of set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).

25.10 Linear extension

A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and yof X, whenever x ≤ y, it is also the case that x ≤* y. A linear extension is an extension that is also a linear (i.e., total)order. Every partial order can be extended to a total order (order-extension principle).[8]

In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability ordersof directed acyclic graphs) are called topological sorting.

108 CHAPTER 25. PARTIALLY ORDERED SET

25.11 In category theory

Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element.More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets areequivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initialobject, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset.Finally, every subcategory of a poset is isomorphism-closed.

25.12 Partial orders in topological spaces

Main article: Partially ordered space

If P is a partially ordered set that has also been given the structure of a topological space, then it is customary toassume that {(a, b) : a ≤ b} is a closed subset of the topological product space P ×P . Under this assumption partialorder relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.[9]

25.13 Interval

For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains atleast the elements a and b.Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a< x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers isempty since there are no integers i such that 1 < i < 2.Sometimes the definitions are extended to allow a > b, in which case the interval is empty.The half-open intervals [a,b) and (a,b] are defined similarly.A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order-ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = {a,b}.This concept of an interval in a partial order should not be confused with the particular class of partial orders knownas the interval orders.

25.14 See also

• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets

• causal set

• comparability graph

• complete partial order

• directed set

• graded poset

• incidence algebra

• lattice

• locally finite poset

• Möbius function on posets

• ordered group

25.15. NOTES 109

• poset topology, a kind of topological space that can be defined from any poset

• Scott continuity - continuity of a function between two partial orders.

• semilattice

• semiorder

• series-parallel partial order

• stochastic dominance

• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive.

• Zorn’s lemma

25.15 Notes[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.

p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hassediagram...

[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory,Partial Orders, Combinatorics. Springer. ISBN 9781848002012.

[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895

[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18

[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.

[6] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). NewYork: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.

[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.

[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1):144–161. doi:10.1090/S0002-9939-1954-0063016-5

25.16 References• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American Mathematical

Society 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7.

• Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.

• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam-bridge University Press. ISBN 0-521-66351-2.

25.17 External links• A001035: Number of posets with n labeled elements in the OEIS

• A000112: Number of posets with n unlabeled elements in the OEIS

Chapter 26

Preorder

Not to be confused with Pre-order.This article is about binary relations. For the graph vertex ordering, see Depth-first search. For other uses, seePreorder (disambiguation).“Quasiorder” redirects here. For irreflexive transitive relations, see strict order.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.All partial orders and equivalence relations are preorders, but preorders are more general.The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders,but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation,the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preordercan be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, isnot always useful or worthwhile, depending on the problem domain being studied.In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, thenotation ← or ≲ is used instead of ≤.To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and theorder relation between pairs of elements corresponding to the directed edges between vertices. The converse is nottrue: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs maycontain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to adirected acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost thedirection markers on the edges of the graph. In general, a preorder may have many disconnected components.

26.1 Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive,i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set that is equipped with a preorder is called a preordered set (or proset).[1]

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category, i.e.as a category with at most one morphism from an object to another. Here the objects correspond to the elementsof P, and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can beunderstood as an enriched category, enriched over the category 2 = (0→1).

110

26.2. EXAMPLES 111

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preorderedclass.

26.2 Examples• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where

x ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorderis the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for everypair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other.In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partiallyordered sets (preorders satisfying an additional anti-symmetry property).

• Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to everyneighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological spacein this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However,the relation between infinite topological spaces and their specialization preorders is not 1-to-1.

• A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergencevia nets is important in topology, where preorders cannot be replaced by partially ordered sets without losingimportant features.

• The relation defined by x ≤ y if f(x) ≤ f(y) , where f is a function into some preorder.

• The relation defined by x ≤ y if there exists some injection from x to y. Injection may be replaced by surjection,or any type of structure-preserving function, such as ring homomorphism, or permutation.

• The embedding relation for countable total orderings.

• The graph-minor relation in graph theory.

• A category with at most one morphism from any object x to any other object y is a preorder. Such categoriesare called thin. In this sense, categories “generalize” preorders by allowing more than one relation betweenobjects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

• Many-one and Turing reductions are preorders on complexity classes.

• The subtyping relations are usually preorders.

• Simulation preorders are preorders (hence the name).

• Reduction relations in abstract rewriting systems.

• The encompassment preorder on the set of terms, defined by s≤t if a subterm of t is a substitution instance ofs.

Example of a total preorder:

• Preference, according to common models.

26.3 Uses

Preorders play a pivotal role in several situations:

• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is inone-to-one correspondence with an Alexandrov topology on that set.

• Preorders may be used to define interior algebras.

• Preorders provide the Kripke semantics for certain types of modal logic.

