Mathematical relations abcdeFrom 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 Simpsons alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
15 Composition of relations 5515.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Coreexive relation 6418.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
19 Demonic composition 6519.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
20 Dense order 6620.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
21 Dependence relation 6721.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
22 Dependency relation 6822.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
23 Directed set 70
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23.1 Equivalent denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
24 Equality (mathematics) 7324.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
24.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
24.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
25 Equivalence class 7725.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
26 Equivalence relation 8226.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
26.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
CONTENTS vii
26.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
26.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
26.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
26.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
26.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8926.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8926.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
27 Euclidean relation 9127.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.2 Relation to transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
28 Exceptional isomorphism 9228.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
28.1.1 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.1.2 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.1.3 Alternating groups and symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.1.4 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.1.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.1.6 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
28.2 Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
29 Quasi-commutative property 9729.1 Applied to matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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29.2 Applied to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 99
29.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
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 y or I can't y' 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 denition, the'accessibility relation' is the idea that modal statements, like 'its 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 diculty 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
1
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 LogicThe 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 rsttwo 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 sucient' 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 'sucient condition' a condition that is good enough for something else. Beingliterate, for instance, is a 'sucient' condition for learning about the accessibility relation.' In other words, its goodenough to be literate in order to learn about the 'accessibility relation,' however it may not be 'necessary' because therelation could be learned in dierent ways (like through speech). Aside from 'necessary and sucient,' 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 rst 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 rst modal axiom, p $ ::p could stand for: 'Its necessary that Iwalk outside if and only if its 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 'Its necessary that I walk outside,' this is the same as saying that 'Its not possible that it is notthe case that I walk outside.' Furthermore, when I say that 'Its not possible that it is not the case that I walk outside,'this is the same as saying that 'Its necessary that I walk outside.'Example 2:p stands for 'The apple is red.'So using the symbols and their meaning described above, the rst modal axiom, p $ ::p could stand for: 'Itsnecessary that the apple is red if and only if its not possible that it is not the case that the apple is red.'And when I say that 'Its necessary that the apple is red,' this is the same as saying that 'Its not possible that it is notthe case that the apple is red.' Furthermore, when I say that 'Its not possible that it is not the case that the apple isred,' this is the same as saying that 'Its 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: 'Itspossible 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 dierent 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 quantication 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 worldsis the 'accessibility relation.'
4 CHAPTER 1. ACCESSIBILITY RELATION
The relation of accessibility can now be dened 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 specic, any formula, axiom like (p _q)! (p _ q) can be translated into aformula of rst-order logic through standard translation. That rst-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 dened 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 specic types of R are:Reexive, or *Axiom (T) above:If R is reexive, every world is accessible to itself. Reexivity 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 its true,which isn't necessarily the case in worlds that aren't accessible to themselves. Without the reexivity condition, Acan be necessary at a world where its false, if that world isn't accessible to itself; thus axiom Tthat A at a worldimplies A is true at that worldfollows from reexivity.Transitive, or *Axiom (4) above:If R is transitive, any world accessible to any world w0 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 w0 accessible to w , including every world w00accessible to any w0 , and every world accessible to any w00 , etc., so when A is true at w , its also true at everyw0 and every w00 , 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 wif and only if, for every world w0 accessible to w , there is a world w00 accessible to w0 at which A is true. If Ais true at a world w0 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 ( w0 ) 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 w0 is accessible to world w , w is accessible to w0 . If A is true at w , then atevery w0 accessible to w , there is a world ( w ) accessible to w0 at which A is true, so A is possible at all w0 ,and thus its necessary at w that A is possible, which is axiom B.
1.4 Philosophical ApplicationsOne 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 exemplied 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 dierent theories of the nature of physical laws will resultin dierent answers to this question. (Notice however that if the objection raised earlier is true, each dierent 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 claimis possible P in the world w00 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 w0 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 w0 = w. We can ask whether the relation is transitive. If I knows nothing that rules out the possibility that w0 = w andknows nothing that rules the possibility that w00 = w0 , it does not follow that I knows nothing which rules out thehypothesis that w00 = 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 reexivity and symmetry will not.
1.5 Computer Science ApplicationsIn modeling a computation, a 'possible world' can be a possible computer state. Given the current computer state, youmight dene the accessible possible worlds to be all future possible computer states, or to be all possible immediatenext computer states (assuming a discrete computer). Either choice denes a particular 'accessibility relation' givingrise to a particular modal logic suited specically 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 rst order logic, Logique et Analyse, No. 117118, pp. 6979,1987.
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 Quantied Modal Logic (subscription required), The Journal of Philosophy,Vol. LXV, No. 5 (1968-03-07), pp. 113126, 1968
List of Logic Systems List of most of the more popular modal logics.
Chapter 2
Allegory (category theory)
In the mathematical eld 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 dierent sorts. Allegories arealso useful in dening 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 rst do S, then doR".
