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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 set
Examples:
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 humans
Counter ex: >= relation, spouse relation
– Use “~” to denote an abstract generic equivalence relation
a~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
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