112 CHAPTER 26. PREORDER

26.4 Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexiveclosure, R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x toy.Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preordertwice, and symmetric by definition.)Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set ofall equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y]if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y.By the construction of ~, this definition is independent of the chosen representatives and the corresponding relationis indeed well-defined. It is readily verified that this yields a partially ordered set.Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 corre-spondence between preorders and pairs (partition, partial order).For a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a≲ b and not b≲ a), or equivalently, usingthe equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial ordercan be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalenceis equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).(Alternatively, for a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The resultis the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive,and if it is, as we have seen, it is the same as before.)Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation " ≲ "; "≤" can beconfusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.Note that with this construction multiple preorders " ≲ " can give the same relation "<", so without more information,such as the equivalence relation, " ≲ " cannot be reconstructed from "<". Possible preorders include the following:

• Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial orderassociated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, sowe don't need the notations ≲ and ~.

• Define a≲ b as “not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a~ b as “neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ isan equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

26.5 Number of preorders

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus thenumber of preorders is the sum of the number of partial orders on every partition. For example:

• for n=3:

• 1 partition of 3, giving 1 preorder• 3 partitions of 2+1, giving 3 × 3 = 9 preorders• 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

• for n=4:

• 1 partition of 4, giving 1 preorder• 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders• 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders

26.6. INTERVAL 113

• 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

26.6 Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains atleast the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a <x and x < b, also written a < x < b. An open interval may be empty even if a < b.Also [a,b) and (a,b] can be defined similarly.

26.7 See also• partial order - preorder that is antisymmetric

• equivalence relation - preorder that is symmetric

• total preorder - preorder that is total

• total order - preorder that is antisymmetric and total

• directed set

• category of preordered sets

• prewellordering

• Well-quasi-ordering

26.8 References[1] For “proset”, see e.g. Eklund, Patrik; Gähler, Werner (1990), “Generalized Cauchy spaces”, Mathematische Nachrichten

147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.

• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9

Chapter 27

Prewellordering

In set theory, a prewellordering is a binary relation ≤ that is transitive, total, and wellfounded (more precisely, therelation x ≤ y ∧ y ≰ x is wellfounded). In other words, if ≤ is a prewellordering on a set X , and if we define ∼ by

x ∼ y ⇐⇒ x ≤ y ∧ y ≤ x

then ∼ is an equivalence relation on X , and ≤ induces a wellordering on the quotient X/ ∼ . The order-type of thisinduced wellordering is an ordinal, referred to as the length of the prewellordering.A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if ϕ : X → Ord is anorm, the associated prewellordering is given by

x ≤ y ⇐⇒ ϕ(x) ≤ ϕ(y)

Conversely, every prewellordering is induced by a unique regular norm (a norm ϕ : X → Ord is regular if, for anyx ∈ X and any α < ϕ(x) , there is y ∈ X such that ϕ(y) = α ).

27.1 Prewellordering property

If Γ is a pointclass of subsets of some collection F of Polish spaces, F closed under Cartesian product, and if ≤ is aprewellordering of some subset P of some element X of F , then ≤ is said to be a Γ -prewellordering of P if therelations <∗ and ≤∗ are elements of Γ , where for x, y ∈ X ,

1. x <∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ {x ≤ y ∧ y ̸≤ x}]

2. x ≤∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ x ≤ y]

Γ is said to have the prewellordering property if every set in Γ admits a Γ -prewellordering.The prewellordering property is related to the stronger scale property; in practice, many pointclasses having theprewellordering property also have the scale property, which allows drawing stronger conclusions.

27.1.1 Examples

Π11 andΣ1

2 both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals,for every n ∈ ω , Π1

2n+1 and Σ12n+2 have the prewellordering property.

27.1.2 Consequences

114

27.2. SEE ALSO 115

Reduction

If Γ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For anyspace X ∈ F and any sets A,B ⊆ X , A and B both in Γ , the union A ∪B may be partitioned into sets A∗, B∗ ,both in Γ , such that A∗ ⊆ A and B∗ ⊆ B .

Separation

If Γ is an adequate pointclass whose dual pointclass has the prewellordering property, then Γ has the separationproperty: For any space X ∈ F and any sets A,B ⊆ X , A and B disjoint sets both in Γ , there is a set C ⊆ Xsuch that both C and its complement X \ C are in Γ , with A ⊆ C and B ∩ C = ∅ .For example, Π1

1 has the prewellordering property, so Σ11 has the separation property. This means that if A and B

are disjoint analytic subsets of some Polish space X , then there is a Borel subset C of X such that C includes A andis disjoint from B .

27.2 See also• Descriptive set theory

• Scale property

• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinalswith a map to the integers

27.3 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 28

Propositional function

A propositional function in logic, is a sentence expressed in a way that would assume the value of true or false, exceptthat within the sentence is a variable (x) that is not defined or specified, which leaves the statement undetermined. Ofcourse, x could also consist of several variables.As a mathematical function, A(x) or A(x1, x2, · · ·, xn), the propositional function is abstracted from predicates orpropositional forms. As an example, let’s imagine the predicate, “x is hot”. The substitution of any entity for x willproduce a specific proposition that can be described as either true or false, even though "x is hot” on its own has novalue as either a true or false statement. However, when you assign x a value, such as lava, the function then has thevalue true; while if you assign x a value like ice, the function then has the value false.Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrotein The Principles of Mathematics (page 106):

"...it has become necessary to take propositional function as a primitive notion.

Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed twotheories to try to get at this question: the zig-zag theory and the ramified theory of types.[1]

A Propositional Function, or a predicate, in a variable x is a sentence p(x) involving x that becomes a propositionwhen we give x a definite value from the set of values it can take.

28.1 References[1] Tiles, Mary (2004). The philosophy of set theory an historical introduction to Cantor’s paradise (Dover ed.). Mineola, N.Y.:

Dover Publications. p. 159. ISBN 978-0-486-43520-6. Retrieved 1 February 2013.

116

28.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 117

28.2 Text and image sources, contributors, and licenses

28.2.1 Text• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914 Contributors: AxelBoldt, Bryan Derksen, Zun-

dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Demonic composition Source: https://en.wikipedia.org/wiki/Demonic_composition?oldid=633771432 Contributors: Michael Hardy,David Eppstein, LokiClock, Classicalecon, AnomieBOT and Anonymous: 2

• Dense order Source: https://en.wikipedia.org/wiki/Dense_order?oldid=635100176 Contributors: EmilJ, Physicistjedi, MarSch, MichaelSlone, SmackBot, Imz, Melchoir, Turms, JAnDbot, VolkovBot, TXiKiBoT, Palnot, Addbot, ב ,.דניאל Pcap, Erik9bot, ZéroBot, HelpfulPixie Bot, Qetuth, Brirush and Anonymous: 6

• Dependence relation Source: https://en.wikipedia.org/wiki/Dependence_relation?oldid=668236798Contributors: Michael Hardy, CharlesMatthews, Jitse Niesen, Josh Parris, Wavelength, Robbjedi, Keegan, Geometry guy and 777sms

• Dependency relation Source: https://en.wikipedia.org/wiki/Dependency_relation?oldid=668236930Contributors: Michael Hardy, WilliamM. Connolley, GPHemsley, Robbot, Wizzy, Goochelaar, Linas, Jsnx, SmackBot, Chris the speller, NickPenguin, Mukake, David Eppstein,Homei, Classicalecon, Addbot, Yobot, WikitanvirBot, Mark viking, W. P. Uzer, Christian Nassif-Haynes, JMP EAX and Anonymous: 4

• Directed set Source: https://en.wikipedia.org/wiki/Directed_set?oldid=663876888Contributors: AxelBoldt, The Anome, SimonP, Patrick,Michael Hardy, AugPi, Nikai, Revolver, Dfeuer, Dysprosia, Tobias Bergemann, Giftlite, Markus Krötzsch, Smimram, Paul August,Varuna, Msh210, Eric Kvaalen, SteinbDJ, Linas, Dionyziz, Salix alba, Margosbot~enwiki, Hairy Dude, Zwobot, Futanari, Reedy, Fitch,Mhss, Vaughan Pratt, CBM, Blaisorblade, Larvy, JoergenB, CommonsDelinker, LokiClock, Don4of4, AlleborgoBot, SieBot, Tom Le-inster, He7d3r, Beroal, Palnot, Plmday, Legobot, Luckas-bot, Obersachsebot, Yaddie, ZéroBot, Haraldbre, Helpful Pixie Bot, Freeze S,Jochen Burghardt, Zoydb, Austrartsua, Some1Redirects4You and Anonymous: 25

• Equality (mathematics) Source: https://en.wikipedia.org/wiki/Equality_(mathematics)?oldid=672048328 Contributors: Toby Bartels,Patrick, Michael Hardy, TakuyaMurata, Looxix~enwiki, Pizza Puzzle, Charles Matthews, Dysprosia, WhisperToMe, Banno, Robbot,RedWolf, Lowellian, Tobias Bergemann, Alan Liefting, Giftlite, Christopher Parham, Recentchanges, Michael Devore, DefLog~enwiki,Chowbok, Smiller933, Shahab, AlexG, Wrp103, Plugwash, Rgdboer, Spoon!, Iltseng, PWilkinson, MPerel, Jumbuck, Msh210, Hu,Japanese Searobin, Simetrical, Linas, MattGiuca, Isnow, Wbeek, Qwertyus, Island, Scottkeir, Jshadias, Pasky, FlaBot, VKokielov, Mar-gosbot~enwiki, Fresheneesz, Chobot, DVdm, Gwernol, Laurentius, Piet Delport, Pnrj, TransUtopian, Reyk, Tinlv7, SmackBot, RDBury,Incnis Mrsi, Melchoir, Blue520, Josephprymak, BiT, Bluebot, Nbarth, Jdthood, Jon Awbrey, Lambiam, Attys, Loadmaster, Mets501,Tauʻolunga, CBM, Sdorrance, Simeon, Gregbard, Cydebot, Benzi455, Blaisorblade, Xantharius, Uv~enwiki, Cj67, Dugwiki, Anti-VandalBot, Malcolm, JAnDbot, Thenub314, Edward321, R'n'B, Anonymous Dissident, PaulTanenbaum, UnitedStatesian, Enigmaman,Vikrant42, Tachikoma’s All Memory, Flyer22, Ctxppc, ClueBot, BodhisattvaBot, SilvonenBot, Addbot, Debresser, Numbo3-bot, Apteva,Legobot, Luckas-bot, Yobot, TaBOT-zerem, Amirobot, Pcap, KamikazeBot, Ningauble, Bryan.burgers, MassimoAr, AnomieBOT, King-pin13, Citation bot, Capricorn42, Kevfest08, NOrbeck, VladimirReshetnikov, Der Falke, FrescoBot, Tkuvho, AmphBot, RedBot, Jauhienij,TobeBot, Belovedeagle, Vrenator, CobraBot, Duoduoduo, Ebe123, ZéroBot, Sungzungkim, D.Lazard, ClueBot NG, Iiii I I I, Wcherowi,Faus, ChrisGualtieri, Brirush, DialaceStarvy, Monkbot, Lizard Pancakes123456789012345678901234567890, Gmalaven, This is a mo-bile phone and Anonymous: 75