2.1 DenitionAn allegory is a category in which
every morphism R:XY is associated with an anti-involution, i.e. a morphism R:YX; and every pair of morphisms R,S:XY with common domain/codomain is associated with an intersection, i.e. amorphism RS:XY
all such that
intersections are idempotent (RR=R), commutative (RS=SR), and associative (RS)T=R(ST); anti-involution distributes over composition ((RS)=SR) and intersection ((RS)=SR); composition is semi-distributive over intersection (R(ST)RSRT, (RS)TRTST); and the modularity law is satised: (RST(RTS)S).
Here, we are abbreviating using the order dened by the intersection: "RS" means "R=RS".A rst example of an allegory is the category of sets and relations. The objects of this allegory are sets, and amorphismXY 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 categoriesIn a category C, a relation between objects X, Y is a span of morphisms XRY that is jointly-monic. Two suchspans XSY and XTY are considered equivalent when there is an isomorphism between S and T that makeeverything commute, and strictly speaking relations are only dened 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
7
8 CHAPTER 2. ALLEGORY (CATEGORY THEORY)
same thing as a monomorphism into XY (or an equivalence class of such). In the presence of pullbacks and a properfactorization system, one can dene the composition of relations. The composition of XRYSZ is found byrst pulling back the cospan RYS and then taking the jointly-monic image of the resulting span XRSZ.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 XXX.Recall that a regular category is a category with nite 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 dened by turning the source/target of the relation around, and intersectionsare intersections of subobjects, computed by pullback.
2.2.2 Maps in allegories, and tabulationsA morphism R in an allegory A is called a map if it is entire (1RR) and deterministic (RR1). 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 CMap(Rel(C)). In particular, a morphism in Map(Rel(Set)) is just an ordinary set function.In an allegory, a morphism R:XY is tabulated by a pair of maps f:ZX, g:ZY if gf=R and ffgg=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, ARel(Map(A)).
2.2.3 Unital allegories and regular categories of mapsA unit in an allegory is an object U for which the identity is the largest morphism UU, 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 allegoryAdditional 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 specically 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 Denitions
3.1.1 Alternating bilinear map
Let S be a set, A be an abelian group, and : S S ! A be a bilinear map. Then is said to be an alternatingbilinear map if
8x 2 S; (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 dened as mapsbetween vector spaces or modules, we distinguish two cases.
Vector spacesLet V be a vector space over a eld K, and : V V ! K be a bilinear form. Then is said to be analternating bilinear form if [1][2]
8x 2 V; (x; x) = 0:
ModulesLet M be a module over a ring R, and : M M ! R be a bilinear form. Then is said to be analternating bilinear form if
8x 2M; (x; x) = 0:
9
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 dened as maps between vector spaces or modules, we distinguish two cases.
Vector spacesLet V be a vector space over a eld K, and : V V ::: V ! K be a multilinear form. Then issaid to be an alternating multilinear form if
8x1; x2; :::; xn 2 V; 8i 2 f1; 2; :::; n 1g; xi = xi+1 =) (x1; x2; :::; xn) = 0:
ModulesLet M be a module over a ring R, and : M M ::: M ! R be a multilinear form. Then issaid to be an alternating multilinear form if [3]
8x1; x2; :::; xn 2M; 8i 2 f1; 2; :::; n 1g; xi = xi+1 =) (x1; x2; :::; xn) = 0:
3.1.4 Alternatization of a bilinear map
Let S be a set, A be an abelian group, and : S S ! A be a bilinear map. 8x; y 2 S; the alternatization of themap is the map
: S S ! A
(x; y) 7! (x; y) (y; x):
3.2 Example The Lie bracket is an alternating bilinear form.
3.3 Properties Every alternating multilinear form is antisymmetric:[4]
8x; y 2 S; (x; y) + (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 dened in a dierent way, see below.Ancestral relations make their rst appearance in Frege's Begrisschrift. Frege later employed them in his Grundge-setze as part of his denition of the nite cardinals. Hence the ancestral was a key part of his search for a logicistfoundation of arithmetic.
4.1 DenitionThe numbered propositions below are taken from his Begrisschrift 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)! PyFrege dened 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 : ` aRb$ 8F [8x(aRx! Fx) ^ 8x8y(Fx ^ xRy ! Fy)! Fb]The ancestral is a transitive relation:
98 : ` (aRb ^ bRc)! aRcLet the notation I(R) denote that R is functional (Frege calls such relations many-one):
115 : ` I(R)$ 8x8y8z[(xRy ^ xRz)! y = z]If R is functional, then the ancestral of R is what nowadays is called connected:
133 : ` (I(R) ^ aRb ^ aRc)! (bRc _ b = c _ cRb)
4.2 Relationship to transitive closureThe Ancestral relation R is equal to the transitive closure R+ of R . Indeed, R is transitive (see 98 above), Rcontains 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 nally, R is contained in R+ (indeed, assume aRb ; take the property Fx to beaR+x ; then the two premises, 8x(aRx ! Fx) and 8x8y(Fx ^ xRy ! Fy) , are obviously satised; therefore,Fb , which means aR+b , by our choice of F ). See also Booloss book below, page 8.
