Mathematical relations acFrom Wikipedia, the free encyclopedia

Contents

1 Accessibility relation 11.1 Description of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Review of (Propositional) Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Four Types of the 'Accessibility Relation' in Formal Semantics . . . . . . . . . . . . . . . . . 41.4 Philosophical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Computer Science Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Allegory (category theory) 72.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Regular categories and allegories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Allegories of relations in regular categories . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Maps in allegories, and tabulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Unital allegories and regular categories of maps . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 More sophisticated kinds of allegory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Alternatization 93.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Alternating bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Alternating bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3 Alternating multilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.4 Alternatization of a bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Ancestral relation 124.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Relationship to transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Antisymmetric relation 145.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Asymmetric relation 166.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Better-quasi-ordering 187.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 Simpson’s alternative definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4 Major theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Bidirectional transformation 208.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Examples of implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9 Bijection 229.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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9.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Bijection, injection and surjection 2810.1 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.2 Surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.3 Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.5.1 Injective and surjective (bijective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.2 Injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.3 Non-injective and surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.4 Non-injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.7 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11 Binary relation 3311.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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12 Cointerpretability 4312.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13 Commutative property 4413.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413.2 Mathematical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 4613.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 47

13.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

13.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Comparability 5314.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2 Comparability graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15 Composition of relations 5515.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.3 Join: another form of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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15.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16 Congruence relation 5716.1 Basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.3 Relation with homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4 Congruences of groups, and normal subgroups and ideals . . . . . . . . . . . . . . . . . . . . . . . 58

16.4.1 Ideals of rings and the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.5 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

17 Contour set 6017.1 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

17.1.1 Contour sets of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

17.2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2.2 Economic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

17.3 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6317.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6317.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

18 Coreflexive relation 6418.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

19 Quasi-commutative property 6519.1 Applied to matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2 Applied to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 67

19.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 1

Accessibility relation

In modal logic, an accessibility relation is a binary relation, written as R between possible worlds.

1.1 Description of Terms

A 'statement' in logic refers to a sentence (with a subject, predicate, and verb) that can be true or false. So, 'Theroom is cold' is a statement because it contains a subject, predicate and verb, and it can be true that 'the room is cold'or false that 'the room is cold.'Generally, commands, beliefs and sentences about probabilities aren't judged as true or false. 'Inhale and exhale' istherefore not a statement in logic because it is a command and cannot be true or false, although a person can obeyor refuse that command. 'I believe I can fly or I can't fly' isn't taken as a statement of truth or falsity, because beliefsdon't say anything about the truth or falsity of the parts of the entire 'and' or 'or' statement and therefore the entire'and' or 'or' statement.A 'possible world' is any possible situation. In every case, a 'possible world' is contrasted with an actual situation.Earth one minute from now is a 'possible world.' The earth as it actually is also a 'possible world.' Hence the oddityof and controversy in contrasting a 'possible' world with an 'actual world' (earth is necessarily possible). In logic,'worlds’ are described as a non-empty set, where the set could consist of anything, depending on what the statementsays.'Modal Logic' is a description of the reasoning in making statements about 'possibility' or 'necessity.' 'It is possiblethat it rains tomorrow' is a statement in modal logic, because it is a statement about possibility. 'It is necessary thatit rains tomorrow' also counts as a statement in modal logic, because it is a statement about 'necessity.' There are atleast six logical axioms or principles that show what people mean whenever they make statements about 'necessity' or'possibility' (described below). For a detailed explanation on modal logic, see here.As described in greater detail below:Necessarily p means that p is true at every 'possible world' w such that R(w∗, w).

Possibly p means that p is true at some possible world w such that R(w∗, w) .'Truth-Value' is whether a statement is true or false. Whether or not a statement is true, in turn, depends on themeanings of words, laws of logic, or experience (observation, hearing, etc.).'Formal Semantics’ refers to the meaning of statements written in symbols. The sentence (□p ∨ □q) → □(p ∨ q)

, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by thesymbol R .

The 'accessibility relation' is a relationship between two 'possible worlds.' More preciselyplease clarify definition, the'accessibility relation' is the idea that modal statements, like 'it’s possible that it rains tomorrow,' may not take thesame truth-value in all 'possible worlds.' On earth, the statement could be true or false. By contrast, in a planet wherewater is non-existent, this statement will always be false.Due to the difficulty in judging if a modal statement is true in every 'possible world,' logicians have derived certainaxioms or principles that show on what basis any statement is true in any 'possible world.' These axioms describing

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2 CHAPTER 1. ACCESSIBILITY RELATION

the relationship between 'possible worlds’ is the 'accessibility relation' in detail.Put another way, these modal axioms describe in detail the 'accessibility relation,' R between two 'worlds.' Thatrelation,R symbolizes that from any given 'possible world' some other 'possible worlds’ may be accessible, and othersmay not be.The 'accessibility relation' has important uses in both the formal/theoretical aspects of modal logic (theories about'modal logic'). It also has applications to a variety of disciplines including epistemology (theories about how peopleknow something is true or false), metaphysics (theories about reality), value theory (theories about morality andethics), and computer science (theories about programmatic manipulation of data).

1.2 Basic Review of (Propositional) Modal Logic

The reasoning behind the 'accessibility relation' uses the basics of 'propositional modal logic' (see modal logic for adetailed discussion). 'Propositional modal logic' is traditional propositional logic with the addition of two key unaryoperators:□ symbolizes the phrase 'It is necessary that...'♢ symbolizes the phrase 'It is possible that...'These operators can be attached to a single sentence to form a new compound sentence.For example, □ can be attached to a sentence such as 'I walk outside.' The new sentence would look like: □ 'I walkoutside.' The entire new sentence would therefore read: 'It is necessary that I walk outside.'But the symbol A can be used to stand for any sentence instead of writing out entire sentences. So any sentence suchas 'I walk outside' or 'I walk outside and I look around' are symbolized by A .Thus for any sentence A (simple or compound), the compound sentences □A and ♢A can be formed. Sentencessuch as 'It is necessary that I walk outside' or 'It is possible that I walk outside' would therefore look like: □ A ♢A .However, the symbols p , q can also be used to stand for any statement of our language. For example, p can standfor 'I walk outside' or 'I walk outside and I look around.' The sentence 'It is necessary that I walk outside' would looklike: □ q . The sentence 'It is possible that I walk outside' would look like: ♢ q .Six Basic Axioms of Modal Logic:There are at least six basic axioms or principles of almost all modal logics or steps in thinking/reasoning. The firsttwo hold in all regular modal logics, and the last holds in all normal modal logics.1st Modal Axiom:

• □p↔ ¬♢¬p (Duality)

The double arrow stands symbolizes 'if and only if,' 'necessary and sufficient' conditions. A 'necessary' condition issomething that must be the case for something else. Being literate, for instance, is a 'necessary' condition for readingabout the 'accessibility relation.' A 'sufficient condition' a condition that is good enough for something else. Beingliterate, for instance, is a 'sufficient' condition for learning about the accessibility relation.' In other words, it’s goodenough to be literate in order to learn about the 'accessibility relation,' however it may not be 'necessary' because therelation could be learned in different ways (like through speech). Aside from 'necessary and sufficient,' the doublearrow represents equivalence between the meaning of two statements, the statement to the left and the statement tothe right of the double arrow.The half square symbols before the diamond and p symbol in the sentence ' □p ↔ ¬♢¬p ' stand for 'it is not thecase, or 'not.'The p symbol stands for any statement such as 'I walk outside.' Therefore it could also stand for 'The apple is Red.'Example 1:The first principle says that any statement involving 'necessity' on the left side of the double arrow is equivalent to thestatement about the negation of 'possibility' on the right.So using the symbols and their meaning, the first modal axiom, □p ↔ ¬♢¬p could stand for: 'It’s necessary that Iwalk outside if and only if it’s not possible that it is not the case that I walk outside.'

1.2. BASIC REVIEW OF (PROPOSITIONAL) MODAL LOGIC 3

And when I say that 'It’s necessary that I walk outside,' this is the same as saying that 'It’s not possible that it is notthe case that I walk outside.' Furthermore, when I say that 'It’s not possible that it is not the case that I walk outside,'this is the same as saying that 'It’s necessary that I walk outside.'Example 2:p stands for 'The apple is red.'So using the symbols and their meaning described above, the first modal axiom, □p ↔ ¬♢¬p could stand for: 'It’snecessary that the apple is red if and only if it’s not possible that it is not the case that the apple is red.'And when I say that 'It’s necessary that the apple is red,' this is the same as saying that 'It’s not possible that it is notthe case that the apple is red.' Furthermore, when I say that 'It’s not possible that it is not the case that the apple isred,' this is the same as saying that 'It’s necessary that the apple is red.'Second Modal Axiom:

• ♢p↔ ¬□¬p (Duality)

Example 1:The second principle says that any statement involving 'possibility' on the left side of the double arrow is the same asthe statement about the negation of 'necessity' on the right.p stands for 'Spring has not arrived.'Using the symbols and their meaning described above, the second modal axiom, ♢p ↔ ¬□¬p could stand for: 'It’spossible that Spring has not arrived if and only if it is not the case that it is necessary that it is not the case that Springhas not arrived.'Essentially, the second axiom says that any statement about possibility called 'X' is the same as a negation or denialof a different statement about necessity 'Y.' The statement about necessity shows the denial of the same originalstatement 'X.'The other axioms can be read and interpreted in the same way, by substituting letters p for any statement and followingthe reasoning. Brackets in a symbolized sentence mean that anything inside the brackets counts as a whole sentence.Any symbol before the brackets therefore applies to the sentence as a whole, not just the letters or an individualsentence.An arrow stands for “then” where the left statement before the arrow is the “if” of the entire sentence.Other Modal Axioms:* □(p ∧ q) ↔ (□p ∧□q)

* (□p ∨□q) → □(p ∨ q)

* □(p → q) → (□p → □q) (Kripke property)Most of the other axioms concerning the modal operators are controversial and not widely agreed upon. Here are themost commonly used and discussed of these:

(T) □p → p

(4) □p → □□p

(5) ♢p → □♢p

(B) p → □♢p

Here, "(T)","(4)","(5)", and "(B)" represent the traditional names of these axioms (or principles).According to the traditional 'possible worlds’ semantics of modal logic, the compound sentences that are formed outof the modal operators are to be interpreted in terms of quantification over possible worlds, subject to the relationof accessibility. A sentence like (□p ∨ □q) → □(p ∨ q) is to be interpreted as true or false in all or some 'possibleworlds.' In turn, the grounds on which the sentence is true (symmetry, transitive property, etc.) in all 'possible worlds’is the 'accessibility relation.'