• Equipollence (geometry) Source: https://en.wikipedia.org/wiki/Equipollence_(geometry)?oldid=674436874Contributors: Michael Hardy,Mdob, Rgdboer, Siddhant, Sadads, Addbot, Omnipaedista, Erik9bot, J.Victor, Specs112, Makhokh and Brad7777

• Equivalence class Source: https://en.wikipedia.org/wiki/Equivalence_class?oldid=667055947Contributors: AxelBoldt, Zundark, Patrick,Michael Hardy, Wshun, Salsa Shark, Revolver, Charles Matthews, Dysprosia, Wolfgang Kufner, Greenrd, Hyacinth, Psychonaut, Naddy,GreatWhiteNortherner, Tobias Bergemann, Giftlite, WiseWoman, Lethe, Fropuff, Fuzzy Logic, Noisy, Tibbetts, Liuyao, Rgdboer,Msh210, MattGiuca, Graham87, Salix alba, Mike Segal, Jameshfisher, Laurentius, Hede2000, Arthur Rubin, Lunch, SmackBot, Mhss,Nbarth, Javalenok, Lhf, Mets501, Andrew Delong, Egriffin, Magioladitis, David Eppstein, VolkovBot, LokiClock, Rjgodoy, Quietbri-tishjim, Dogah, Henry Delforn (old), Sjn28, Classicalecon, Watchduck, Kausikghatak, Addbot, WikiDreamer Bot, Calle, Rinke 80,Erik9bot, HJ Mitchell, WillNess, Igor Yalovecky, Quondum, D.Lazard, Herebo, Wcherowi, Rpglover64, ChrisGualtieri, Brirush, Markviking, A4b3c2d1e0f, Riddleh, Verdana Bold, Addoergosum and Anonymous: 45

• Equivalence relation Source: https://en.wikipedia.org/wiki/Equivalence_relation?oldid=672050542 Contributors: AxelBoldt, Zundark,Toby Bartels, PierreAbbat, Ryguasu, Stevertigo, Patrick, Michael Hardy, Wshun, Dominus, TakuyaMurata, William M. Connolley, AugPi,Silverfish, Ideyal, Revolver, Charles Matthews, Dysprosia, Hyacinth, Fibonacci, Phys, McKay, GPHemsley, Robbot, Fredrik, Romanm,COGDEN, Ashley Y, Bkell, Tobias Bergemann, Tosha, Giftlite, Arved, ShaunMacPherson, Lethe, Herbee, Fropuff, LiDaobing, AlexG,Paul August, Elwikipedista~enwiki, FirstPrinciples, Rgdboer, Spearhead, Smalljim, SpeedyGonsales, Obradovic Goran, Haham hanuka,Kierano, Msh210, Keenan Pepper, PAR, Jopxton, Oleg Alexandrov, Linas, MFH, BD2412, Salix alba, Alexb@cut-the-knot.com, Mark

