Equivalence classFrom Wikipedia, the free encyclopedia

Contents

1 Asymmetric relation 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Binary relation 32.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Divisor 133.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Further notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Equality (mathematics) 184.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Equivalence class 225.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Equivalence relation 276.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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6.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Homogeneous space 367.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Intransitivity 408.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Material conditional 439.1 Definitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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9.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10 Matrilineality 4810.1 Early human kinship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2 Matrilineal surname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.3 Cultural patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.3.1 Clan names vs. surnames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.3.2 Care of children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.3.3 A feminist and patriarchal relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.4 Matrilineality in specific ethnic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.4.1 In America . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.4.2 In Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.4.3 In Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.4.4 In Oceania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10.5 Matrilineal identification within Judaism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.6 In mythology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 Partition of a set 6111.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.4 Refinement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12 Preorder 6912.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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12.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13 Quotient category 7313.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

14 Quotient ring 7514.1 Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

14.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.6 Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

15 Quotient space (linear algebra) 7915.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

15.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

15.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

16 Quotient space (topology) 8216.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

16.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

16.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vi CONTENTS

17 Semigroup 8617.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

17.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

17.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.8 Semigroup methods in partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.13Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

18 Set (mathematics) 9418.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9518.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9518.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

18.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9718.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

18.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

18.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10018.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10018.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

19 Subset 106

CONTENTS vii

19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

20 Symmetric relation 11020.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

20.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11020.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

20.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 11120.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11120.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

21 Total relation 11321.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11321.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11321.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11321.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

22 Transitive closure 11522.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

23 Transitive relation 11923.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11923.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

23.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12023.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12023.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

23.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12023.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

viii CONTENTS

23.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

23.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

24 Weak ordering 12224.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12324.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

24.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12324.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

24.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

24.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

24.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12724.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12724.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 128

24.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12824.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13124.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Asymmetric relation

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

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

In mathematical notation, this is:

a, b X, aRb (bRa)

1.1 Examples

An 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 irreflexive) is also notasymmetric.Asymmetric is not the same thing as not symmetric": a relation can be neither symmetric nor asymmetric, such as, or can be both, only in the case of the empty relation (vacuously).

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

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

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

1.3 See also Symmetric relation

1

https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Inequality_(mathematics)https://en.wikipedia.org/wiki/Tarski%2527s_axiomatization_of_the_realshttps://en.wikipedia.org/wiki/Tarski%2527s_axiomatization_of_the_realshttps://en.wikipedia.org/wiki/Total_relationhttps://en.wikipedia.org/wiki/Strict_subsethttps://en.wikipedia.org/wiki/Strict_partial_orderhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Intransitivityhttps://en.wikipedia.org/wiki/Rock-paper-scissorshttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Symmetric_relationhttps://en.wikipedia.org/wiki/Vacuous_truthhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Binary_relation#Restrictionhttps://en.wikipedia.org/wiki/Inverse_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Symmetric_relation

2 CHAPTER 1. ASYMMETRIC RELATION

Antisymmetric relation

Symmetry

Symmetry in mathematics

1.4 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.

https://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Symmetryhttps://en.wikipedia.org/wiki/Symmetry_in_mathematicshttps://en.wikipedia.org/wiki/David_Grieshttps://en.wikipedia.org/wiki/Fred_B._Schneiderhttp://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA273http://books.google.com/books?id=_H_nJdagqL8C&pg=PA158http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf

Chapter 2

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.

2.1 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.

3

https://en.wikipedia.org/wiki/Finitary_relationhttps://en.wikipedia.org/wiki/Theory_of_relationshttps://en.wikipedia.org/wiki/Relation#Mathematicshttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Ordered_pairhttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Cartesian_producthttps://en.wikipedia.org/wiki/Divideshttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Divisibilityhttps://en.wikipedia.org/wiki/Inequality_(mathematics)https://en.wikipedia.org/wiki/Equality_(mathematics)https://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Congruence_(geometry)https://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Orthogonalhttps://en.wikipedia.org/wiki/Linear_algebrahttps://en.wikipedia.org/wiki/Linear_algebrahttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Computer_sciencehttps://en.wikipedia.org/wiki/Finitary_relationhttps://en.wikipedia.org/wiki/Tuplehttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Class_(mathematics)https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Russell%2527s_paradoxhttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Cartesian_producthttps://en.wikipedia.org/wiki/Domain_(mathematics)https://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Graph_of_a_functionhttps://en.wikipedia.org/wiki/Indicator_function

4 CHAPTER 2. BINARY RELATION

2.1.1 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.

2.1.2 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".

2.2 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

2.2. SPECIAL TYPES OF BINARY RELATIONS 5

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

6 CHAPTER 2. 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.

2.2.1 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]

2.3 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.

https://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Surjective_functionhttps://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Partial_equivalence_relationhttps://en.wikipedia.org/wiki/Partial_equivalence_relationhttps://en.wikipedia.org/wiki/Kernel_(set_theory)https://en.wikipedia.org/wiki/Automata_theoryhttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Directed_graphhttps://en.wikipedia.org/wiki/Loop_(graph_theory)https://en.wikipedia.org/wiki/Power_sethttps://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Binary_relation#Operations_on_binary_relationshttps://en.wikipedia.org/wiki/Relation_algebrahttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Coreflexive_relationhttps://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relationshttps://en.wikipedia.org/wiki/Symmetric_relation

2.4. OPERATIONS ON BINARY RELATIONS 7

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.