4 CHAPTER 1. ACCESSIBILITY RELATION

The relation of accessibility can now be defined as an (uninterpreted) relation R(w1, w2) that holds between 'possibleworlds’ w1 and w2 only when w2 is accessible from w1 .Additionally, to make things more specific, any formula, axiom like (□p ∨□q) → □(p ∨ q) can be translated into aformula of first-order logic through standard translation. That first-order logic formula or sentence makes the meaningof the boxes and diamonds in modal logic explicit.

1.3 The Four Types of the 'Accessibility Relation' in Formal Semantics

'Formal semantics’ studies the meaning of statements written in symbols. The sentence (□p ∨ □q) → □(p ∨ q) ,for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by thesymbol R , where R takes the form of the 'necessity relation' described below.So, the 'accessibility relation,' R can take on at least four forms, that is, the 'accessibility relation' can be describedin at least four ways.Each type is either about 'possibility' or 'necessity' where 'possibility' and 'necessity' is defined as:

• (TS) Necessarily p means that p is true at every 'possible world' w such that R(w∗, w) .

• Possibly p means that p is true at some possible world w such that R(w∗, w) .

The four types of R will be a variation of these two general types. They will specify on what conditions a statementis true either in every possible world, or some possible. The four specific types of R are:Reflexive, or *Axiom (T) above:If R is reflexive, every world is accessible to itself. Reflexivity guarantees that any world at which A is true is aworld from which there is an accessible world at which A is true, and thus A is possible at worlds where it’s true,which isn't necessarily the case in worlds that aren't accessible to themselves. Without the reflexivity condition, Acan be necessary at a world where it’s false, if that world isn't accessible to itself; thus axiom T—that □A at a worldimplies A is true at that world—follows from reflexivity.Transitive, or *Axiom (4) above:If R is transitive, any world accessible to any world w′ accessible to world w is also accessible to w . Transitively,□A is true at a world w only when A is true at every world w′ accessible to w , including every world w′′

accessible to any w′ , and every world accessible to any w′′ , etc., so when □A is true at w , it’s also true at everyw′ and every w′′ , etc., which means □□A is also true at w , which is axiom 4.Euclidean or *Axiom (5) above:If R is euclidean, any two worlds accessible to a given world are accessible to each other. □♢A is true at a world w

if and only if, for every world w′ accessible to w , there is a world w′′ accessible to w′ at which A is true. If A

is true at a world w′ accessible to w , then if that world is accessible to every other world accessible to w , it willbe true that for every world accessible to w there is an accessible world ( w′ ) at which A is true, so ♢A is true atall worlds accessible to w . The euclidean property thus entails that ♢A implies □♢A , which is axiom 5.Symmetric or *Axiom (B) above:If R is symmetric, then if world w′ is accessible to world w , w is accessible to w′ . If A is true at w , then atevery w′ accessible to w , there is a world ( w ) accessible to w′ at which A is true, so A is possible at all w′ ,and thus it’s necessary at w that A is possible, which is axiom B.

1.4 Philosophical Applications

One of the applications of 'possible worlds’ semantics and the 'accessibility relation' is to physics. Instead of justtalking generically about 'necessity (or logical necessity),' the relation in physics deals with 'nomological necessity.'The fundamental translational schema (TS) described earlier can be exemplified as follows for physics:

• (TSN)P is nomologically necessary means thatP is true at all possible worlds that are nomologically accessible

1.5. COMPUTER SCIENCE APPLICATIONS 5

from the actual world. In other words, P is true at all possible worlds that obey the physical laws of the actualworld.

The interesting thing to observe is that instead of having to ask, now, “Does nomological necessity satisfy the axiom(5)?", that is, “Is something that is nomologically possible nomologically necessarily possible?", we can ask instead:“Is the nomological accessibility relation euclidean?" And different theories of the nature of physical laws will resultin different answers to this question. (Notice however that if the objection raised earlier is true, each different theoryof the nature of physical laws would be 'possible' and 'necessary,' since the euclidean concept depends on the ideaabout 'possibility' and 'necessity'). The theory of Lewis, for example, is asymmetric. His counterpart theory alsorequires an intransitive relation of accessibility because it is based on the notion of similarity and similarity is generallyintransitive. For example, a pile of straw with one less handful of straw may be similar to the whole pile but a pilewith two (or more) less handfuls may not be. So x can be necessarily P without x being necessarily necessarily P. On the other hand, Saul Kripke has an account of de re modality which is based on (metaphysical) identity acrossworlds and is therefore transitive.Another interpretation of the 'accessibility relation' with a physical meaning was given in Gerla 1987 where the claim“is possible P in the world w′′ is interpreted as “it is possible to transform w into a world in which P is true”. So,the properties of the modal operators depend on the algebraic properties of the set of admissible transformations.There are other applications of the 'accessibility relation' in philosophy. In epistemology, one can, instead of talkingabout nomological accessibility, talk about epistemic accessibility. A world w′ is epistemically accessible from wfor an individual I in w if and only if I does not know something which would rule out the hypothesis that w′ = w. We can ask whether the relation is transitive. If I knows nothing that rules out the possibility that w′ = w andknows nothing that rules the possibility that w′′ = w′ , it does not follow that I knows nothing which rules out thehypothesis that w′′ = w . To return to our earlier example, one may not be able to distinguish a pile of sand from thesame pile with one less handful and one may not be able to distinguish the pile with one less handful from the samepile with two less handfuls of sand, but one may still be able to distinguish the original pile from the pile with twoless handfuls of sand.Yet another example of the use of the 'accessibility relation' is in deontic logic. If we think of obligatoriness as truthin all morally perfect worlds, and permissibility as truth in some morally perfect world, then we will have to restrictout universe to include only morally perfect worlds. But, in that case, we will have left out the actual world. A betteralternative would be to include all the metaphysically possible worlds but restrict the 'accessibility relation' to morallyperfect worlds. Transitivity and the euclidean property will hold, but reflexivity and symmetry will not.

1.5 Computer Science Applications

In modeling a computation, a 'possible world' can be a possible computer state. Given the current computer state, youmight define the accessible possible worlds to be all future possible computer states, or to be all possible immediate“next” computer states (assuming a discrete computer). Either choice defines a particular 'accessibility relation' givingrise to a particular modal logic suited specifically for theorems about the computation.

1.6 See also

• Modal logic

• Possible worlds

• Propositional attitude

• Modal depth

1.7 References

• Gerla, G.; Transformational semantics for first order logic, Logique et Analyse, No. 117–118, pp. 69–79,1987.

6 CHAPTER 1. ACCESSIBILITY RELATION

• Fitelson, Brandon; Notes on “Accessibility” and Modality, 2003.

• Brown, Curtis; Propositional Modal Logic: A Few First Steps, 2002.

• Kripke, Saul; Naming and Necessity, Oxford, 1980.

• Lewis, David K.; Counterpart Theory and Quantified Modal Logic (subscription required), The Journal of Philosophy,Vol. LXV, No. 5 (1968-03-07), pp. 113–126, 1968

• List of Logic Systems List of most of the more popular modal logics.

Chapter 2

Allegory (category theory)

In the mathematical field category theory, an allegory is a category that has some of the structure of the category ofsets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in thissense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories arealso useful in defining and investigating certain constructions in category theory, such as exact completions.In this article we adopt the convention that morphisms compose from right to left, so RS means “first do S, then doR".

2.1 Definition

An allegory is a category in which

• every morphism R:X→Y is associated with an anti-involution, i.e. a morphism R°:Y→X; and

• every pair of morphisms R,S:X→Y with common domain/codomain is associated with an intersection, i.e. amorphism R∩S:X→Y

all such that

• intersections are idempotent (R∩R=R), commutative (R∩S=S∩R), and associative (R∩S)∩T=R∩(S∩T);

• anti-involution distributes over composition ((RS)°=S°R°) and intersection ((R∩S)°=S°∩R°);

• composition is semi-distributive over intersection (R(S∩T)⊆RS∩RT, (R∩S)T⊆RT∩ST); and

• the modularity law is satisfied: (RS∩T⊆(R∩TS°)S).

Here, we are abbreviating using the order defined by the intersection: "R⊆S" means "R=R∩S".A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and amorphismX→Y is a binary relation between X and Y. Composition of morphisms is composition of relations; intersection ofmorphisms is intersection of relations.