118 CHAPTER 28. PROPOSITIONAL FUNCTION

J, Epitome83, Chobot, Algebraist, Roboto de Ajvol, YurikBot, Wavelength, RussBot, Nils Grimsmo, BOT-Superzerocool, Googl, Larry-LACa, Arthur Rubin, Pred, Cjfsyntropy, Draicone, RonnieBrown, SmackBot, Adam majewski, Melchoir, Stifle, Srnec, Gilliam, Kurykh,Concerned cynic, Foxjwill, Vanished User 0001, Michael Ross, Jon Awbrey, Jim.belk, Feraudyh, CredoFromStart, Michael Kinyon,JHunterJ, Vanished user 8ij3r8jwefi, Mets501, Rschwieb, Captain Wacky, JForget, CRGreathouse, CBM, 345Kai, Gregbard, Doctor-matt, PepijnvdG, Tawkerbot4, Xantharius, Hanche, BetacommandBot, Thijs!bot, Egriffin, Rlupsa, WilliamH, Rnealh, Salgueiro~enwiki,JAnDbot, Thenub314, Magioladitis, VoABot II, JamesBWatson, MetsBot, Robin S, Philippe.beaudoin, Pekaje, Pomte, Interwal, Cpiral,GaborLajos, Policron, Taifunbrowser, Idioma-bot, Station1, Davehi1, Billinghurst, Geanixx, AlleborgoBot, SieBot, BotMultichill, This,that and the other, Henry Delforn (old), Aspects, OKBot, Bulkroosh, C1wang, Classicalecon, Wmli, Kclchan, Watchduck, Hans Adler,Qwfp, Cdegremo, Palnot, XLinkBot, Gerhardvalentin, Libcub, LaaknorBot, CarsracBot, Dyaa, Legobot, Luckas-bot, Yobot, Ht686rg90,Gyro Copter, Andy.melnikov, ArthurBot, Xqbot, GrouchoBot, Lenore, RibotBOT, Antares5245, Sokbot3000, Anthonystevens2, ARan-domNicole, Tkuvho, SpaceFlight89, TobeBot, Miracle Pen, EmausBot, ReneGMata, AvicBot, Vanished user fois8fhow3iqf9hsrlgkjw4tus,TyA, Donner60, Gottlob Gödel, ClueBot NG, Bethre, Helpful Pixie Bot, Mark Arsten, ChrisGualtieri, Rectipaedia, YFdyh-bot, Noix07,Adammwagner, Damonamc and Anonymous: 108

• Euclidean relation Source: https://en.wikipedia.org/wiki/Euclidean_relation?oldid=557928957 Contributors: Toby Bartels, Giftlite,EmilJ, PAR, Salix alba, Lhf, Turms, Gregbard, Egriffin, David Eppstein, Robertgreer, Cdegremo, Yangtseyangtse, Helpful Pixie Botand Anonymous: 4

• Exceptional isomorphism Source: https://en.wikipedia.org/wiki/Exceptional_isomorphism?oldid=629067192 Contributors: MichaelHardy, Charles Matthews, Tobias Bergemann, Rjwilmsi, Koavf, Wavelength, SmackBot, Nbarth, Tamfang, David Eppstein, Citation bot,Twri, Jamontaldi, Teddyktchan and Anonymous: 5

• Fiber (mathematics) Source: https://en.wikipedia.org/wiki/Fiber_(mathematics)?oldid=638573433Contributors: Chinju, Charles Matthews,Oleg Alexandrov, Christopher Thomas, MarSch, LkNsngth, Jon Awbrey, Krasnoludek, JRSpriggs, CBM, Kilva, OrenBochman, Camrn86,LokiClock, Dmcq, JP.Martin-Flatin, Addbot, Ptbotgourou, Ciphers, Erik9bot, Artem M. Pelenitsyn, ZéroBot, Beaumont877, Qetuth,SillyBunnies and Anonymous: 8

• Finitary relation Source: https://en.wikipedia.org/wiki/Finitary_relation?oldid=674490128 Contributors: Damian Yerrick, AxelBoldt,The Anome, Tarquin, Jan Hidders, Patrick, Michael Hardy, Wshun, Kku, Ellywa, Andres, Charles Matthews, Dcoetzee, Hyacinth, Rob-bot, Romanm, MathMartin, Tobias Bergemann, Alan Liefting, Marc Venot, Giftlite, Almit39, Zfr, Starx, PhotoBox, Erc, ArnoldRein-hold, Paul August, Elwikipedista~enwiki, Randall Holmes, Obradovic Goran, Oleg Alexandrov, Woohookitty, Mangojuice, MichielHelvensteijn, Isnow, Qwertyus, Dpr, MarSch, Salix alba, Oblivious, Mathbot, Jrtayloriv, Chobot, YurikBot, Hairy Dude, Dmharvey,RussBot, Muu-karhu, Zwobot, Bota47, Arthur Rubin, Reyk, Netrapt, Claygate, JoanneB, Pred, GrinBot~enwiki, SmackBot, Mmernex,Unyoyega, Nbarth, DHN-bot~enwiki, Tinctorius, Jon Awbrey, Henning Makholm, Lambiam, Dfass, Newone, Aeons, CRGreathouse,Gregbard, King Bee, Kilva, Escarbot, Salgueiro~enwiki, Nosbig, JAnDbot, .anacondabot, Tarif Ezaz, VoABot II, Jonny Cache, Der-Hexer, Mike.lifeguard, And Dedicated To, Aervanath, VolkovBot, Rponamgi, The Tetrast, Mscman513, GirasoleDE, Newbyguesses,SieBot, Phe-bot, Paolo.dL, Siorale, Skeptical scientist, Sheez Louise, Mild Bill Hiccup, DragonBot, Cenarium, Palnot, Cat Dancer WS,Kal-El-Bot, Addbot, MrOllie, ChenzwBot, Ariel Black, SpBot, Yobot, Ptbotgourou, Bgttgb, QueenCake, Dinnertimeok, AnomieBOT,JRB-Europe, Xqbot, Nishantjr, Howard McCay, Paine Ellsworth, Throw it in the Fire, RandomDSdevel, Miracle Pen, Straightontillmorn-ing, ZéroBot, Cackleberry Airman, Paulmiko, Tijfo098, Mister Stan, Deer*lake, Frietjes, ChrisGualtieri, Fuebar, Brirush, Mark viking,Andrei Petre, KasparBot, Some1Redirects4You and Anonymous: 52

