Equivalence classFrom Wikipedia, the free encyclopedia

Contents

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

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

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

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

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

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

2 Equality (mathematics) 112.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3 Equivalence class 153.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Equivalence relation 204.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Homogeneous space 29

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5.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Intransitivity 336.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Matrilineality 367.1 Early human kinship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Matrilineal surname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 Cultural patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.3.1 Clan names vs. surnames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.2 Care of children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.3 A feminist and patriarchal relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.4 Matrilineality in specific ethnic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4.1 In America . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4.2 In Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.4.3 In Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.4.4 In Oceania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.5 Matrilineal identification within Judaism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.6 In mythology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8 Partition of a set 498.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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8.4 Refinement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

9 Quotient category 579.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10 Quotient ring 5910.1 Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

10.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.6 Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

11 Quotient space (linear algebra) 6311.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

11.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12 Quotient space (topology) 6612.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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12.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13 Reflexive relation 7013.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

14 Semigroup 7414.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

14.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.8 Semigroup methods in partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.13Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

15 Set (mathematics) 8215.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8315.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8315.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

15.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

15.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

15.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

vi CONTENTS

15.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

15.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16 Symmetric relation 9416.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

16.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 9516.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

17 Transitive closure 9717.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

18 Transitive relation 10118.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10118.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

18.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10218.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

CONTENTS vii

18.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

18.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 104

18.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10718.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Chapter 1

Binary relation

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

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A A. More generally, a binary relation between two sets A and B is a subsetof A B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include 4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", anddivides 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 onZZZ 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 Russells paradox.

1.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X Y for the set of pairs of G.The order of the elements in each pair 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.

1

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2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, 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 functions 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 surjectivityor being ontoas a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

1.1.2 Example

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

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

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

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

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

1.2 Special types of binary relations

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

Uniqueness properties:

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

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

https://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Range_(mathematics)https://en.wikipedia.org/wiki/Surjectionhttps://en.wikipedia.org/wiki/Restriction_(mathematics)https://en.wikipedia.org/wiki/Composition_of_relationshttps://en.wikipedia.org/wiki/Inverse_relationhttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Partial_function

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

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

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

Totality properties:

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

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

Uniqueness and totality properties:

https://en.wikipedia.org/wiki/Multivalued_function

4 CHAPTER 1. BINARY RELATION

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

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

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

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

1.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR1R.[11]

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

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

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

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term rectangular to denote any heterogeneous relation whatsoever.[6]

1.3 Relations over a set

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

reflexive: for all x in X it holds that xRx. For example, greater than or equal to () is a reflexive relation butgreater 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.

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1.4. OPERATIONS ON BINARY RELATIONS 5

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

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

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

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

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

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

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

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

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

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

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

1.4 Operations on binary relations

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

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

Intersection: R S X Y, defined as R S = { (x, y) | (x, y) R and (x, y) S }.

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

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

https://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Asymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Total_relationhttps://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relationshttps://en.wikipedia.org/wiki/Trichotomy_(mathematics)https://en.wikipedia.org/wiki/Euclidean_relationhttps://en.wikipedia.org/wiki/Euclidean_relationhttps://en.wikipedia.org/wiki/Euclidean_relationhttps://en.wikipedia.org/wiki/Binary_relation#difunctionalhttps://en.wikipedia.org/wiki/Binary_relation#difunctionalhttps://en.wikipedia.org/wiki/Class_(set_theory)https://en.wikipedia.org/wiki/Ordinal_numberhttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Partial_equivalence_relationhttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Total_orderhttps://en.wikipedia.org/wiki/Least_elementhttps://en.wikipedia.org/wiki/Well-orderhttps://en.wikipedia.org/wiki/Composition_of_relationshttps://en.wikipedia.org/wiki/Composition_of_functions

6 CHAPTER 1. BINARY RELATION

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

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

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

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

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

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

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

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

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

1.4.1 Complement

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

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

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

If a relation is symmetric, the complement is too.

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

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

The complement of the inverse has these same properties.