2.2 Regular categories and allegories

2.2.1 Allegories of relations in regular categories

In a category C, a relation between objects X, Y is a span of morphisms X←R→Y that is jointly-monic. Two suchspans X←S→Y and X←T→Y are considered equivalent when there is an isomorphism between S and T that makeeverything commute, and strictly speaking relations are only defined up to equivalence (one may formalise this eitherusing equivalence classes or using bicategories). If the category C has products, a relation between X and Y is the

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8 CHAPTER 2. ALLEGORY (CATEGORY THEORY)

same thing as a monomorphism into X×Y (or an equivalence class of such). In the presence of pullbacks and a properfactorization system, one can define the composition of relations. The composition of X←R→Y←S→Z is found byfirst pulling back the cospan R→Y←S and then taking the jointly-monic image of the resulting span X←R←·→S→Z.Composition of relations will be associative if the factorization system is appropriately stable. In this case one canconsider a category Rel(C), with the same objects as C, but where morphisms are relations between the objects. Theidentity relations are the diagonals X→X×X.Recall that a regular category is a category with finite limits and images in which covers are stable under pullback. Aregular category has a stable regular epi/mono factorization system. The category of relations for a regular categoryis always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersectionsare intersections of subobjects, computed by pullback.

2.2.2 Maps in allegories, and tabulations

A morphism R in an allegory A is called a map if it is entire (1⊆R°R) and deterministic (RR°⊆1). Another way ofsaying this: a map is a morphism that has a right adjoint in A, when A is considered, using the local order structure,as a 2-category. Maps in an allegory are closed under identity and composition. Thus there is a subcategory Map(A)of A, with the same objects but only the maps as morphisms. For a regular category C, there is an isomorphism ofcategories C≅Map(Rel(C)). In particular, a morphism in Map(Rel(Set)) is just an ordinary set function.In an allegory, a morphism R:X→Y is tabulated by a pair of maps f:Z→X, g:Z→Y if gf°=R and f°f∩g°g=1. Anallegory is called tabular if every morphism has a tabulation. For a regular category C, the allegory Rel(C) is alwaystabular. On the other hand, for any tabular allegory A, the category Map(A) of maps is a locally regular category: ithas pullbacks, equalizers and images that are stable under pullback. This is enough to study relations in Map(A) and,in this setting, A≅Rel(Map(A)).

2.2.3 Unital allegories and regular categories of maps

A unit in an allegory is an object U for which the identity is the largest morphism U→U, and such that from everyother object there is an entire relation to U. An allegory with a unit is called unital. Given a tabular allegory A, thecategory Map(A) is a regular category (it has a terminal object) if and only if A is unital.

2.2.4 More sophisticated kinds of allegory

Additional properties of allegories can be axiomatized. Distributive allegories have a union-like operation that issuitably well-behaved, and division allegories have a generalization of the division operation of relation algebra.Power allegories are distributive division allegories with additional powerset-like structure. The connection betweenallegories and regular categories can be developed into a connection between power allegories and toposes.

2.3 References• Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland.ISBN 978-0-444-70368-2.

• Peter Johnstone (2003). Sketches of an Elephant: A Topos Theory Compendium. Oxford Science Publications.OUP. ISBN 0-19-852496-X.

Chapter 3

Alternatization

In mathematics, more specifically in multilinear algebra, the notion of alternatization (or alternatisation in BritishEnglish) is used to pass from any map to an alternating map.An alternating map is a multilinear map (e.g., a bilinear map or a multilinear form) that is equal to zero for everytuple with two adjacent elements that are equal.

3.1 Definitions

3.1.1 Alternating bilinear map

Let S be a set, A be an abelian group, and α : S × S → A be a bilinear map. Then α is said to be an alternatingbilinear map if

∀x ∈ S, α(x, x) = 0.

3.1.2 Alternating bilinear form

An alternating bilinear form is a special case of alternating bilinear map. As bilinear forms can be defined as mapsbetween vector spaces or modules, we distinguish two cases.

Vector spacesLet V be a vector space over a field K, and α : V × V → K be a bilinear form. Then α is said to be analternating bilinear form if [1][2]

∀x ∈ V, α(x, x) = 0.

ModulesLet M be a module over a ring R, and α : M ×M → R be a bilinear form. Then α is said to be analternating bilinear form if

∀x ∈M, α(x, x) = 0.

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10 CHAPTER 3. ALTERNATIZATION

3.1.3 Alternating multilinear form

An alternatingmultilinear form generalizes the concept of alternating bilinear form to n dimensions. Asmultilinearforms can be defined as maps between vector spaces or modules, we distinguish two cases.

Vector spacesLet V be a vector space over a field K, and α : V × V × ...× V → K be a multilinear form. Then α issaid to be an alternating multilinear form if

∀x1, x2, ..., xn ∈ V, ∀i ∈ {1, 2, ..., n− 1}, xi = xi+1 =⇒ α(x1, x2, ..., xn) = 0.

ModulesLet M be a module over a ring R, and α : M ×M × ... ×M → R be a multilinear form. Then α issaid to be an alternating multilinear form if [3]

∀x1, x2, ..., xn ∈M, ∀i ∈ {1, 2, ..., n− 1}, xi = xi+1 =⇒ α(x1, x2, ..., xn) = 0.

3.1.4 Alternatization of a bilinear map

Let S be a set, A be an abelian group, and α : S × S → A be a bilinear map. ∀x, y ∈ S, the alternatization of themap α is the map

β : S × S → A

(x, y) 7→ α(x, y)− α(y, x).

3.2 Example

• The Lie bracket is an alternating bilinear form.

3.3 Properties

• Every alternating multilinear form is antisymmetric:[4]

∀x, y ∈ S, α(x, y) + α(y, x) = 0

• If the characteristic of the ring R is not equal to 2, then every antisymmetric multilinear form is alternating.[5]

• The alternatization of an alternating map is its double.

• The alternatization of a symmetric map is zero.

• The alternatization of a bilinearmap is bilinear. Most notably, the alternatization of any cocycle is bilinear. Thisfact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternatingbilinear forms on a lattice.

• There may be non-bilinear maps whose alternatization is bilinear.

3.4. SEE ALSO 11

3.4 See also• Bilinear map

• Map (mathematics)

• Multilinear algebra

• Multilinear map

3.5 Notes[1] Rotman 1995, page 235.

[2] Cohn 2003, page 298.

[3] Lang 2002, page 511.

[4] Rotman 1995, page 235.

[5] Rotman 1995, page 235.

3.6 References• Cohn, P.M. (2003). Basic Algebra: Groups, Rings and Fields. Springer. ISBN 1-85233-587-4. OCLC248833275.

• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.

• Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 (4thed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.

Chapter 4

Ancestral relation

In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitiveclosure, however defined in a different way, see below.Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundge-setze as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicistfoundation of arithmetic.

4.1 Definition

The numbered propositions below are taken from his Begriffsschrift and recast in contemporary notation.A property P is called R-hereditary if, whenever x is P and xRy holds, then y is also P:

(Px ∧ xRy) → Py

Frege defined b to be an R-ancestor of a, written aR*b, if b has every R-hereditary property that all objects x suchthat aRx have:

76 : ⊢ aR∗b↔ ∀F [∀x(aRx→ Fx) ∧ ∀x∀y(Fx ∧ xRy → Fy) → Fb]

The ancestral is a transitive relation:

98 : ⊢ (aR∗b ∧ bR∗c) → aR∗c

Let the notation I(R) denote that R is functional (Frege calls such relations “many-one”):

115 : ⊢ I(R) ↔ ∀x∀y∀z[(xRy ∧ xRz) → y = z]

If R is functional, then the ancestral of R is what nowadays is called connected:

133 : ⊢ (I(R) ∧ aR∗b ∧ aR∗c) → (bR∗c ∨ b = c ∨ cR∗b)

4.2 Relationship to transitive closure

The Ancestral relation R∗ is equal to the transitive closure R+ of R . Indeed, R∗ is transitive (see 98 above), R∗

contains R (indeed, if aRb then, of course, b has every R-hereditary property that all objects x such that aRx have,because b is one of them), and finally, R∗ is contained in R+ (indeed, assume aR∗b ; take the property Fx to beaR+x ; then the two premises, ∀x(aRx → Fx) and ∀x∀y(Fx ∧ xRy → Fy) , are obviously satisfied; therefore,Fb , which means aR+b , by our choice of F ). See also Boolos’s book below, page 8.

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4.3. DISCUSSION 13

4.3 Discussion

Principia Mathematica made repeated use of the ancestral, as does Quine’s (1951) Mathematical Logic.However, it is worth noting that the ancestral relation cannot be defined in first-order logic. It is controversial whethersecond-order logic is really “logic” at all. Quine famously claimed that it was not, despite his reliance upon it forhis 1951 book (which largely retells Principia in abbreviated form, for which second-order logic is required to fit itstheorems).

4.4 See also• Begriffsschrift

• Gottlob Frege

• Transitive closure

4.5 References• George Boolos, 1998. Logic, Logic, and Logic. Harvard Univ. Press.

• Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton Univ. Press.

• Willard Van Orman Quine, 1951 (1940). Mathematical Logic. Harvard Univ. Press. ISBN 0-674-55451-5.

4.6 External links• Stanford Encyclopedia of Philosophy: "Frege’s Logic, Theorem, and Foundations for Arithmetic" -- by EdwardN. Zalta. Section 4.2.

Chapter 5

Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of distinct elements of X each ofwhich is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X

if R(a,b) and R(b,a), then a = b,

or, equivalently,

if R(a,b) with a ≠ b, then R(b,a) must not hold.

