The relation R in the set Z of integers given by R ={(a, b) : 2 divides a-b } is-

Equivalence relation

R is reflexive, as 2 divides (a-a) for all a∈ Z. Further, if (a, b)∈ R then 2 divides (a-b). Therefore ,2 divides (b-a) also. Hence (b, a)∈ R, which shows that R is symmetric. Similarly, if (a,b) ∈ RR and (b, c)∈ R, then (a-b) and (b-c) are divisible by 2. Now, (a-c) = (a-b) + (b-c) is even. So, (a-c) is divisible by 2, This shows that R is transitive. Thus, R is an equivalence relation in Z.