2.4 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

8 CHAPTER 2. 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.

2.4.1 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.

2.4.2 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

2.5. SETS VERSUS CLASSES 9

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.

2.4.3 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.

2.5 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.

2.6 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

10 CHAPTER 2. 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).

2.7 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.

2.8 See also

Confluence (term rewriting)

Hasse diagram

Incidence structure

Logic of relatives

Order theory

Triadic relation

2.9 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

2.10. REFERENCES 11

[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.

2.10 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

12 CHAPTER 2. BINARY RELATION

2.11 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 3

Divisor

Divisible redirects here. For divisibility of groups, see Divisible group.For the second operand of a division, see Division (mathematics). For divisors in algebraic geometry, see Divisor(algebraic geometry). For divisibility in the ring theory, see Divisibility (ring theory).In mathematics a divisor of an integer n , also called a factor of n , is an integer that can be multiplied by some

The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

other integer to produce n .

3.1 Definition

Two versions of the definition of a divisor are commonplace:

For integers m and n , it is said that m divides n , m is a divisor of n , or n is a multiple of m , and this iswritten asm | n,

if there exists an integer k such thatmk = n .[1] Under this definition, the statement 0 | 0 holds.

As before, but with the additional constraintm = 0 .[2] Under this definition, the statement 0 | 0 does not hold.

In the remainder of this article, which definition is applied is indicated where this is significant.

13

https://en.wikipedia.org/wiki/Divisible_grouphttps://en.wikipedia.org/wiki/Division_(mathematics)https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)https://en.wikipedia.org/wiki/Divisibility_(ring_theory)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Cuisenaire_rods

14 CHAPTER 3. DIVISOR

3.2 General

Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example,there are six divisors of 4; they are 1, 2, 4, 1, 2, and 4, but only the positive ones (1, 2, and 4) would usually bementioned.1 and 1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.[3] Every integeris a divisor of 0.[4] Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd.1, 1, n and n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as anon-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, whilethe units 1 and 1 and prime numbers have no non-trivial divisors.There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits.The generalization can be said to be the concept of divisibility in any integral domain.

3.3 Examples

7 is a divisor of 42 because 7 6 = 42 , so we can say 7 | 42 . It can also be said that 42 is divisible by 7, 42is a multiple of 7, 7 divides 42, or 7 is a factor of 42.

The non-trivial divisors of 6 are 2, 2, 3, 3.

The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

5 | 0 , because 5 0 = 0 .

The set of all positive divisors of 60, A = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} , partially ordered by divisi-bility, has the Hasse diagram:

3.4 Further notions and facts

There are some elementary rules:

If a | b and b | c , then a | c , i.e. divisibility is a transitive relation.

If a | b and b | a , then a = b or a = b .

If a | b and a | c , then a | (b+ c) holds, as does a | (b c) .[5] However, if a | b and c | b , then (a+ c) | bdoes not always hold (e.g. 2 | 6 and 3 | 6 but 5 does not divide 6).

If a | bc , and gcd (a, b) = 1 , then a | c . This is called Euclids lemma.If p is a prime number and p | ab then p | a or p | b .A positive divisor of n which is different from n is called a proper divisor or an aliquot part of n . A number thatdoes not evenly divide n but leaves a remainder is called an aliquant part of n .An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positiveinteger which has exactly two positive factors: 1 and itself.Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of thefundamental theorem of arithmetic.A number n is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisorsis less than n , and abundant if this sum exceeds n .The total number of positive divisors of n is a multiplicative function d(n) , meaning that when two numbersm and nare relatively prime, then d(mn) = d(m)d(n) . For instance, d(42) = 8 = 222 = d(2)d(3)d(7) ; the eightdivisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicativefunction: if the two numbersm and n share a common divisor, then it might not be true that d(mn) = d(m) d(n)

https://en.wikipedia.org/wiki/Negative_numberhttps://en.wikipedia.org/wiki/Even_and_odd_numbershttps://en.wikipedia.org/wiki/Even_and_odd_numbershttps://en.wikipedia.org/wiki/Composite_numberhttps://en.wikipedia.org/wiki/Unit_(ring_theory)https://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Divisibility_rulehttps://en.wikipedia.org/wiki/Generalization_(logic)https://en.wikipedia.org/wiki/Integral_domainhttps://en.wikipedia.org/wiki/Multiple_(mathematics)https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Hasse_diagramhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Greatest_common_divisorhttps://en.wikipedia.org/wiki/Euclid%2527s_lemmahttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Prime_factorhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetichttps://en.wikipedia.org/wiki/Perfect_numberhttps://en.wikipedia.org/wiki/Deficient_numberhttps://en.wikipedia.org/wiki/Abundant_numberhttps://en.wikipedia.org/wiki/Multiplicative_functionhttps://en.wikipedia.org/wiki/Relatively_prime