As a simple example, the divisibility order on the natural numbers is an antisymmetric relation. And what antisym-metry means here is that the only way each of two numbers can be divisible by the other is if the two are, in fact, thesame number; equivalently, if n and m are distinct and n is a factor of m, then m cannot be a factor of n.In mathematical notation, this is:

∀a, b ∈ X, R(a, b) ∧R(b, a) ⇒ a = b

or, equivalently,

∀a, b ∈ X, R(a, b) ∧ a ̸= b⇒ ¬R(b, a).

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalitiesx ≤ y and y ≤ x hold then x and y must be equal. Similarly, the subset order ⊆ on the subsets of any given set isantisymmetric: given two sets A and B, if every element in A also is in B and every element in B is also in A, then Aand B must contain all the same elements and therefore be equal:

A ⊆ B ∧B ⊆ A⇒ A = B

Partial and total orders are antisymmetric by definition. A relation can be both symmetric and antisymmetric (e.g.,the equality relation), and there are relations which are neither symmetric nor antisymmetric (e.g., the “preys on”relation on biological species).Antisymmetry is different from asymmetry, which requires both antisymmetry and irreflexivity.

5.1 Examples

The relation "x is even, y is odd” between a pair (x, y) of integers is antisymmetric:

14

5.2. SEE ALSO 15

Every asymmetric relation is also an antisymmetric relation.

5.2 See also• Symmetric relation

• Asymmetric relation

• Symmetry in mathematics

5.3 References• Weisstein, Eric W., “Antisymmetric Relation”, MathWorld.

• Lipschutz, Seymour; Marc Lars Lipson (1997). Theory and Problems of Discrete Mathematics. McGraw-Hill.p. 33. ISBN 0-07-038045-7.

Chapter 6

Asymmetric relation

In mathematics an asymmetric relation is a binary relation on a set X where:

• For all a and b in X, if a is related to b, then b is not related to a.[1]

In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ ¬(bRa)

6.1 Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski’s axioms characterizingthe real numbers R is that < over R is asymmetric.An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2}and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, everytransitive asymmetric relation is a strict partial order.Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation isthe rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ xand both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also notasymmetric.Asymmetric is not the same thing as “not symmetric": a relation can be neither symmetric nor asymmetric, such as≤, or can be both, only in the case of the empty relation (vacuously).

6.2 Properties• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]

• Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < fromthe reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.

• A transitive relation is asymmetric if and only if it is irreflexive:[3] if a R b and b R a, transitivity gives a R a,contradicting irreflexivity.

6.3 See also• Symmetric relation

16

6.4. REFERENCES 17

• Antisymmetric relation

• Symmetry

• Symmetry in mathematics

6.4 References[1] Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.

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

[3] 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”.

Chapter 7

Better-quasi-ordering

In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array.Every bqo is well-quasi-ordered.

7.1 Motivation

Though wqo is an appealing notion, many important infinitary operations do not preserve wqoness. An exampledue to Richard Rado illustrates this.[1] In a 1965 paper Crispin Nash-Williams formulated the stronger notion ofbqo in order to prove that the class of trees of height ω is wqo under the topological minor relation.[2] Since then,many quasi-orders have been proven to be wqo by proving them to be bqo. For instance, Richard Laver establishedFraïssé's conjecture by proving that the class of scattered linear order types is bqo.[3] More recently, Carlos Martinez-Ranero has proven that, under the Proper Forcing Axiom, the class of Aronszajn lines is bqo under the embeddabilityrelation.[4]

7.2 Definition

It is common in bqo theory to write ∗x for the sequence x with the first term omitted. Write [ω]<ω for the set offinite, strictly increasing sequences with terms in ω , and define a relation ◁ on [ω]<ω as follows: s ◁ t if and only ifthere is u such that s is a strict initial segment of u and t = ∗u . Note that the relation ◁ is not transitive.A block is a subset B of [ω]<ω that contains an initial segment of every infinite subset of

∪B . For a quasi-order Q

a Q -pattern is a function from a block B into Q . A Q -pattern f : B → Q is said to be bad if f(s) ̸≤Q f(t) forevery pair s, t ∈ B such that s ◁ t ; otherwise f is good. A quasi-order Q is better-quasi-ordered (bqo) if there is nobad Q -pattern.In order to make this definition easier to work with, Nash-Williams defines a barrier to be a block whose elementsare pairwise incomparable under the inclusion relation⊂ . A Q -array is aQ -pattern whose domain is a barrier. Byobserving that every block contains a barrier, one sees that Q is bqo if and only if there is no bad Q -array.

7.3 Simpson’s alternative definition

Simpson introduced an alternative definition of bqo in terms of Borel maps [ω]ω → Q , where [ω]ω , the set of infinitesubsets of ω , is given the usual (product) topology.[5]

Let Q be a quasi-order and endow Q with the discrete topology. A Q -array is a Borel function [A]ω → Q forsome infinite subset A of ω . A Q -array f is bad if f(X) ̸≤Q f(∗X) for every X ∈ [A]ω ; f is good otherwise.The quasi-order Q is bqo if there is no bad Q -array in this sense.

18

7.4. MAJOR THEOREMS 19

7.4 Major theorems

Many major results in bqo theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson’spaper[5] as follows. See also Laver’s paper,[6] where the Minimal Bad Array Lemma was first stated as a result. Thetechnique was present in Nash-Williams’ original 1965 paper.Suppose (Q,≤Q) is a quasi-order. A partial ranking ≤′ of Q is a well-founded partial ordering of Q such thatq ≤′ r → q ≤Q r . For bad Q -arrays (in the sense of Simpson) f : [A]ω → Q and g : [B]ω → Q , define:

g ≤∗ f if B ⊆ A and g(X) ≤′ f(X) every for X ∈ [B]ω

g <∗ f if B ⊆ A and g(X) <′ f(X) every for X ∈ [B]ω

We say a bad Q -array g is minimal bad (with respect to the partial ranking ≤′ ) if there is no bad Q -array f suchthat f <∗ g . Note that the definitions of≤∗ and<′ depend on a partial ranking≤′ ofQ . Note also that the relation<∗ is not the strict part of the relation ≤∗ .Theorem (Minimal Bad Array Lemma). Let Q be a quasi-order equipped with a partial ranking and suppose f is abad Q -array. Then there is a minimal bad Q -array g such that g ≤∗ f .

7.5 See also• Well-quasi-ordering

• Well-order

7.6 References[1] Rado, Richard (1954). “Partial well-ordering of sets of vectors”. Mathematika 1 (2): 89–95. doi:10.1112/S0025579300000565.

MR 0066441.

[2] Nash-Williams, C. St. J. A. (1965). “On well-quasi-ordering infinite trees”. Mathematical Proceedings of the CambridgePhilosophical Society 61 (3): 697–720. Bibcode:1965PCPS...61..697N. doi:10.1017/S0305004100039062. ISSN 0305-0041. MR 0175814.

[3] Laver, Richard (1971). “On Fraisse’s Order TypeConjecture”. TheAnnals ofMathematics 93 (1): 89–111. doi:10.2307/1970754.JSTOR 1970754.

[4] Martinez-Ranero, Carlos (2011). “Well-quasi-ordering Aronszajn lines”. Fundamenta Mathematicae 213 (3): 197–211.doi:10.4064/fm213-3-1. ISSN 0016-2736. MR 2822417.

[5] Simpson, Stephen G. (1985). “BQOTheory and Fraïssé's Conjecture”. InMansfield, Richard; Weitkamp, Galen. RecursiveAspects of Descriptive Set Theory. The Clarendon Press, Oxford University Press. pp. 124–38. ISBN 978-0-19-503602-2.MR 786122.

[6] Laver, Richard (1978). “Better-quasi-orderings and a class of trees”. In Rota, Gian-Carlo. Studies in foundations andcombinatorics. Academic Press. pp. 31–48. ISBN 978-0-12-599101-8. MR 0520553.

Chapter 8

Bidirectional transformation

In computer programming, bidirectional transformations (bx) are programs in which a single piece of code canbe run in several ways, such that the same data are sometimes considered as input, and sometimes as output. Forexample, a bx run in the forward direction might transform input I into output O, while the same bx run backwardwould take as input versions of I and O and produce a new version of I as its output.Bidirectional model transformations are an important special case in which a model is input to such a program.Some bidirectional languages are bijective. The bijectivity of a language is a severe restriction of its bidirectionality,[1]because a bijective language is merely relating two different ways to present the very same information.More general is a lens language, in which there is a distinguished forward direction (“get”) that takes a concrete inputto an abstract output, discarding some information in the process: the concrete state includes all the information thatis in the abstract state, and usually some more. The backward direction (“put”) takes a concrete state and an abstractstate and computes a new concrete state. Lenses are required to obey certain conditions to ensure sensible behaviour.The most general case is that of symmetric bidirectional transformations. Here the two states that are related typicallyshare some information, but each also includes some information that is not included in the other.

8.1 Usage

Bidirectional transformations can be used to:

• Maintain several sources of information consistent[2]

• Provide an 'abstract view' to easily manipulate data and write them back to their source

8.2 Vocabulary

A bidirectional program which obeys certain round-trip laws is called a lens.

