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 Parent relation on set Human > (greater than) relation on set Z (all integers) Example: student records s1 = set of students with 0< GPA <= 1 s2 = set of students with 1 < GPA <= 2 etc Question: Are x and y in the same set? Yes, if find(x) == find(y).

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 a generic equivalence relation a~b Example: student records s1 = set of students with 0< GPA <= 1 s2 = set of students with 1 < GPA <= 2 etc Question: Are x and y in the same set? Yes, if find(x) == find(y).

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 Example: student records s1 = set of students with 0< GPA <= 1 s2 = set of students with 1 < GPA <= 2 etc Question: Are x and y in the same set? Yes, if find(x) == find(y).