1.4.2 Restriction

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

https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Inverse_relationhttps://en.wikipedia.org/wiki/Duality_(order_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Irreflexivehttps://en.wikipedia.org/wiki/Transitive_closurehttps://en.wikipedia.org/wiki/Transitive_reductionhttps://en.wikipedia.org/wiki/Preorderhttps://en.wikipedia.org/wiki/Reflexive_transitive_symmetric_closurehttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Strict_weak_orderhttps://en.wikipedia.org/wiki/Restriction_(mathematics)https://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Irreflexive_relationhttps://en.wikipedia.org/wiki/Symmetric_relationhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Asymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Total_relationhttps://en.wikipedia.org/wiki/Binary_relation#Relations_over_a_sethttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Total_orderhttps://en.wikipedia.org/wiki/Strict_weak_orderhttps://en.wikipedia.org/wiki/Strict_weak_order#Total_preordershttps://en.wikipedia.org/wiki/Equivalence_relation

1.5. SETS VERSUS CLASSES 7

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

1.4.3 Algebras, categories, and rewriting systems

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

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

1.5 Sets versus classes

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

1.6 The number of binary relations

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

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

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

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

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

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

https://en.wikipedia.org/wiki/Completeness_(order_theory)https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Upper_boundhttps://en.wikipedia.org/wiki/Supremumhttps://en.wikipedia.org/wiki/Algebraic_structurehttps://en.wikipedia.org/wiki/Relation_algebrahttps://en.wikipedia.org/wiki/Relation_algebrahttps://en.wikipedia.org/wiki/Relational_algebrahttps://en.wikipedia.org/wiki/Finitary_relationhttps://en.wikipedia.org/wiki/Finite_sethttps://en.wikipedia.org/wiki/Many-sortedhttps://en.wikipedia.org/wiki/Category_of_relationshttps://en.wikipedia.org/wiki/Abstract_rewriting_systemhttps://en.wikipedia.org/wiki/Abstract_rewriting_systemhttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Russell%2527s_paradoxhttps://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theoryhttps://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theoryhttps://en.wikipedia.org/wiki/Proper_classhttps://oeis.org/A002416https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://en.wikipedia.org/wiki/Partially_ordered_set#Strict_and_non-strict_partial_ordershttps://en.wikipedia.org/wiki/Partition_of_a_sethttps://en.wikipedia.org/wiki/Bell_number

8 CHAPTER 1. BINARY RELATION

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

1.7 Examples of common binary relations

order relations, including strict orders:

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

equivalence relations:

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

dependency relation, a finite, symmetric, reflexive relation.

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

1.8 See also

Confluence (term rewriting)

Hasse diagram

Incidence structure

Logic of relatives

Order theory

Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 13301331. 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. 299300. ISBN 978-0-387-74758-3.

https://en.wikipedia.org/wiki/Binary_relation#Complementhttps://en.wikipedia.org/wiki/4-tuplehttps://en.wikipedia.org/wiki/Binary_relation#Operations_on_binary_relationshttps://en.wikipedia.org/wiki/Order_relationhttps://en.wikipedia.org/wiki/Strict_orderhttps://en.wikipedia.org/wiki/Greater_thanhttps://en.wikipedia.org/wiki/Less_thanhttps://en.wikipedia.org/wiki/Divideshttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Equality_(mathematics)https://en.wikipedia.org/wiki/Parallel_(geometry)https://en.wikipedia.org/wiki/Affine_spacehttps://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Dependency_relationhttps://en.wikipedia.org/wiki/Independency_relationhttps://en.wikipedia.org/wiki/Confluence_(term_rewriting)https://en.wikipedia.org/wiki/Hasse_diagramhttps://en.wikipedia.org/wiki/Incidence_structurehttps://en.wikipedia.org/wiki/Logic_of_relativeshttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Triadic_relationhttp://books.google.co.uk/books?id=azS2ktxrz3EC&pg=PA1331&hl=en&sa=X&ei=glo6T_PmC9Ow8QPvwYmFCw&ved=0CGIQ6AEwBg#v=onepage&f=falsehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-262-59020-4https://en.wikipedia.org/wiki/Patrick_Suppeshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-61630-4https://en.wikipedia.org/wiki/Raymond_Smullyanhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-486-47484-7https://en.wikipedia.org/wiki/Azriel_Levyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-42079-5https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-74758-3

1.10. REFERENCES 9

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. xxi. 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. 1921. 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. 2122. ISBN 978-3-89675-629-9.