• Foundational relation Source: https://en.wikipedia.org/wiki/Foundational_relation?oldid=659203908 Contributors: Michael Hardy,BiH and IkamusumeFan

• Hypostatic abstraction Source: https://en.wikipedia.org/wiki/Hypostatic_abstraction?oldid=582042811Contributors: Bevo, MisfitToys,El C, Diego Moya, Versageek, Jeffrey O. Gustafson, Magister Mathematicae, DoubleBlue, TeaDrinker, Brandmeister (old), Closedmouth,C.Fred, Rajah9, JasonMR, Jon Awbrey, Inhahe, JzG, Slakr, CBM, Gogo Dodo, Hut 8.5, Brigit Zilwaukee, Yolanda Zilwaukee, Karrade,Mike V, The Tetrast, Rjd0060, Wolf of the Steppes, Doubtentry, Icharus Ixion, Hans Adler, Buchanan’s Navy Sec, Mr. Peabody’s Boy,Overstay, Marsboat, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, NavyPierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, AnomieBOT, Paine Ellsworth,Gamewizard71, PhnomPencil and Anonymous: 3

• Idempotence Source: https://en.wikipedia.org/wiki/Idempotence?oldid=675763490 Contributors: Damian Yerrick, AxelBoldt, Zundark,Taw, Patrick, Chas zzz brown, Michael Hardy, Ahoerstemeier, Stevan White, Andres, Zarius, Revolver, Charles Matthews, Dysprosia, JitseNiesen, Glimz~enwiki, Hyacinth, Robbot, Mattblack82, Altenmann, Tobias Bergemann, Giftlite, Fropuff, Mboverload, DefLog~enwiki,Joseph Myers, Tothebarricades.tk, Andreas Kaufmann, Gcanyon, Mormegil, Rgdboer, Kwamikagami, Nigelj, Sleske, Obradovic Goran,Mdd, Abdulqabiz, Dirac1933, Bookandcoffee, Simetrical, Igny, MFH, Grammarbot, MarkHudson, Tokigun, FlaBot, Mathbot, Vonkje,DVdm, Roboto de Ajvol, Angus Lepper, RobotE, Hairy Dude, Michael Slone, Gaius Cornelius, FuzzyBSc, Aeusoes1, Deskana, Trova-tore, Sangwine, Sfnhltb, Crasshopper, Tomisti, Kompik, Evilbu, SmackBot, Elronxenu, Octahedron80, Nbarth, Syberghost, Cybercobra,Lambiam, Robofish, Loadmaster, JHunterJ, MedeaMelana, Rschwieb, Alex Selby, Dreftymac, Happy-melon, CRGreathouse, CmdrObot,Gregbard, Julian Mendez, Thijs!bot, Kilva, Konradek, DB Durham NC, RichardVeryard, RobHar, Gioto, Ouc, JAnDbot, Deflective, Ran-dal Oulton, Danculley, Vanish2, Albmont, Brewhaha@edmc.net, David Eppstein, Emw, Cander0000, Stolsvik, Catskineater, Robin S,Gwern, Bostonvaulter, TomyDuby, Krishnachandranvn, Policron, JohnBlackburne, TXiKiBoT, Una Smith, Broadbot, Jamelan, SieBot,LungZeno, Roujo, Svick, Stjarnblom, Niceguyedc, DragonBot, Alexbot, He7d3r, Herbert1000, Rswarbrick, Qwfp, Gniemeyer, Addbot,MrOllie, Download, Legobot, Luckas-bot, Yobot, AnomieBOT, Götz, JackieBot, NOrbeck, Kaoru Itou, RandomDSdevel, Pinethicket,Serols, Ms7821, Duoduoduo, Timtempleton, Peoplemerge, Wham Bam Rock II, Quondum, Dumbier, Deltahedron, Jochen Burghardt,00prometheus, JaconaFrere, Acominym, Loraof and Anonymous: 83

• Idempotent relation Source: https://en.wikipedia.org/wiki/Idempotent_relation?oldid=666347335Contributors: Michael Hardy, Bearcat,Deltahedron, Jochen Burghardt and Flokam

• Intransitivity Source: https://en.wikipedia.org/wiki/Intransitivity?oldid=676080474 Contributors: Michael Hardy, Booyabazooka, Rp,6birc, Radicalsubversiv, Andres, Charles Matthews, Ruakh, Giftlite, Andris, Quickwik, D6, Xezbeth, Paul August, Jnestorius, Spoon!,Polluks, SmackBot, Melchoir, Mauls, MisterHand, Lambiam, Iamagloworm, Dinkumator, CRGreathouse, CmdrObot, CBM, Thomas-meeks, Ael 2, Thijs!bot, VoABot II, Cnilep, Ddxc, Anchor Link Bot, PixelBot, DumZiBoT, Pa68, Addbot, Forich, WissensDürster,Undsoweiter, HRoestBot, ChronoKinetic, RjwilmsiBot, Chharvey, Diakov and Anonymous: 9

