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Equivalence relations• Binary relations:

– Let S1 and S2 be two sets, and R be a (binary relation) from S1 to S2

– Not every x in S1 and y in S2 have such relation– If R holds for a in S1 and b in S2, denote as aRb or R(a, b)

Examples: Spouse relation from set Men to set Women

– S1 and S2 can be the same setExamples:Parent relation on set Human> (greater than) relation on set Z (all integers)

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Equivalence relations (cont)• Properties of binary relations:

– Let R be a binary relation on set S– R is reflexive: if aRa for all a in S

Ex: = relation, >= relation– R is symmetric: aRb iff bRa

Ex: = relation, spouse relation– R is transitive: if aRb and bRc, then aRc

Ex: = relation, >= relation, ancestor relation– R is an equivalence relation if it is reflexive, symmetric,

and transitive.Ex. = relation, relative relation among humansCounter ex: >= relation, spouse relation

– Use “~” to denote an abstract generic equivalence relationa~b

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Equivalence relations (cont)• Equivalence classes

– Let ~ be a equivalence relation defined on set S– S can be partitioned into disjoint subsets such that

• If a ~ b, then a and b are in one subset• If a and b are in two different subsets, then a ~ b does not

hold– Each of such subsets is called an equivalence class (with

respect to relation ~), denoted C1, C2, ...• All elements in an equivalence class relate to each other by ~• No elements in different equivalence classes relate to each

other by ~– Equivalence classes can be represented as disjoint sets