8.3 Equivalence Relations

Definition

Given a partition of a set A, the relation induced by the partition, R, is defined on A as follows: For every x, y ∈ A,

• x R y ↔ there is a subset A_i of the partition such that both x and y are in A_i.

Theorem 8.3.1

Let A be a set with a partition and let R be the relation induced by the partition. Then R is reflexive, symmetric and transitive.

Definition

Let A be a set and R a relation on A. R is an equivalence relation if and only if R is reflexive, symmetric and transitive.

Definition

Suppose A is a set and R is an equivalence relation on A. For each element a in A, the equivalence class of a, denoted [a] and called the class of a for short, is the set of all elements x in A such that x is related to a by R. In symbols:

• [a] = {x ∈ A | x R a}

Lemma 8.3.2

Suppose A is a set, R is an equivalence relation on A, and a and b are elements of A. If a R b, then [a] = [b].

Lemma 8.3.3

If A is a set, R is an equivalence relation on A, and a and b are elements of A, then either [a] ∩ [b] = ∅ or [a] = [b].

Theorem 8.3.4

If A is a set and R is an equivalence relation on A, then the distinct equivalence classes of R form a partition of A; that is, the union of the equivalence classes is all of A, and the intersection of any two distinct classes is empty.

Definition

Suppose R is an equivalence relation on a set A and S is an equivalence class of R. A representative of the class S is any element a such that [a] = S.

In Class Exercises

• 3. A = {0, 1, 2, 3, 4} and R = {(0,0),(0,4),(1,1),(1,3),(2,2), (3,1),(3,3),(4,0),(4,4)}. Find [0], [1], [2] and [3]. List the distinct equivalence classes.
• 13. A is the set of all strings of length 4 in a's and b's. R is defined on A as follows: For all strings s and t in A, s R t ↔ s has the same first two characters as t. Find the distinct equivalence classes of R.
• 21. R is the relation defined on Z as follows: for every m, n ∈ Z, m R n ↔ 7m - 5n is even. Prove that the relation is reflexive and symmetric (it is also transitive!). Describe the distinct equivalence classes.
• 42b. Let A be the set of all ordered pairs of integers for which the second element of the pair is nonzero. Symbolically: A = Z ⨯ (Z - {0}). Define a relation R on A as follows: For all pairs (a,b) and (c,d) in A, (a,b) R (c,d) ↔ ad = bc. Prove that R is symmetric.