• Inverse relation Source: https://en.wikipedia.org/wiki/Inverse_relation?oldid=657149498 Contributors: Patrick, Charles Matthews, Al-tenmann, Tobias Bergemann, Giftlite, Kaldari, Spiffy sperry, El C, Versageek, Jeffrey O. Gustafson, MFH, YurikBot, Paul Erik, Knowled-geOfSelf, C.Fred, Nbarth, Faaaa, Nixeagle, Jon Awbrey, Mearnhardtfan, Coredesat, Slakr, DBooth, CBM, Thomasmeeks, Gogo Dodo,

28.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 119

Mojo Hand, Hut 8.5, MetsBot, David Eppstein, Real World Apple, Ars Tottle, VolkovBot, Iamthedeus, Maelgwnbot, Classicalecon,Rjd0060, Overstay, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, FlowerMound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, Myst-Bot, Omphaloskeptor, Addbot, Quercus solaris, Luckas-bot, Yobot, Pcap, Materialscientist, MathHisSci, ویکی ,علی John of Reading,SporkBot, Mark viking, JMP EAX and Anonymous: 12

• Inverse trigonometric functions Source: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions?oldid=676298808 Contribu-tors: XJaM, Patrick, Michael Hardy, Stevenj, Dysprosia, Fibonacci, Robbot, Tobias Bergemann, Giftlite, Anville, SURIV, Daniel,levine,Pmanderson, Abdull, Discospinster, Osrevad, Zenohockey, Army1987, Alansohn, Anthony Appleyard, Wtmitchell, StradivariusTV, Ar-mando, Gerbrant, Emallove, R.e.b., Kri, Glenn L, Salvatore Ingala, Chobot, Visor, DVdm, Algebraist, YurikBot, Wavelength, Sceptre,Hede2000, KSmrq, Grafen, Int 80h, NorsemanII, Bamse, RDBury, Maksim-e~enwiki, Thelukeeffect, Eskimbot, Mhss, Mirokado, JC-Santos, PrimeHunter, Deathanatos, V1adis1av, Saippuakauppias, Lambiam, Eridani, Ian Vaughan, ChaoticLlama, CapitalR, JRSpriggs,Conrad.Irwin, HenningThielemann, Fommil, Rian.sanderson, Palmtree3000, Zalgo, Thijs!bot, Spikedmilk, Nonagonal Spider, EdJohn-ston, Luna Santin, Hannes Eder, JAnDbot, Ricardo sandoval, Jetstreamer, JNW, Albmont, Gamkiller, JoergenB, Ac44ck, Gwern,Isamil, Mythealias, Pomte, Knorlin, TungstenWolfram, Hennessey, Patrick, Bobianite, BentonMiller, Sigmundur, DavidCBryant, AlanU. Kennington, VolkovBot, Indubitably, LokiClock, Justtysen, VasilievVV, Riku92mr, Anonymous Dissident, Corvus coronoides, Dmcq,Vertciel, Logan, CagedKiller360, Aly89, AlanUS, ClueBot, JoeHillen, Bender2k14, Cenarium, Leandropls, Kiensvay, Nikhilkrgvr,Aaron north, Dthomsen8, DaL33T, Addbot, Fgnievinski, Iceblock, Zarcadia, Sleepaholic, Jasper Deng, Zorrobot, Luckas-bot, Yobot,Tohd8BohaithuGh1, Ptbotgourou, TaBOT-zerem, AnomieBOT, JackieBot, Nickweedon, Geek1337~enwiki, Diego Queiroz, Txebixev,St.nerol, Hdullin, GrouchoBot, Uniwersalista, SassoBot, Prari, Nixphoeni, D'ohBot, Kusluj, Emjayeff, Number Googol, Adammer-linsmith, TjBot, Jowa fan, EmausBot, ModWilson, Velowiki, X-4-V-I, Wham Bam Rock II, ZéroBot, Michael.YX.Wu, Isaac Euler,Tolly4bolly, Jay-Sebastos, Colin.campbell.27, Maschen, ChuispastonBot, Rmashhadi, ClueBot NG, Hdreuter, Helpful Pixie Bot, KLBot2,Vagobot, Crh23, YatharthROCK, StevinSimon, Tfr000, Modalanalytiker, Pratyya Ghosh, Ahmed Magdy Hosny, Brirush, Yardimsever,Wamiq, Jerming, Blackbombchu, Pqnlrn, DTL LAPOS, De Riban5, Monkbot, Arsenal CR7, Cdserio99 and Anonymous: 168

