Equivalence Class 6

Equivalence classFrom Wikipedia, the free encyclopedia

Contents

1 Binary relation 11.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Equivalence class 112.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Equivalence relation 163.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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3.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Homogeneous space 254.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Partition of a set 295.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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5.4 Renement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Quotient category 376.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7 Quotient ring 397.1 Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.6 Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8 Quotient space (linear algebra) 438.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9 Quotient space (topology) 469.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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9.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 Reexive relation 5010.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.3 Number of reexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

11 Semigroup 5411.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.8 Semigroup methods in partial dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.13Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12 Set (mathematics) 6212.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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12.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

12.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13 Symmetric relation 7413.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

13.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

13.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 7513.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

14 Transitive relation 7714.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

14.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

14.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

14.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

14.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 80

14.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 1

Binary relation

Relation (mathematics)" redirects here. For a more general notion of relation, see nitary 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 dened 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 denition

A binary relation R is usually dened 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 dened 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 specied or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

1

2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?According to the denition above, two relations with identical graphs but dierent domains or dierent codomainsare considered dierent. For example, ifG = f(1; 2); (1; 3); (2; 7)g , 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 dened as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then dened as the set of all x such that there exists at least oney such that (x; y) 2 R , the range of R is dened as the set of all y such that there exists at least one x such that(x; y) 2 R , and the eld of R is the union of its domain and its range.[2][3][4]A special case of this dierence 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 rst components. This dierence 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 denitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodenitions usually matters only in very formal contexts, like category theory.

1.1.2 ExampleExample: 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 rst 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 dierent relations could have the same graph. For example: the relation

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

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

1.2 Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be dierent 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-denite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=5 and z=+5.

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

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

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

Totality properties:

left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is dierentfrom the denition 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:


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 DifunctionalLess commonly encountered is the notion of difunctional (or regular) relation, dened 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 dene 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 dene 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 justied 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 setIf 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:

reexive: for all x in X it holds that xRx. For example, greater than or equal to () is a reexive relation butgreater than (>) is not.

irreexive (or strict): for all x in X it holds that not xRx. For example, > is an irreexive relation, but is not. coreexive: for all x and y in X it holds that if xRy then x = y. An example of a coreexive 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 reexive and coreexive relation.

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

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

1.4. OPERATIONS ON BINARY RELATIONS 5

antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, is anti-symmetric (so is >, butonly because the condition in the denition 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 irreexive.[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 irreexive 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 denition for total is dierent 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 reexive 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 denition 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 reexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reexive. A relation that is only symmetric and transitive (without necessarily beingreexive) is called a partial equivalence relation.A relation that is reexive, 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 relationsIf 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, dened as R S = { (x, y) | (x, y) R or (x, y) S }. For example, is the union of >and =.

Intersection: R S X Y, dened 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), dened 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.


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, dened 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:

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

Reexive reduction: R , dened as R = R \ { (x, x) | x X } or the largest irreexive relation over Xcontained in R.

Transitive closure: R +, dened 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 , dened as a minimal relation having the same transitive closure as R. Reexive transitive closure: R *, dened as R * = (R +) =, the smallest preorder containing R. Reexive transitive symmetric closure: R , dened as the smallest equivalence relation over X containingR.

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

The complement S is dened 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 reexive relation is irreexive 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 RestrictionThe 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 reexive, irreexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of is parent of is is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

1.5. SETS VERSUS CLASSES 7

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

1.4.3 Algebras, categories, and rewriting systemsVarious 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 nitary relations (and in practice also niteand 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 classesCertain mathematical relations, such as equal to, member of, and subset of, cannot be understood to be binaryrelations as dened 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 specic 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 dened 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 modication 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 denition one can for instance dene 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 irreexive relations is the same as that of reexive relations. The number of strict partial orders (irreexive 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.


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 ane spaces) is in bijection with isomorphy

dependency relation, a nite, symmetric, reexive relation. independency relation, a symmetric, irreexive relation which is the complement of some dependency relation.

1.8 See also Conuence (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.

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

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


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

Chapter 2

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 dened 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 rst 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 denition 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 dened 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.