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

[9] Ms, Stephan (2007), Reasoning on Spatial Semantic Integrity Constraints, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 1923, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285302, 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 Bchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 3537. ISBN 978-1-4613-8853-1.

[16] M. E. Mller (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] 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). 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: 3033..

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

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

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

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

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

https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-6164-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-67995-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-13-460643-9https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-460643-9https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-89675-629-9https://en.wikipedia.org/wiki/Gunther_Schmidthttps://en.wikipedia.org/wiki/Special:BookSources/9780521762687https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-3-540-74788-8_18https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-211-82971-4https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-3-540-27778-1_15https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-23990-1https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-6588-0https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-6588-0https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-3-662-44124-4_7https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-662-44123-7https://en.wikipedia.org/wiki/Julius_Richard_B%C3%BCchihttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-4613-8853-1https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-19021-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-67995-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-534-39900-2http://books.google.com/books?id=_H_nJdagqL8C&pg=PA158http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdfhttp://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdfhttps://en.wikipedia.org/wiki/Special:BookSources/0125976801https://en.wikipedia.org/wiki/Alfred_Tarskihttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-8218-1041-3https://en.wikipedia.org/wiki/Special:BookSources/3110152487https://en.wikipedia.org/wiki/Gunther_Schmidthttps://en.wikipedia.org/wiki/Special:BookSources/9780521762687

10 CHAPTER 1. BINARY RELATION

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

http://www.encyclopediaofmath.org/index.php?title=p/b016380https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4

Chapter 2

Equality (mathematics)

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

2.1 Etymology

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

2.2 Types of equalities

2.2.1 Identities

Main article: Identity (mathematics)

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

2.2.2 Equalities as predicates

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

2.2.3 Congruences

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

2.2.4 Equations

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

11

https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Mathematical_expressionhttps://en.wikipedia.org/wiki/Mathematical_objecthttps://en.wikipedia.org/wiki/Equals_signhttps://en.wikipedia.org/wiki/Etymologyhttps://en.wiktionary.org/wiki/aequalis#Latinhttps://en.wiktionary.org/wiki/aequus#Latinhttps://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Variable_(mathematics)https://en.wikipedia.org/wiki/Proposition_(mathematics)https://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Predicate_(mathematical_logic)https://en.wikipedia.org/wiki/Truth_valuehttps://en.wikipedia.org/wiki/Computer_programminghttps://en.wikipedia.org/wiki/Relational_operatorhttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Geometric_shapehttps://en.wikipedia.org/wiki/Equationhttps://en.wikipedia.org/wiki/Unit_circle

12 CHAPTER 2. EQUALITY (MATHEMATICS)

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

2.2.5 Equivalence relations

Main article: Equivalence relation

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

2.3 Logical formalizations of equality

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

x = y x z y z, andx = y z x z y.[1]

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

2.4 Logical formulations

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

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

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

2.5 Some basic logical properties of equality

The substitution property states:

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

https://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Symmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Peano_axiomshttps://en.wikipedia.org/wiki/Axiom_of_extensionalityhttps://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttps://en.wikipedia.org/wiki/Azriel_L%C3%A9vyhttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Substitution_(algebra)https://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Undecidable_problemhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Arithmetic_operationhttps://en.wikipedia.org/wiki/Logarithmhttps://en.wikipedia.org/wiki/Exponential_functionhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Identity_(philosophy)https://en.wikipedia.org/wiki/Identity_of_indiscernibleshttps://en.wikipedia.org/wiki/Axiomhttps://en.wikipedia.org/wiki/Property_(philosophy)https://en.wikipedia.org/wiki/Given_anyhttps://en.wikipedia.org/wiki/Material_conditionalhttps://en.wikipedia.org/wiki/Predicate_(mathematics)https://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Theoremhttps://en.wikipedia.org/wiki/For_anyhttps://en.wikipedia.org/wiki/Material_conditionalhttps://en.wikipedia.org/wiki/Well-formed_formula

2.6. RELATION WITH EQUIVALENCE AND ISOMORPHISM 13

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

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

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

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

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

The reflexive property states:

For any quantity a, a = a.

