Let be the set of all integers

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let

Z

be the set of all integers. Consider the relation

R

on the Cartesian product

Z \times Z

defined by

(a, b) \; R \; (c, d)

if and only if

a + d = b + c

. (i) Prove that

R

is reflexive, symmetric and transitive on

Z \times Z

. (ii) Hence, justify that

R

is an equivalence relation on

Z \times Z

. (iii) Determine the equivalence class of the ordered pair

(3,5)

. (iv) Describe in set-builder form the distinct equivalence classes generated by

R