2.1 Notation and formal denitionAn equivalence relation is a binary relation ~ satisfying three properties:[1]

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

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).

11

12 CHAPTER 2. EQUIVALENCE CLASS

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

[a] = fx 2 X j a xgof elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specically 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 denes 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 integersdened 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 identied, 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).

2.2 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 identied 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 dierence 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 dene 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 beidentied with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal denition of the set of rational numbers.[5] The same construction can be generalized to the eldof 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 innity.

2.3 PropertiesEvery 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:

2.4. GRAPHICAL REPRESENTATION 13

x y [x] = [y] [x] \ [y] 6= ;:

2.4 Graphical representationAny 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]

2.5 InvariantsIf ~ 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-dened 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 nite groups. Some authors use compatible with ~" or just respects ~" instead of invariantunder ~".Any function f : X Y itself denes 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 .

2.6 Quotient space in topologyIn 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 denition of invariants of equivalence relations given above.

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

14 CHAPTER 2. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs Homogeneous space, the quotient space of Lie groups. Transversal (combinatorics)

2.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 7778

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

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

[8] Devlin 2004, p. 123

2.9 References Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8 Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

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

Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematicians Toolbox, Freeman, ISBN 978-0-7167-3050-7

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

Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole) Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall Lay (2001), Analysis with an introduction to proof, Prentice Hall Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

2.10. FURTHER READING 15

Cupillari, The Nuts and Bolts of Proofs, Wadsworth Bond, Introduction to Abstract Mathematics, Brooks/Cole Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall Ash, A Primer of Abstract Mathematics, MAA

Chapter 3

Equivalence relation

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

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

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

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

a ~ a. (Reexivity)

if a ~ b then b ~ a. (Symmetry)

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

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is dened as[a] = fb 2 X j a bg .

3.3 Examples

3.3.1 Simple example

Let the set fa; b; cg have the equivalence relation f(a; a); (b; b); (c; c); (b; c); (c; b)g . The following sets are equivalenceclasses of this relation:[a] = fag; [b] = [c] = fb; cg .The set of all equivalence classes for this relation is ffag; fb; cgg .

16

3.4. CONNECTIONS TO OTHER RELATIONS 17

3.3.2 Equivalence relations

The following are all equivalence relations:

Has the same birthday as on the set of all people. Is similar to on the set of all triangles. Is congruent to on the set of all triangles. Is congruent to, modulo n" on the integers. Has the same image under a function" on the elements of the domain of the function. Has the same absolute value on the set of real numbers Has the same cosine on the set of all angles.

3.3.3 Relations that are not equivalences The relation "" between real numbers is reexive and transitive, but not symmetric. For example, 7 5 doesnot imply that 5 7. It is, however, a partial order.

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

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

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

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

A strict partial order is irreexive, transitive, and asymmetric. A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reexive if and onlyif for all a X, there exists a b X such that a ~ b.

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

18 CHAPTER 3. EQUIVALENCE RELATION

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

3.6 Equivalence class, quotient set, partitionLet a; b 2 X . Some denitions:

3.6.1 Equivalence classMain article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := fx 2 X j a xg denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

3.6.2 Quotient setMain article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/ := f[x] j x 2 Xg , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

3.6.3 ProjectionMain article: Projection (relational algebra)

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

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

3.6.4 Equivalence kernelThe equivalence kernel of a function f is the equivalence relation ~ dened by x y () f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

3.6.5 PartitionMain article: Partition of a set

3.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 19

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

Counting possible partitions

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

Bn =1

e

1Xk=0

kn

k!;

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

3.7 Fundamental theorem of equivalence relationsA key result links equivalence relations and partitions:[2][3][4]

An equivalence relation ~ on a set X partitions X.

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

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

3.8 Comparing equivalence relationsIf ~ and are two equivalence relations on the same set S, and a~b implies ab for all a,b S, then is said to be acoarser relation than ~, and ~ is a ner relation than . Equivalently,

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

~ is ner than if the partition created by ~ is a renement of the partition created by .