• Near sets Source: https://en.wikipedia.org/wiki/Near_sets?oldid=662493389 Contributors: Michael Hardy, Bearcat, Gandalf61, Giftlite,Rjwilmsi, Algebraist, SmackBot, MartinPoulter, NSH001, Sebras, Salih, Philip Trueman, JL-Bot, SchreiberBike, Tassedethe, Jarble,Alvin Seville, Erik9bot, FrescoBot, NSH002, NearSetAccount, AManWithNoPlan, Frietjes, Helpful Pixie Bot, BG19bot, Christopher-JamesHenry and Anonymous: 2

• Partial equivalence relation Source: https://en.wikipedia.org/wiki/Partial_equivalence_relation?oldid=672263741 Contributors: Zun-dark, Charles Matthews, Malcohol, Smimram, Rgdboer, Kierano, Cjoev, Alex Bakharev, AndrewWTaylor, SmackBot, Bluebot, Alansmithee, J. Finkelstein, Mets501, David Eppstein, R'n'B, Functor salad, Palnot, AnomieBOT, Erik9bot, Helpful Pixie Bot, Some1Redirects4Youand Anonymous: 3

• Partial function Source: https://en.wikipedia.org/wiki/Partial_function?oldid=670578722 Contributors: AxelBoldt, Jan Hidders, Ede-maine, Michael Hardy, Pizza Puzzle, Charles Matthews, Timwi, Zoicon5, Altenmann, MathMartin, Henrygb, Tobias Bergemann, Tosha,Connelly, Giftlite, Lethe, Waltpohl, DRE, Paul August, Rgdboer, Pearle, Sligocki, Schapel, Artur adib, LunaticFringe, Oleg Alexandrov,^demon, Mpatel, MFH, Salix alba, Dougluce, FlaBot, VKokielov, Jameshfisher, Vonkje, Hairy Dude, Grubber, Trovatore, Kooky, ArthurRubin, Crystallina, Jbalint, SmackBot, Incnis Mrsi, Reedy, Ppntori, Chris the speller, Bluebot, Tsca.bot, TheArchivist, Cícero, Wvbailey,Etatoby, CBM, Blaisorblade, Dvandersluis, Dricherby, Albmont, Timlevin, Ravikaushik, Joshua Issac, Dmcq, Paolo.dL, , Classicale-con, ClueBot, Gulmammad, Maheshexp, El bot de la dieta, Lightbot, Legobot, Luckas-bot, Yobot, Pcap, AnomieBOT, Isheden, Brook-swift, NoJr0xx, Nicolas Perrault III, SkpVwls, Unbitwise, AvicAWB, Quondum, This lousy T-shirt, YannickN, Jergas, Tomtom2357, SJDefender, JMP EAX, Shaelja and Anonymous: 41

• Partially ordered set Source: https://en.wikipedia.org/wiki/Partially_ordered_set?oldid=672760099 Contributors: Bryan Derksen, Zun-dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus,Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De-fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, OlegAlexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, JosephSilverman, Sanguinity,Modify, RDBury, Incnis Mrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki,Mike Fikes, A. Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE,Tbleher, A3nm, David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielE-merson, TXiKiBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush,Megaloxantha, Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead,Watchduck, ComputerGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz,Legobot, Kilom691, AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ri-cardo Ferreira de Oliveira, Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot,Chharvey, The man who was Friday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jak-shap, Paolo Lipparini, ElphiBot, Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987,Victor Lesyk, Some1Redirects4You and Anonymous: 79

• Preorder Source: https://en.wikipedia.org/wiki/Preorder?oldid=677053946 Contributors: AxelBoldt, Patrick, Repton, Delirium, An-dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krötzsch, Lethe, Fropuff, Vadmium, De-fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba,Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack-Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, Magioladitis,David Eppstein, Jwuthe2, PaulTanenbaum, SieBot, Thehotelambush, MenoBot, Functor salad, He7d3r, Sun Creator, Cenarium, 1ForThe-Money, Palnot, Мыша, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, ComputScientist, BrideOfKrip-kenstein, Notedgrant, WikitanvirBot, Lclem, Dfabera, SporkBot, Paolo Lipparini, RichardMills65, Khazar2, Lerutit, Jochen Burghardt,Reatlas, Damonamc and Anonymous: 26

• Prewellordering Source: https://en.wikipedia.org/wiki/Prewellordering?oldid=488790287Contributors: Zundark, Patrick, Charles Matthews,EmilJ, Oleg Alexandrov, NickBush24, Trovatore, Tetracube, That Guy, From That Show!, SmackBot, Nbarth, Henry Delforn (old), Pal-not, Citation bot and Anonymous: 3

• Propositional function Source: https://en.wikipedia.org/wiki/Propositional_function?oldid=668907292 Contributors: Michael Hardy,TakuyaMurata, Jason Quinn, Rgdboer, Linas, Thekohser, Arthur Rubin, Gregbard, Uncle uncle uncle, Izno, Addbot, Montblanc1988,Orenburg1, Helpful Pixie Bot, YFdyh-bot, JYBot and Anonymous: 3

120 CHAPTER 28. PROPOSITIONAL FUNCTION

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122 CHAPTER 28. PROPOSITIONAL FUNCTION

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