8.3 Examples of implementations

• Boomerang is a programming language which allows to write lenses to process text data formats bidirectionally

• Augeas is a configuration management library whose lens language is inspired by the Boomerang project

• biXid is a programming language to process XML data bidirectionally[3]

• XSugar allows to translate from XML to non-XML formats[4]

20

8.4. SEE ALSO 21

8.4 See also• Bidirectionalization

8.5 References[1] http://grace.gsdlab.org/images/e/e2/Nate-short.pdf

[2] http://www.cs.cornell.edu/~{}jnfoster/papers/grace-report.pdf

[3] http://arbre.is.s.u-tokyo.ac.jp/~{}hahosoya/papers/bixid.pdf

[4] http://www.brics.dk/xsugar/

8.6 External links• GRACE International Meeting on Bidirectional Transformations

• Bidirectional Transformations: The Bx Wiki

Chapter 9

Bijection

X 1

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4

YD

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C

A

A bijective function, f: X→ Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical

22

9.1. DEFINITION 23

terms, a bijective function f: X→ Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the setX to the set Y has an inverse function from Y toX. IfX and Y are finite sets, then the existenceof a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of infinite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the definitions of isomorphism, homeomorphism,diffeomorphism, permutation group, and projective map.

9.1 Definition

For more details on notation, see Function (mathematics) § Notation.

For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both “one-to-one” and “onto”.

9.2 Examples

9.2.1 Batting line-up of a baseball team

Consider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The “pairing”is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere inthe list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.

9.2.2 Seats and students of a classroom

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

4. No seat had more than one student in it.

24 CHAPTER 9. BIJECTION

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

9.3 More mathematical examples and some non-examples• For any set X, the identity function 1X: X→ X, 1X(x) = x, is bijective.

• The function f: R→ R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R→ R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

• The function f: R → (-π/2, π/2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of π/2 by this arctan function.

• The exponential function, g: R→ R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =−1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ ≡ (0,+∞) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

• The function h: R→ R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+

0 ≡ [0,+∞) , then h becomes bijective; its inverseis the positive square root function.

9.4 Inverses

A bijection f with domainX (“functionally” indicated by f: X→Y) also defines a relation starting in Y and going toX(by turning the arrows around). The process of “turning the arrows around” for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being defined takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

9.5 Composition

The composition g ◦ f of two bijections f: X → Y and g: Y → Z is a bijection. The inverse of g ◦ f is (g ◦ f)−1 =

(f−1) ◦ (g−1) .Conversely, if the composition g ◦ f of two functions is bijective, we can only say that f is injective and g is surjective.

9.6 Bijections and cardinality

If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of “same number ofelements” (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number,a way to distinguish the various sizes of infinite sets.

9.7. PROPERTIES 25

X1

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C

A

ZP

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R

A bijection composed of an injection (left) and a surjection (right).

9.7 Properties

• A function f: R→ R is bijective if and only if its graph meets every horizontal and vertical line exactly once.

• If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

• Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f−1(B)| = |B|.

• If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

• For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that set—namely, n!.

9.8 Bijections and category theory

Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

26 CHAPTER 9. BIJECTION

9.9 Generalization to partial functions

The notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of Aand likewise B′⊆B.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is theMöbius transformation simply defined on the complex plane, rather than its completion to the extendedcomplex plane.[5]

9.10 Contrast withThis list is incomplete; you can help by expanding it.

• Multivalued function

9.11 See also• Injective function

• Surjective function

• Bijection, injection and surjection

• Symmetric group

• Bijective numeration

• Bijective proof

• Cardinality

• Category theory

• Ax–Grothendieck theorem

9.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 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] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). “Groups and semigroups: connections and contrasts”. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson,M.V. (1998). “TheMöbius InverseMonoid”. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.

9.13. REFERENCES 27

9.13 References

This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

• Wolf (1998). Proof, Logic and Conjecture: A Mathematician’s Toolbox. Freeman.

• 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 (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.

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

• Morash. Bridge to Abstract Mathematics. Random House.

• Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.

• 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.

• Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.

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

• 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.

9.14 External links• Hazewinkel, Michiel, ed. (2001), “Bijection”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Bijection”, MathWorld.

• Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 10

Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in whicharguments (input expressions from the domain) and images (output expressions from the codomain) are related ormapped to each other.A function maps elements from its domain to elements in its codomain. Given a function f : A→ B

• The function is injective (one-to-one) if every element of the codomain is mapped to by at most one elementof the domain. An injective function is an injection. Notationally:

∀x, y ∈ A, f(x) = f(y) ⇒ x = y.

Or, equivalently (using logical transposition),∀x, y ∈ A, x ̸= y ⇒ f(x) ̸= f(y).

• The function is surjective (onto) if every element of the codomain is mapped to by at least one element of thedomain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection.Notationally:

∀y ∈ B, ∃x ∈ A that such y = f(x).

• The function is bijective (one-to-one and onto or one-to-one correspondence) if every element of thecodomain is mapped to by exactly one element of the domain. (That is, the function is both injective andsurjective.) A bijective function is a bijection.

An injective function need not be surjective (not all elements of the codomain may be associated with arguments),and a surjective function need not be injective (some images may be associated with more than one argument). Thefour possible combinations of injective and surjective features are illustrated in the diagrams to the right.

10.1 Injection

Main article: Injective functionFor more details on notation, see Function (mathematics) § Notation.A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument.Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is aninjection. The formal definition is the following.

The function f : A→ B is injective iff for all a, b ∈ A , we have f(a) = f(b) ⇒ a = b.

• A function f : A→ B is injective if and only if A is empty or f is left-invertible; that is, there is a function g :f(A) → A such that g o f = identity function on A. Here f(A) is the image of f.

28

10.2. SURJECTION 29

X1

2

3

YD

B

C

A

ZP

Q

R

S

Injective composition: the second function need not be injective.

• Since every function is surjective when its codomain is restricted to its image, every injection induces a bijectiononto its image. More precisely, every injection f : A→ B can be factored as a bijection followed by an inclusionas follows. Let fR : A→ f(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusionmap from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.

• The composition of two injections is again an injection, but if g o f is injective, then it can only be concludedthat f is injective. See the figure at right.

• Every embedding is injective.

10.2 Surjection

Main article: Surjective functionA function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, everyelement in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to itscodomain. A surjective function is a surjection. The formal definition is the following.

The function f : A→ B is surjective iff for all b ∈ B , there is a ∈ A such that f(a) = b.

• A function f : A→ B is surjective if and only if it is right-invertible, that is, if and only if there is a function g:B→ A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)

• By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on aquotient of its domain. More precisely, every surjection f : A→ B can be factored as a non-bijection followedby a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be theprojection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-definedfunction given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.

• The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concludedthat g is surjective. See the figure.

30 CHAPTER 10. BIJECTION, INJECTION AND SURJECTION

X1

2

3

4

YD

B

C

A

ZP

Q

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Surjective composition: the first function need not be surjective.

10.3 Bijection

Main article: Bijective functionA function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one corre-spondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. Thisequivalent condition is formally expressed as follow.

The function f : A→ B is bijective iff for all b ∈ B , there is a unique a ∈ A such that f(a) = b.

• A function f : A → B is bijective if and only if it is invertible, that is, there is a function g: B → A such thatg o f = identity function on A and f o g = identity function on B. This function maps each image to its uniquepreimage.

• The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concludedthat f is injective and g is surjective. (See the figure at right and the remarks above regarding injections andsurjections.)

• The bijections from a set to itself form a group under composition, called the symmetric group.

10.4 Cardinality

Suppose you want to define what it means for two sets to “have the same number of elements”. One way to do this isto say that two sets “have the same number of elements” if and only if all the elements of one set can be paired withthe elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we candefine two sets to “have the same number of elements” if there is a bijection between them. We say that the two setshave the same cardinality.Likewise, we can say that set A “has fewer than or the same number of elements” as set B if there is an injectionfrom A to B . We can also say that set A “has fewer than the number of elements” in set B if there is an injectionfrom A to B but not a bijection between A and B .

10.5. EXAMPLES 31

X1

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Bijective composition: the first function need not be surjective and the second function need not be injective.

10.5 Examples

It is important to specify the domain and codomain of each function since by changing these, functions which wethink of as the same may have different jectivity.

10.5.1 Injective and surjective (bijective)

• For every set A the identity function idA and thus specifically R → R : x 7→ x .

• R+ → R+ : x 7→ x2 and thus also its inverse R+ → R+ : x 7→√x .

• The exponential function exp : R → R+ : x 7→ ex and thus also its inverse the natural logarithm ln : R+ →R : x 7→ lnx

10.5.2 Injective and non-surjective

• The exponential function exp : R → R : x 7→ ex

10.5.3 Non-injective and surjective

• R → R : x 7→ (x− 1)x(x+ 1) = x3 − x

• R → [−1, 1] : x 7→ sin(x)

10.5.4 Non-injective and non-surjective

• R → R : x 7→ sin(x)

32 CHAPTER 10. BIJECTION, INJECTION AND SURJECTION

10.6 Properties• For every function f, subsetA of the domain and subsetB of the codomainwe haveA⊂ f −1(f(A)) and f(f −1(B))⊂ B. If f is injective we have A = f −1(f(A)) and if f is surjective we have f(f −1(B)) = B.

• For every function h : A→ C we can define a surjection H : A→ h(A) : a → h(a) and an injection I : h(A)→C : a → a. It follows that h = I ∘ H. This decomposition is unique up to isomorphism.

10.7 Category theory

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms,and isomorphisms, respectively.

10.8 History

This terminology was originally coined by the Bourbaki group.

10.9 See also• Bijective function

• Horizontal line test

• Injective module

• Injective function

• Permutation

• Surjective function

10.10 External links• Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 11

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.

11.1 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 ofG 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.

33

34 CHAPTER 11. BINARY RELATION

11.1.1 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, ifG = {(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.

11.1.2 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".

11.2 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.

11.2. SPECIAL TYPES OF BINARY RELATIONS 35

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:

36 CHAPTER 11. 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.

11.2.1 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]

11.3 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.