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

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


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

The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi )R or (xi,xi)R, i = 1, ..., n1.

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

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

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

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

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

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

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

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

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

3.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 21

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b (ab1 H). The equivalence classes of ~also called the orbits of the action of H on Gare the right cosetsof H in G. Interchanging a and b yields the left cosets.Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that x A g G ([g(x)] = [x]), because G satises the following fourconditions:

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

Existence of identity function. The identity function, I(x)=x, is an obvious element of G; Existence of inverse function. Every bijective function g has an inverse g1, such that gg1 = I; Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

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

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

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

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

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

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

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


Reexive and transitive: The relation on N. Or any preorder; Symmetric and transitive: The relation R on N, dened as aRb ab 0. Or any partial equivalence relation; Reexive and symmetric: The relation R on Z, dened as aRb "a b is divisible by at least one of 2 or 3.Or any dependency relation.

Properties denable in rst-order logic that an equivalence relation may or may not possess include:

The number of equivalence classes is nite or innite; The number of equivalence classes equals the (nite) natural number n; All equivalence classes have innite cardinality; The number of elements in each equivalence class is the natural number n.

3.12 Euclidean relationsEuclid's The Elements includes the following Common Notion 1":

Things which equal the same thing also equal one another.

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

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

The following theorem connects Euclidean relations and equivalence relations:

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

Proof for a left-Euclidean relation

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

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

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

3.13 See also Partition of a set Equivalence class Up to Conjugacy class Topological conjugacy

3.14. NOTES 23

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

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

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

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 2932, Marcel Dekker

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

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

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

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

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



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

3.15 References Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8. Castellani, E., 2003, Symmetry and equivalence in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reections. Cambridge Univ. Press: 422-433.

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

Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint. John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31. Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdening equivalence, pp 4850, John Wiley & Sons.

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

Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009 Equivalence relation at PlanetMath Binary matrices representing equivalence relations at OEIS.


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

Chapter 4

Homogeneous space

A torus. The standard torus is homogeneous under its dieomorphism and homeomorphism groups, and the at torus is homogeneousunder its dieomorphism, homeomorphism, and isometry groups.

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneousspace for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements ofG are called the symmetries of X. A special case of this is when the group G in question is the automorphism groupof the space X here automorphism group can mean isometry group, dieomorphism group, or homeomorphismgroup. In this case X is homogeneous if intuitively X looks locally the same at each point, either in the sense ofisometry (rigid geometry), dieomorphism (dierential geometry), or homeomorphism (topology). Some authorsinsist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not.Thus there is a group action of G on X which can be thought of as preserving some geometric structure on X, andmaking X into a single G-orbit.

4.1 Formal denitionLet X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X.[1]Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category,

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26 CHAPTER 4. HOMOGENEOUS SPACE

then the elements of G are assumed to act as automorphisms in the same category. Thus the maps on X eected byG are structure preserving. A homogeneous space is a G-space on which G acts transitively.Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:

: G! AutC(X)

into the group of automorphisms of the object X in the category C. The pair (X, ) denes a homogeneous spaceprovided (G) is a transitive group of symmetries of the underlying set of X.

4.1.1 ExamplesFor example, if X is a topological space, then group elements are assumed to act as homeomorphisms on X. Thestructure of a G-space is a group homomorphism : G Homeo(X) into the homeomorphism group of X.Similarly, if X is a dierentiable manifold, then the group elements are dieomorphisms. The structure of a G-spaceis a group homomorphism : G Dieo(X) into the dieomorphism group of X.Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the exampleslisted below.Concrete examples include:

Isometry groups

Positive curvature:

1. Sphere (orthogonal group): Sn1 = O(n)/O(n 1)2. Oriented sphere (special orthogonal group): Sn1 = SO(n)/SO(n 1)3. Projective space (projective orthogonal group): Pn1 = PO(n)/PO(n 1)

Flat (zero curvature):

1. Euclidean space (Euclidean group, point stabilizer is orthogonal group): An E(n)/O(n)

Negative curvature:

1. Hyperbolic space (orthochronous Lorentz group, point stabilizer orthogonal group, corresponding to hyperboloidmodel): Hn O+(1, n)/O(n)

2. Oriented hyperbolic space: SO+(1, n)/SO(n)3. Anti-de Sitter space: AdS = O(2, n)/O(1, n)

Others

Ane space (for ane group, point stabilizer general linear group): An = A(n, K)/GL(n, k). Grassmannian: Gr(r; n) = O(n)/(O(r) O(n r))

4.2 GeometryFrom the point of view of the Erlangen program, onemay understand that all points are the same, in the geometry ofX. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenthcentury.

4.3. HOMOGENEOUS SPACES AS COSET SPACES 27

Thus, for example, Euclidean space, ane space and projective space are all in natural ways homogeneous spacesfor their respective symmetry groups. The same is true of the models found of non-Euclidean geometry of constantcurvature, such as hyperbolic space.A further classical example is the space of lines in projective space of three dimensions (equivalently, the spaceof two-dimensional subspaces of a four-dimensional vector space). It is simple linear algebra to show that GL4 actstransitively on those. We can parameterize them by line co-ordinates: these are the 22 minors of the 42 matrix withcolumns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometryof Julius Plcker.

4.3 Homogeneous spaces as coset spacesIn general, if X is a homogeneous space, and Ho is the stabilizer of some marked point o in X (a choice of origin),the points of X correspond to the left cosets G/Ho, and the marked point o corresponds to the coset of the identity.Conversely, given a coset space G/H, it is a homogeneous space for G with a distinguished point, namely the coset ofthe identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.In general, a dierent choice of origin o will lead to a quotient of G by a dierent subgroup Ho which is related toHo by an inner automorphism of G. Specically,

Ho0 = gHog1 (1)

where g is any element of G for which go = o. Note that the inner automorphism (1) does not depend on which suchg is selected; it depends only on g modulo Ho.If the action of G on X is continuous, then H is a closed subgroup of G. In particular, if G is a Lie group, then H isa Lie subgroup by Cartans theorem. Hence G/H is a smooth manifold and so X carries a unique smooth structurecompatible with the group action.If H is the identity subgroup {e}, then X is a principal homogeneous space.One can go further to double coset spaces, notably CliordKlein forms \G/H, where is a discrete subgroup (ofG) acting properly discontinuously.

4.4 ExampleFor example in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional generallinear group, GL(4), dened by conditions on the matrix entries

h13 = h14 = h23 = h24 = 0,

by looking for the stabilizer of the subspace spanned by the rst two standard basis vectors. That shows that X hasdimension 4.Since the homogeneous coordinates given by theminors are 6 in number, this means that the latter are not independentof each other. In fact a single quadratic relation holds between the six minors, as was known to nineteenth-centurygeometers.This example was the rst known example of a Grassmannian, other than a projective space. There are many furtherhomogeneous spaces of the classical linear groups in common use in mathematics.

4.5 Prehomogeneous vector spacesThe idea of a prehomogeneous vector space was introduced by Mikio Sato.It is a nite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of Gthat is open for the Zariski topology (and so, dense). An example is GL(1) acting on a one-dimensional space.

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The denition is more restrictive than it initially appears: such spaces have remarkable properties, and there is aclassication of irreducible prehomogeneous vector spaces, up to a transformation known as castling.

4.6 Homogeneous spaces in physicsCosmology using the general theory of relativity makes use of the Bianchi classication system. Homogeneous spacesin relativity represent the space part of background metrics for some cosmological models; for example, the threecases of the FriedmannLematreRobertsonWalker metric may be represented by subsets of the Bianchi I (at),V (open), VII (at or open) and IX (closed) types, while the Mixmaster universe represents an anisotropic exampleof a Bianchi IX cosmology.[2]

A homogeneous space of N dimensions admits a set of 12N(N + 1) Killing vectors.[3] For three dimensions, thisgives a total of six linearly independent Killing vector elds; homogeneous 3-spaces have the property that one mayuse linear combinations of these to nd thre

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