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

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

The transitive property states:

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

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

2.6 Relation with equivalence and isomorphism

See also: Equivalence relation and Isomorphism

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

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

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

A 7 1,B 7 2,C 7 3.

However, there are other choices of isomorphism, such as

A 7 3,B 7 2,C 7 1,

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

https://en.wikipedia.org/wiki/First-order_logichttps://en.wikipedia.org/wiki/Schema_(logic)https://en.wikipedia.org/wiki/Functional_predicatehttps://en.wikipedia.org/wiki/Functional_predicatehttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Division_by_zerohttps://en.wikipedia.org/wiki/0_(number)https://en.wikipedia.org/wiki/For_anyhttps://en.wikipedia.org/wiki/Mathematical_proofhttps://en.wikipedia.org/wiki/For_anyhttps://en.wikipedia.org/wiki/Material_conditionalhttps://en.wikipedia.org/wiki/For_anyhttps://en.wikipedia.org/wiki/Material_conditionalhttps://en.wikipedia.org/wiki/And_(logic)https://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Approximationhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Difference_(mathematics)https://en.wikipedia.org/wiki/Almost_everywherehttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Fraction_(mathematics)https://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Quotient_sethttps://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Isomorphism#Relation_with_equalityhttps://en.wikipedia.org/wiki/Category_theory

14 CHAPTER 2. EQUALITY (MATHEMATICS)

2.7 See also Equals sign

Inequality

Logical equality

Extensionality

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

[2] (Mazur 2007)

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

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

https://en.wikipedia.org/wiki/Equals_signhttps://en.wikipedia.org/wiki/Inequality_(mathematics)https://en.wikipedia.org/wiki/Logical_equalityhttps://en.wikipedia.org/wiki/Extensionalityhttps://en.wikipedia.org/wiki/Azriel_L%C3%A9vyhttps://en.wikipedia.org/wiki/Equality_(mathematics)#CITEREFMazur2007https://en.wikipedia.org/wiki/Barry_Mazurhttp://www.math.harvard.edu/~mazur/preprints/when_is_one.pdfhttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Garrett_Birkhoff

Chapter 3

Equivalence class

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

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

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

3.1 Notation and formal definition

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

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

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

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

15

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16 CHAPTER 3. EQUIVALENCE CLASS

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

[a] = {x X | a x}

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

3.2 Examples If X is the set of all cars, and ~ is the equivalence relation has the same color as. then one particular equivalenceclass consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

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

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

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

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

3.3 Properties

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

It follows from the properties of an equivalence relation that

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

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

https://en.wikipedia.org/wiki/Surjective_maphttps://en.wikipedia.org/wiki/Injective_maphttps://en.wikipedia.org/wiki/Section_(category_theory)https://en.wikipedia.org/wiki/Canonical_formhttps://en.wikipedia.org/wiki/Modular_arithmetichttps://en.wikipedia.org/wiki/Remainderhttps://en.wikipedia.org/wiki/Euclidean_divisionhttps://en.wikipedia.org/wiki/Cardinalityhttps://en.wikipedia.org/wiki/Modular_arithmetichttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Even_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Field_of_fractionshttps://en.wikipedia.org/wiki/Field_of_fractionshttps://en.wikipedia.org/wiki/Integral_domainhttps://en.wikipedia.org/wiki/Euclidean_planehttps://en.wikipedia.org/wiki/Parallel_lineshttps://en.wikipedia.org/wiki/Parallel_(geometry)#Reflexive_varianthttps://en.wikipedia.org/wiki/Parallel_(geometry)#Reflexive_varianthttps://en.wikipedia.org/wiki/Point_at_infinityhttps://en.wikipedia.org/wiki/Disjoint_setshttps://en.wikipedia.org/wiki/Partition_of_a_set

3.4. GRAPHICAL REPRESENTATION 17

x y

[x] = [y]

[x] [y] = .

3.4 Graphical representation

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

3.5 Invariants

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

3.6 Quotient space in topology

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

3.7 See also Equivalence partitioning, a method for devisin