3.4. FURTHER NOTIONS AND FACTS 15

0 10 20n divisors 0 10 20n 30 divisors124610 1220 2430 3640 4850 606070 72

8080 849090 96

100100 108110 112 120120 126130 132

140140 144150150 156160160 168170 180180

190 192200

210210 216220230 240240250 252260 264

270270280280

288290300300

310 312320320 324

330330 336340350 360360370380390 396400410

420420430 432440

450450460

468470480480

490500

504510520

528530540540

550560560

570 576580

588590600600

610 612620 624

630630640

648650660660

670 672680 684690

700700710 720720730740750 756760770

780780790 792800810820830 840840850860 864870880890

900900910920 924930 936940950

960960970980

9909901000

Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers arein bold.

. The sum of the positive divisors of n is another multiplicative function (n) (e.g. (42) = 96 = 3 4 8 =(2)(3)(7) = 1+2+3+6+7+14+21+42 ). Both of these functions are examples of divisor functions.If the prime factorization of n is given by

n = p11 p22 p

kk

https://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Highly_composite_numberhttps://en.wikipedia.org/wiki/Divisor_functionhttps://en.wikipedia.org/wiki/Prime_factorization

16 CHAPTER 3. DIVISOR

then the number of positive divisors of n is

d(n) = (1 + 1)(2 + 1) (k + 1),

and each of the divisors has the form

p11 p22 p

kk

where 0 i i for each 1 i k.For every natural n , d(n) < 2n .Also,[6]

d(1) + d(2) + + d(n) = n lnn+ (2 1)n+O(n).

where is EulerMascheroni constant. One interpretation of this result is that a randomly chosen positive integer nhas an expected number of divisors of about lnn .

3.5 In abstract algebra

Given the definition for which 0 | 0 holds, the relation of divisibility turns the set N of non-negative integers into apartially ordered set: a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. Themeet operation is given by the greatest common divisor and the join operation by the least common multiple.This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z .

3.6 See also Arithmetic functions

https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttps://en.wikipedia.org/wiki/Non-negativehttps://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Greatest_common_divisorhttps://en.wikipedia.org/wiki/Least_common_multiplehttps://en.wikipedia.org/wiki/Duality_(order_theory)https://en.wikipedia.org/wiki/Lattice_of_subgroupshttps://en.wikipedia.org/wiki/Cyclic_grouphttps://en.wikipedia.org/wiki/Arithmetic_functions

3.7. NOTES 17

Divisibility rule

Divisor function

Euclids algorithm

Fraction (mathematics)

Table of divisors A table of prime and non-prime divisors for 11000

Table of prime factors A table of prime factors for 11000

Unitary divisor

3.7 Notes[1] for instance, Sims 1984, p. 42 or Durbin 1992, p. 61

[2] Herstein 1986, p. 26

[3] This statement either requires 0|0 or needs to be restricted to nonzero integers.

[4] This statement either requires 0|0 or needs to be restricted to nonzero integers.

[5] a | b, a | c b = ja, c = ka b + c = (j + k)a a | (b + c) . Similarly, a | b, a | c b = ja, c = ka b c = (j k)a a | (b c)

[6] Hardy, G. H.; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264.ISBN 0-19-853171-0.

3.8 References Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0-471-51001-7.

Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7;section B.

Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1

ystein Ore, Number Theory and its History, McGrawHill, NY, 1944 (and Dover reprints).

Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN0-471-09846-9

https://en.wikipedia.org/wiki/Divisibility_rulehttps://en.wikipedia.org/wiki/Divisor_functionhttps://en.wikipedia.org/wiki/Euclid%2527s_algorithmhttps://en.wikipedia.org/wiki/Fraction_(mathematics)https://en.wikipedia.org/wiki/Table_of_divisorshttps://en.wikipedia.org/wiki/Table_of_prime_factorshttps://en.wikipedia.org/wiki/Unitary_divisorhttps://en.wikipedia.org/wiki/Divisor#CITEREFSims1984https://en.wikipedia.org/wiki/Divisor#CITEREFDurbin1992https://en.wikipedia.org/wiki/Divisor#CITEREFHerstein1986https://en.wikipedia.org/wiki/G._H._Hardyhttp://archive.org/stream/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright#page/n279/mode/2uphttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-19-853171-0http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.htmlhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-51001-7https://en.wikipedia.org/wiki/Richard_K._Guyhttps://en.wikipedia.org/wiki/Springer_Verlaghttps://en.wikipedia.org/wiki/Special:BookSources/0387208607https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-02-353820-1https://en.wikipedia.org/wiki/%C3%98ystein_Orehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-09846-9

Chapter 4

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".

4.1 Etymology

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

4.2 Types of equalities

4.2.1 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.

4.2.2 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.

4.2.3 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.

4.2.4 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

18

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

4.3. LOGICAL FORMALIZATIONS OF EQUALITY 19

distinguishes an equation from an identity or other use of the equality