12
4.3. DISCUSSION 13
4.3 DiscussionPrincipia Mathematica made repeated use of the ancestral, as does Quines (1951) Mathematical Logic.However, it is worth noting that the ancestral relation cannot be dened in rst-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 t itstheorems).
4.4 See also Begrisschrift 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: "Freges 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:
8a; b 2 X; R(a; b) ^R(b; a) ) a = b
or, equivalently,
8a; b 2 X; R(a; b) ^ a 6= 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 denition. 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 onrelation on biological species).Antisymmetry is dierent from asymmetry, which requires both antisymmetry and irreexivity.
5.1 ExamplesThe 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:
8a; b 2 X; aRb ) :(bRa)
6.1 ExamplesAn example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarskis 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 irreexive) 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 irreexive.[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 irreexive:[3] if a R b and b R a, transitivity gives a R a,contradicting irreexivity.
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] Flaka, V.; Jeek, 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 innitary 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 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 establishedFrass'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 Denition
It is common in bqo theory to write x for the sequence x with the rst term omitted. Write [!]7.3 Simpsons alternative denition
Simpson introduced an alternative denition of bqo in terms of Borel maps [!]! ! Q , where [!]! , the set of innitesubsets of ! , 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]! ! Q forsome innite subset A of ! . A Q -array f is bad if f(X) 6Q f(X) for every X 2 [A]! ; 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 theoremsMany major results in bqo theory are consequences of the Minimal Bad Array Lemma, which appears in Simpsonspaper[5] as follows. See also Lavers paper,[6] where the Minimal Bad Array Lemma was rst stated as a result. Thetechnique was present in Nash-Williams original 1965 paper.Suppose (Q;Q) is a quasi-order. A partial ranking 0 of Q is a well-founded partial ordering of Q such thatq 0 r ! q Q r . For bad Q -arrays (in the sense of Simpson) f : [A]! ! Q and g : [B]! ! Q , dene:
g f if B A and g(X) 0 f(X) every for X 2 [B]!
g
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 dierent 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 UsageBidirectional 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 VocabularyA 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 conguration 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
2
3
4
YD
B
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 rst 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 nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.
9.1 DenitionFor more details on notation, see Function (mathematics) Notation.
For a pairing between X and Y (where Y need not be dierent 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 teamConsider 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 pairingis given by which player is in what position in this order. Property (1) is satised since each player is somewhere inthe list. Property (2) is satised 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 classroomIn 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 (-/2, /2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-/2, /2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-/2, /2) was made larger to include an integer multiple of /2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of /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;+1) , 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;+1) , then h becomes bijective; its inverseis the positive square root function.
9.4 InversesA bijection f with domainX (functionally indicated by f: XY) also denes 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 satises 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 dened 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 CompositionThe 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 =(f1) (g1) .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 cardinalityIf X and Y are nite 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 denition of same number ofelements (equinumerosity), and generalising this denition to innite sets leads to the concept of cardinal number,a way to distinguish the various sizes of innite sets.
9.7. PROPERTIES 25
X1
2
3
YD
B
C
A
ZP
Q
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 |f1(B)| = |B|.
If X and Y are nite 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 nite 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 setnamely, n!.
9.8 Bijections and category theoryBijections 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 functionsThe 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 undened for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. dened everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]
Another way of dening 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:AB, where A is a subset of Aand likewise BB.[3]
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is theMbius transformation simply dened 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 AxGrothendieck theorem
9.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satises 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). TheMbius InverseMonoid. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.
9.13. REFERENCES 27
9.13 ReferencesThis 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 Mathematicians 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:
8x; y 2 A; f(x) = f(y)) x = y:Or, equivalently (using logical transposition),8x; y 2 A; x 6= y ) f(x) 6= 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:
8y 2 B; 9x 2 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 InjectionMain 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 denition is the following.
The function f : A! B is injective i for all a; b 2 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
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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 gure at right.
Every embedding is injective.
10.2 SurjectionMain 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 denition is the following.
The function f : A! B is surjective i for all b 2 B , there is a 2 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 xed image, every surjection induces a bijection dened 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-denedfunction 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 gure.
30 CHAPTER 10. BIJECTION, INJECTION AND SURJECTION
X1
2
3
4
YD
B
C
A
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Q
R
Surjective composition: the rst function need not be surjective.
10.3 BijectionMain 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 i for all b 2 B , there is a unique a 2 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 gure 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 CardinalitySuppose you want to dene 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 candene 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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3
YD
B
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A
ZP
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Bijective composition: the rst function need not be surjective and the second function need not be injective.
10.5 ExamplesIt is important to specify the domain and codomain of each function since by changing these, functions which wethink of as the same may have dierent jectivity.
10.5.1 Injective and surjective (bijective)
For every set A the identity function idA and thus specically R! R : x 7! x .
R+ ! R+ : x 7! x2 and thus also its inverse R+ ! R+ : x 7! px .
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 dene 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 theoryIn the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms,and isomorphisms, respectively.
10.8 HistoryThis 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 nitary relation. For a morecombinatorial viewpoint, see theo
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