11.4. OPERATIONS ON BINARY RELATIONS 37

• 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.

11.4 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”.

38 CHAPTER 11. 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.

11.4.1 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.

11.4.2 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.

11.5. SETS VERSUS CLASSES 39

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.

11.4.3 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.

11.5 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.

11.6 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.

40 CHAPTER 11. 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).

11.7 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.

11.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

11.9 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; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

11.10. REFERENCES 41

[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.

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

42 CHAPTER 11. BINARY RELATION

11.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 12

Cointerpretability

In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretablein another such theory S, when the language of S can be translated into the language of T in such a way that S provesevery formula whose translation is a theorem of T. The “translation” here is required to preserve the logical structureof formulas.This concept, in a sense dual to interpretability, was introduced by Dzhaparidze (1993), who also proved that, fortheories of Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalentto Σ1 -conservativity.

12.1 See also• Cotolerance

• interpretability logic.

• Tolerance (in logic)

12.2 References• Dzhaparidze, Giorgie (1993), “A generalized notion of weak interpretability and the corresponding modallogic”,Annals of Pure andApplied Logic 61 (1-2): 113–160, doi:10.1016/0168-0072(93)90201-N,MR1218658.

• Japaridze, Giorgi; de Jongh, Dick (1998), “The logic of provability”, in Buss, Samuel R., Handbook of ProofTheory, Studies in Logic and the Foundations of Mathematics 137, Amsterdam: North-Holland, pp. 475–546,doi:10.1016/S0049-237X(98)80022-0, MR 1640331.

43

Chapter 13

Commutative property

For other uses, see Commute (disambiguation).In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

=

==

This image illustrates that addition is commutative.

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar asthe name of the property that says “3 + 4 = 4 + 3” or “2 × 5 = 5 × 2”, the property can also be used in more advancedsettings. The name is needed because there are operations, such as division and subtraction that do not have it (forexample, “3 − 5 ≠ 5 − 3”), such operations are not commutative, or noncommutative operations. The idea that simpleoperations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumedand the property was not named until the 19th century when mathematics started to become formalized.

13.1 Common uses

The commutative property (or commutative law) is a property generally associatedwith binary operations and functions.If the commutative property holds for a pair of elements under a certain binary operation then the two elements aresaid to commute under that operation.

44

13.2. MATHEMATICAL DEFINITIONS 45

13.2 Mathematical definitions

Further information: Symmetric function

The term “commutative” is used in several related senses.[1][2]

1. A binary operation ∗ on a set S is called commutative if:x ∗ y = y ∗ x for all x, y ∈ S

An operation that does not satisfy the above property is called non-commutative.2. One says that x commutes with y under ∗ if:x ∗ y = y ∗ x

3. A binary function f : A×A→ B is called commutative if:f(x, y) = f(y, x) for all x, y ∈ A

13.3 Examples

13.3.1 Commutative operations in everyday life• Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Eitherway, the result (having both socks on), is the same.

• The commutativity of addition is observed when paying for an item with cash. Regardless of the order the billsare handed over in, they always give the same total.

13.3.2 Commutative operations in mathematics

Two well-known examples of commutative binary operations:[1]

• The addition of real numbers is commutative, since

y + z = z + y for all y, z ∈ R

For example 4 + 5 = 5 + 4, since both expressions equal 9.

• The multiplication of real numbers is commutative, since

yz = zy for all y, z ∈ R

For example, 3 × 5 = 5 × 3, since both expressions equal 15.

• Some binary truth functions are also commutative, since the truth tables for the functions are the same whenone changes the order of the operands.

For example, the logical biconditional function p↔ q is equivalent to q↔ p. This function is also writtenas p IFF q, or as p ≡ q, or as Epq.The last form is an example of the most concise notation in the article on truth functions, which lists thesixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp;Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

• Further examples of commutative binary operations include addition and multiplication of complex numbers,addition and scalar multiplication of vectors, and intersection and union of sets.

46 CHAPTER 13. COMMUTATIVE PROPERTY

b

b

aaa+

b

The addition of vectors is commutative, because a⃗+ b⃗ = b⃗+ a⃗ .

13.3.3 Noncommutative operations in everyday life

• Concatenation, the act of joining character strings together, is a noncommutative operation. For example

EA+ T = EAT ̸= TEA = T + EA

• Washing and drying clothes resembles a noncommutative operation; washing and then drying produces amarkedly different result to drying and then washing.

• Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientationthan when the rotations are performed in the opposite order.

• The twists of the Rubik’s Cube are noncommutative. This can be studied using group theory.

• Also thought processes are noncommutative: A person asked a question (A) and then a question (B) may givedifferent answers to each question than a person asked first (B) and then (A), because asking a question maychange the person’s state of mind.

13.4. HISTORY AND ETYMOLOGY 47

13.3.4 Noncommutative operations in mathematics

Some noncommutative binary operations:[3]

• Subtraction is noncommutative, since 0− 1 ̸= 1− 0

• Division is noncommutative, since 1/2 ̸= 2/1

• Some truth functions are noncommutative, since the truth tables for the functions are different when one changesthe order of the operands.

For example, the truth tables for f (A,B) = A Λ ¬B (A AND NOT B) and f (B,A) = B Λ ¬A are

• Matrix multiplication is noncommutative since

[0 20 1

]=

[1 10 1

]·[0 10 1

]̸=

[0 10 1

]·[1 10 1

]=

[0 10 1

]• The vector product (or cross product) of two vectors in three dimensions is anti-commutative, i.e., b × a = −(a× b).

13.4 History and etymology

The first known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commuta-tive property of multiplication to simplify computing products.[4][5] Euclid is known to have assumed the commutativeproperty of multiplication in his book Elements.[6] Formal uses of the commutative property arose in the late 18th andearly 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative propertyis a well known and basic property used in most branches of mathematics.The first recorded use of the term commutative was in a memoir by François Servois in 1814,[7][8] which used theword commutatives when describing functions that have what is now called the commutative property. The word is acombination of the French word commuter meaning “to substitute or switch” and the suffix -ative meaning “tendingto” so the word literally means “tending to substitute or switch.” The term then appeared in English in PhilosophicalTransactions of the Royal Society in 1844.[7]

48 CHAPTER 13. COMMUTATIVE PROPERTY

13.5 Propositional logic

13.5.1 Rule of replacement

In truth-functional propositional logic, commutation,[9][10] or commutativity[11] refer to two valid rules of replacement.The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

(P ∨Q) ⇔ (Q ∨ P )

and

(P ∧Q) ⇔ (Q ∧ P )

where "⇔ " is a metalogical symbol representing “can be replaced in a proof with.”

13.5.2 Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logicalequivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functionaltautologies.

Commutativity of conjunction (P ∧Q) ↔ (Q ∧ P )

Commutativity of disjunction (P ∨Q) ↔ (Q ∨ P )

Commutativity of implication (also called the law of permutation(P → (Q→ R)) ↔ (Q→ (P → R))

Commutativity of equivalence (also called the complete commutative law of equivalence)(P ↔ Q) ↔ (Q↔ P )

13.6 Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the com-mutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity ofwell-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitlyassumed) in proofs.[12][13][14]

13.7 Mathematical structures and commutativity

• A commutative semigroup is a set endowed with a total, associative and commutative operation.

• If the operation additionally has an identity element, we have a commutative monoid

• An abelian group, or commutative group is a group whose group operation is commutative.[13]

• Acommutative ring is a ringwhosemultiplication is commutative. (Addition in a ring is always commutative.)[15]

• In a field both addition and multiplication are commutative.[16]

13.8. RELATED PROPERTIES 49

13.8 Related properties

13.8.1 Associativity

Main article: Associative property

The associative property is closely related to the commutative property. The associative property of an expressioncontaining two or more occurrences of the same operator states that the order operations are performed in does notaffect the final result, as long as the order of terms doesn't change. In contrast, the commutative property states thatthe order of the terms does not affect the final result.Most commutative operations encountered in practice are also associative. However, commutativity does not implyassociativity. A counterexample is the function

f(x, y) =x+ y

2,

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, forexample, f(−4, f(0,+4)) = −1 but f(f(−4, 0),+4) = +1 ). More such examples may be found in Commutativenon-associative magmas.

13.8.2 Symmetry

-10-8-6-4-2 0 2 4 6 8 10

Graph showing the symmetry of the addition function

Main article: Symmetry in mathematics

50 CHAPTER 13. COMMUTATIVE PROPERTY

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as abinary function then the resulting function is symmetric across the line y = x. As an example, if we let a function frepresent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seenin the image on the right.For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, thenaRb⇔ bRa .

13.9 Non-commuting operators in quantum mechanics

Main article: Canonical commutation relation

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x(meaning multiply by x), and d

dx . These two operators do not commute as may be seen by considering the effect oftheir compositions x d

dx and ddxx (also called products of operators) on a one-dimensional wave function ψ(x) :

xd

dxψ = xψ′ ̸= d

dxxψ = ψ + xψ′

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do notcommute, then that pair of variables are mutually complementary, which means they cannot be simultaneously mea-sured or known precisely. For example, the position and the linear momentum in the x-direction of a particle arerepresented respectively by the operators x and −iℏ ∂

∂x (where ℏ is the reduced Planck constant). This is the sameexample except for the constant −iℏ , so again the operators do not commute and the physical meaning is that theposition and linear momentum in a given direction are complementary.

