Question

Show that the relation R in the set Z of integers given byR = {(a, b) : 2 divides a – b}is an equivalence relation.​

Answer 1

$\huge\star\mathfrak\pink{{Answer:-}}$

$\huge\star\mathfrak\red{{Solution:-}}$

Solution R is reflexive, as 2 divides (a – a) for all a ∈ Z. Further, if (a, b) ∈ R, then2 divides a – b. Therefore, 2 divides b – a. Hence, (b, a) ∈ R, which shows that R issymmetric. Similarly, if (a, b) ∈ R and (b, c) ∈ R, then a – b and b – c are divisible by2. Now, a – c = (a – b) + (b – c) is even (Why?). So, (a – c) is divisible by 2. Thisshows that R is transitive. Thus, R is an equivalence relation in Z.