1. 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).

2. 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).

3. 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).