13.10 See also

• Anticommutativity

• Associative Property

• Binary operation

• Centralizer or Commutant

• Commutative diagram

• Commutative (neurophysiology)

• Commutator

• Distributivity

• Parallelogram law

• Particle statistics (for commutativity in physics)

• Quasi-commutative property

• Trace monoid

• Truth function

• Truth table

13.11. NOTES 51

13.11 Notes

[1] Krowne, p.1

[2] Weisstein, Commute, p.1

[3] Yark, p.1

[4] Lumpkin, p.11

[5] Gay and Shute, p.?

[6] O'Conner and Robertson, Real Numbers

[7] Cabillón and Miller, Commutative and Distributive

[8] O'Conner and Robertson, Servois

[9] Moore and Parker

[10] Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

[11] Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

[12] Axler, p.2

[13] Gallian, p.34

[14] p. 26,87

[15] Gallian p.236

[16] Gallian p.250

13.12 References

13.12.1 Books

• Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2.

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.

• Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

• Gallian, Joseph (2006). Contemporary Abstract Algebra, 6e. Boston, Mass.: Houghton Mifflin. ISBN 0-618-51471-6.

Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

• Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry, 2e. Prentice Hall. ISBN0-13-067342-0.

Abstract algebra theory. Uses commutativity property throughout book.

• Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

52 CHAPTER 13. COMMUTATIVE PROPERTY

13.12.2 Articles

• http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy OfAncient Egypt - A Response To Robert Palter. Unpublished manuscript.

Article describing the mathematical ability of ancient civilizations.

• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text.London: British Museum Publications Limited. ISBN 0-7141-0944-4

Translation and interpretation of the Rhind Mathematical Papyrus.

13.12.3 Online resources

• Hazewinkel, Michiel, ed. (2001), “Commutativity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Krowne, Aaron, Commutative at PlanetMath.org., Accessed 8 August 2007.

Definition of commutativity and examples of commutative operations

• Weisstein, Eric W., “Commute”, MathWorld., Accessed 8 August 2007.

Explanation of the term commute

• Yark. Examples of non-commutative operations at PlanetMath.org., Accessed 8 August 2007

Examples proving some noncommutative operations

• O'Conner, J J and Robertson, E F. MacTutor history of real numbers, Accessed 8 August 2007

Article giving the history of the real numbers

• Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 22 November 2008

Page covering the earliest uses of mathematical terms

• O'Conner, J J and Robertson, E F. MacTutor biography of François Servois, Accessed 8 August 2007

Biography of Francois Servois, who first used the term

Chapter 14

Comparability

See also: Comparison (mathematics)

In mathematics, any two elements x and y of a set P that is partially ordered by a binary relation ≤ are comparablewhen either x ≤ y or y ≤ x. If it is not the case that x and y are comparable, then they are called incomparable.A totally ordered set is exactly a partially ordered set in which every pair of elements is comparable.It follows immediately from the definitions of comparability and incomparability that both relations are symmetric,that is x is comparable to y if and only if y is comparable to x, and likewise for incomparability.

14.1 Notation

Comparability is denoted by the symbol ⊥, and incomparability by the symbol ||.[1] Thus, for any pair of elements xand y of a partially ordered set, exactly one of

• x ⊥ y and

• x || y

is true.

14.2 Comparability graphs

The comparability graph of a partially ordered set P has as vertices the elements of P and has as edges preciselythose pairs {x, y} of elements for which x ⊥ y.[2]

14.3 Classification

When classifying mathematical objects (e.g., topological spaces), two criteria are said to be comparable when theobjects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they arecomparable under the partial order ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobrietycriteria are not.

14.4 See also

• Strict weak ordering, a partial ordering in which incomparability is a transitive relation

53

54 CHAPTER 14. COMPARABILITY

14.5 References

“PlanetMath: partial order”. Retrieved 6 April 2010.

[1] Trotter, William T. (1992), Combinatorics and Partially Ordered Sets:Dimension Theory, Johns Hopkins Univ. Press, p. 3

[2] Gilmore, P. C.; Hoffman, A. J. (1964), “A characterization of comparability graphs and of interval graphs”, CanadianJournal of Mathematics 16: 539–548, doi:10.4153/CJM-1964-055-5.

Chapter 15

Composition of relations

In mathematics, the composition of binary relations is a concept of forming a new relation S ∘ R from two givenrelations R and S, having as its most well-known special case the composition of functions.

15.1 Definition

If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition S ◦R is the relation

S ◦R = {(x, z) ∈ X × Z | ∃y ∈ Y : (x, y) ∈ R ∧ (y, z) ∈ S}.

In other words, S ◦ R ⊆ X × Z is defined by the rule that says (x, z) ∈ S ◦ R if and only if there is an elementy ∈ Y such that xR y S z (i.e. (x, y) ∈ R and (y, z) ∈ S ).In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. The convention chosen here issuch that function composition (with the usual notation) is obtained as a special case, when R and S are functionalrelations. Some authors[1] prefer to write ◦l and ◦r explicitly when necessary, depending whether the left or the rightrelation is the first one applied.A further variation encountered in computer science is the Z notation: ◦ is used to denote the traditional (right)composition, but ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition.[2][3] This useof semicolon coincides with the notation for function composition used (mostly by computer scientists) in Categorytheory,[4] as well as the notation for dynamic conjunction within linguistic dynamic semantics.[5] The semicolonnotation (with this semantic) was introduced by Ernst Schröder in 1895.[6]

The binary relations R ⊆ X × Y are sometimes regarded as the morphisms R : X → Y in a category Rel whichhas the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. Thecategory Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of thisis found in the theory of allegories.

15.2 Properties

Composition of relations is associative.The inverse relation of S ∘ R is (S ∘ R)−1 = R−1 ∘ S−1. This property makes the set of all binary relations on a set asemigroup with involution.The compose of (partial) functions (i.e. functional relations) is again a (partial) function.If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R.If R and S are surjective, then S ∘ R is surjective, which conversely implies only the surjectivity of S.The set of binary relations on a set X (i.e. relations from X to X) together with (left or right) relation compositionforms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element.

55

56 CHAPTER 15. COMPOSITION OF RELATIONS

15.3 Join: another form of composition

Main article: Join (relational algebra)

Other forms of composition of relations, which apply to general n-place relations instead of binary relations, arefound in the join operation of relational algebra. The usual composition of two binary relations as defined here canbe obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middlecomponent.

15.4 See also• Binary relation

• Relation algebra

• Demonic composition

• Function composition

• Join (SQL)

• Logical matrix

15.5 Notes[1] Kilp, Knauer & Mikhalev, p. 7

[2] ISO/IEC 13568:2002(E), p. 23

[3] http://www.fileformat.info/info/unicode/char/2a3e/index.htm

[4] http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6

[5] http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog

[6] Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 24. ISBN 978-0-521-63107-5.A free HTML version of the book is available at http://www.cs.man.ac.uk/~{}pt/Practical_Foundations/

15.6 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

Chapter 16

Congruence relation

For the term as used in elementary geometry, see congruence (geometry).

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic struc-ture (such as a group, ring, or vector space) that is compatible with the structure. Every congruence relation has acorresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

16.1 Basic example

The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positiveinteger n , two integers a and b are called congruent modulo n , written

a ≡ b (mod n)

if a− b is divisible by n (or equivalently if a and b have the same remainder when divided by n ).for example, 37 and 57 are congruent modulo 10 ,

37 ≡ 57 (mod 10)

since 37 − 57 = −20 is a multiple of 10, or equivalently since both 37 and 57 have a remainder of 7 when dividedby 10 .Congruence modulo n (for a fixed n ) is compatible with both addition and multiplication on the integers. That is,if

a1 ≡ a2 (mod n) and b1 ≡ b2 (mod n)

then

a1 + b1 ≡ a2 + b2 (mod n) and a1b1 ≡ a2b2 (mod n)

The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the pointof view of abstract algebra, congruence modulo n is a congruence relation on the ring of integers, and arithmeticmodulo n occurs on the corresponding quotient ring.

16.2 Definition

The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions ofcongruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common

57

58 CHAPTER 16. CONGRUENCE RELATION

theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraicstructure, in the sense that the operations are well-defined on the equivalence classes.For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfyingcertain axioms. If G is a group with operation ∗, a congruence relation on G is an equivalence relation ≡ on theelements of G satisfying

g1 ≡ g2 and h1 ≡ h2 ⇒ g1 ∗ h1 ≡ g2 ∗ h2

for all g1, g2, h1, h2 ∈ G. For a congruence on a group, the equivalence class containing the identity element is alwaysa normal subgroup, and the other equivalence classes are the cosets of this subgroup. Together, these equivalenceclasses are the elements of a quotient group.When an algebraic structure includes more than one operation, congruence relations are required to be compatiblewith each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on aring must satisfy

r1 + s1 ≡ r2 + s2 and r1s1 ≡ r2s2

whenever r1 ≡ r2 and s1 ≡ s2. For a congruence on a ring, the equivalence class containing 0 is always a two-sidedideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.The general notion of a congruence relation can be given a formal definition in the context of universal algebra, afield which studies ideas common to all algebraic structures. In this setting, a congruence relation is an equivalencerelation ≡ on an algebraic structure that satisfies

μ(a1, a2, ..., an) ≡ μ(a1′, a2′, ..., an′)

for every n-ary operation μ, and all elements a1,...,an,a1′,...,an′ satisfying ai ≡ ai′ for each i.

16.3 Relation with homomorphisms

If ƒ: A→ B is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear mapbetween vector spaces), then the relation R defined by

a1 R a2 if and only if ƒ(a1) = ƒ(a2)

is a congruence relation. By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphicto the quotient of A by this congruence.

