7.2 EQUIVALENCE RELATIONS 235- Find the domain and range of each of the following relations:
(a) R, and R,, Example 1 (b) Rs, Example 2
(c) R7 and R8, Example 3 (d) R ,, and R, ,, Example 4
(e) Rls and Rl69 Example 6 (f) R~,={(~,Y)ER~R~Y=&-~)
ts) R22 = ((1, 5), (1,6), (2,6), (2,7), (3,7), (39% (4,8), (4,9), (5,919 (5, 10))
8. Find R - ' for each of the relations:
(a) R,, Example 1 (6) Rs, Example 2
(c) R,, Example 3 (d) R, ,, Example 4
(e) R12 and R14, Example^5
9. Prove (a), (b), and (c) of Theorem 3.
10. (a) Prove that a relation R on a nonempty set A is reflexive if and only if
I, c R.
(b) Prove that if a reflexive relation R on nonempty set A is both symmetric and
antisymmetric, then R = I,, the identity relation on A.
*(c) Prove that a relation R on a set A is symmetric if and only if R = R-'.
(d) Prove that if a nonempty relation R on a set A is antisymmetric, then
R n R-' = I,. [This is the unproved part of (e), Theorem 3.1
(e) Prove that if R is any nonempty relation on a set A, then the relations R n R- '
and R u R-' are symmetric. - Given a relation R on a set A, define, for each x E A, the set [x] by the rule
[XI = (y E Al(x, y) E R). Note that [x] consists of all elements of A to which x is
related by the relation R. Notice also that [x] is clearly a subset of A for each
x E A. Let A = (1,2,3,4,5). Calculate [I], [2], [3], [4], and [5] for each of the
following relations on A:
7.2 Equivalence Relations
An equivalence relation is a special kind of relation on a set; whenever two
elements x and y of a set A are related by an equivalence relation on A,
there is some property that x and y share in common, some point of view
from which x and y can be regarded as indistinguishable.EXAMPLE 1 (a) Consider the relation on R, R,, = {(x, y) (x2 = y2), from
Exercise 5, Article 7.1. Two real numbers are related by R,, if and only
if they have the same absolute value. We will soon see that R,, is an
equivalence relation on R.