16.4 Congruences of groups, and normal subgroups and ideals

In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group(with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever:

1. Given any element a of G, a ~ a (reflexivity);

2. Given any elements a and b of G, if a ~ b, then b ~ a (symmetry);

3. Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c (transitivity);

4. Given any elements a, a' , b, and b' of G, if a ~ a' and b ~ b' , then a * b ~ a' * b' ;

5. Given any elements a and a' of G, if a ~ a' , then a−1 ~ a' −1 (this can actually be proven from the other four,so is strictly redundant).

16.5. UNIVERSAL ALGEBRA 59

Conditions 1, 2, and 3 say that ~ is an equivalence relation.A congruence ~ is determined entirely by the set {a ∈ G : a ~ e} of those elements of G that are congruent to theidentity element, and this set is a normal subgroup. Specifically, a ~ b if and only if b−1 * a ~ e. So instead of talkingabout congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruencecorresponds uniquely to some normal subgroup of G.

16.4.1 Ideals of rings and the general case

A similar trick allows one to speak of kernels in ring theory as ideals instead of congruence relations, and in moduletheory as submodules instead of congruence relations.The most general situation where this trick is possible is with Omega-groups (in the general sense allowing operatorswith multiple arity). But this cannot be done with, for example, monoids, so the study of congruence relations playsa more central role in monoid theory.

16.5 Universal algebra

The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct productA × A that is both an equivalence relation on A and a subalgebra of A × A.The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. For a givencongruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion,the quotient algebra. The function that maps every element of A to its equivalence class is a homomorphism, and thekernel of this homomorphism is ~.The lattice Con(A) of all congruence relations on an algebra A is algebraic.

16.6 See also• Table of congruences

• Linear congruence theorem

• Congruence lattice problem

16.7 References• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5discusses congruency of matrices.)

Chapter 17

Contour set

In mathematics, contour sets generalize and formalize the everyday notions of

• everything superior to something

• everything superior or equivalent to something

• everything inferior to something

• everything inferior or equivalent to something.

17.1 Formal definitions

Given a relation on pairs of elements of set X

≽ ⊆ X2

and an element x of X

x ∈ X

The upper contour set of x is the set of all y that are related to x :

{y ϶ y ≽ x}

The lower contour set of x is the set of all y such that x is related to them:

{y ϶ x ≽ y}

The strict upper contour set of x is the set of all y that are related to x without x being in this way related to anyof them:

{y ϶ (y ≽ x) ∧ ¬(x ≽ y)}

The strict lower contour set of x is the set of all y such that x is related to them without any of them being in thisway related to x :

{y ϶ (x ≽ y) ∧ ¬(y ≽ x)}

60

17.2. EXAMPLES 61

The formal expressions of the last two may be simplified if we have defined

≻ = {(a, b) ϶ (a ≽ b) ∧ ¬(b ≽ a)}

so that a is related to b but b is not related to a , in which case the strict upper contour set of x is

{y ϶ y ≻ x}

and the strict lower contour set of x is

{y ϶ x ≻ y}

17.1.1 Contour sets of a function

In the case of a function f() considered in terms of relation ▷ , reference to the contour sets of the function is implicitlyto the contour sets of the implied relation

(a ≽ b) ⇐ [f(a) ▷ f(b)]

17.2 Examples

17.2.1 Arithmetic

Consider a real number x , and the relation ≥ . Then

• the upper contour set of x would be the set of numbers that were greater than or equal to x ,

• the strict upper contour set of x would be the set of numbers that were greater than x ,

• the lower contour set of x would be the set of numbers that were less than or equal to x , and

• the strict lower contour set of x would be the set of numbers that were less than x .

Consider, more generally, the relation

(a ≽ b) ⇐ [f(a) ≥ f(b)]

Then

• the upper contour set of x would be the set of all y such that f(y) ≥ f(x) ,

• the strict upper contour set of x would be the set of all y such that f(y) > f(x) ,

• the lower contour set of x would be the set of all y such that f(x) ≥ f(y) , and

• the strict lower contour set of x would be the set of all y such that f(x) > f(y) .

It would be technically possible to define contour sets in terms of the relation

(a ≽ b) ⇐ [f(a) ≤ f(b)]

62 CHAPTER 17. CONTOUR SET

though such definitions would tend to confound ready understanding.In the case of a real-valued function f() (whose arguments might or might not be themselves real numbers), referenceto the contour sets of the function is implicitly to the contour sets of the relation

(a ≽ b) ⇐ [f(a) ≥ f(b)]

Note that the arguments to f() might be vectors, and that the notation used might instead be

[(a1, a2, . . .) ≽ (b1, b2, . . .)] ⇐ [f(a1, a2, . . .) ≥ f(b1, b2, . . .)]

17.2.2 Economic

In economics, the set X could be interpreted as a set of goods and services or of possible outcomes, the relation ≻as strict preference, and the relationship ≽ as weak preference. Then

• the upper contour set, or better set,[1] of x would be the set of all goods, services, or outcomes that were atleast as desired as x ,

• the strict upper contour set of x would be the set of all goods, services, or outcomes that were more desiredthan x ,

• the lower contour set, or worse set,[1] of x would be the set of all goods, services, or outcomes that were nomore desired than x , and

• the strict lower contour set of x would be the set of all goods, services, or outcomes that were less desired thanx .

Such preferences might be captured by a utility function u() , in which case

• the upper contour set of x would be the set of all y such that u(y) ≥ u(x) ,

• the strict upper contour set of x would be the set of all y such that u(y) > u(x) ,

• the lower contour set of x would be the set of all y such that u(x) ≥ u(y) , and

• the strict lower contour set of x would be the set of all y such that u(x) > u(y) .

17.3 Complementarity

On the assumption that≽ is a total ordering ofX , the complement of the upper contour set is the strict lower contourset.

X2\ {y ϶ y ≽ x} = {y ϶ x ≻ y}

X2\ {y ϶ x ≻ y} = {y ϶ y ≽ x}

and the complement of the strict upper contour set is the lower contour set.

X2\ {y ϶ y ≻ x} = {y ϶ x ≽ y}

X2\ {y ϶ x ≽ y} = {y ϶ y ≻ x}

17.4. SEE ALSO 63

17.4 See also• Epigraph

• Hypograph

17.5 References[1] Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilib-

rium Theory. Springer. p. 35.

17.6 Bibliography• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M62471995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)

Chapter 18

Coreflexive relation

In mathematics, a coreflexive relation is a binary relation that is a subset of the identity relation.[1] Thus if a is relatedto b (aRb) then a is equal to b (a = b), but if c is equal to d (c = d) it does not necessarily hold that c is related to d(cRd).In mathematical notation, this is:

∀a, b ∈ X, aRb⇒ a = b.

The identity relation is coreflexive by definition. Any relation that is coreflexive is thus a subset of the identity relation.For example, consider the relation R as “equal to and odd”. Over the set of positive integers, the relationship R holdsover the pairs {(1, 1), (3, 3), ...} but does not hold over {(2, 2), (4, 4), ...}.

18.1 Notes[1] Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions

to Hash Tables. In Mathematics of Program Construction (p. 337).

64

Chapter 19

Quasi-commutative property

In mathematics, the quasi-commutative property is an extension or generalization of the general commutativeproperty. This property is used in certain specific applications with various definitions.

19.1 Applied to matrices

Two matrices p and q are said to have the commutative property whenever

pq = qp

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices x and y

xy − yx = z

satisfy the quasi-commutative property whenever z satisfies the following properties:

xz = zx

yz = zy

An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In thismechanics, p and q are infinite matrices corresponding respectively to the momentum and position variables of aparticle.[1] These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unitmatrix, where ħ is the reduced Planck constant.

19.2 Applied to functions

A function f, defined as follows:

f : X × Y → X

is said to be quasi-commutative[2] if for all x ∈ X and for all y1, y2 ∈ Y ,

f(f(x, y1), y2) = f(f(x, y2), y1)

65

66 CHAPTER 19. QUASI-COMMUTATIVE PROPERTY

19.3 See also• Commutative property

• Accumulator (cryptography)

19.4 References[1] Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.

[2] Benaloh, J., & De Mare, M. (1994, January). One-way accumulators: A decentralized alternative to digital signatures. InAdvances in Cryptology—EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg.

19.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 67

19.5 Text and image sources, contributors, and licenses

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han, Roisterer, Urhixidur, Spug, Firsfron, Nivaca, Bgwhite, KSchutte, Imz, Mhss, Lacatosias, Mets501, Simeon, Gregbard, Egriffin,Classicalecon, Auntof6, Alexbot, Sun Creator, Dthomsen8, Addbot, Frehley, Tempodivalse, MikeyMouse10, ISpamThisSite3, Widr,Cabbagist and Anonymous: 28

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• Alternatization Source: https://en.wikipedia.org/wiki/Alternatization?oldid=674572806Contributors: Rich Farmbrough, Tompw, Vipul,BD2412, Malcolma, Elonka, Mclay1, David Eppstein, Haseldon, LokiClock, JP.Martin-Flatin, Erik9bot, Paine Ellsworth, Quondum,Solomon7968 and Anonymous: 1

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68 CHAPTER 19. QUASI-COMMUTATIVE PROPERTY

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• Coreflexive relation Source: https://en.wikipedia.org/wiki/Coreflexive_relation?oldid=316718782 Contributors: Michael Hardy, Xez-beth, H8erade, Ctylikkehl, Kms15 and Anonymous: 1

• Quasi-commutative property Source: https://en.wikipedia.org/wiki/Quasi-commutative_property?oldid=643466557Contributors: MichaelHardy, Dirac66, Marius siuram and OccultZone

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19.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 69

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

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