Question

Check if R : Z → Z, R = {(a, b) | 2 divides a – b} is an equivalence relation.

Answer 1

(i) Since 2 divides a – a, (a, a) ∈ R

∴ R is reflexive. .

(ii) Let (a, b) ∈ R

Then 2 divides a – b

∴ 2 divides b – a

∴ (b, a) ∈ R

∴ R is symmetric.

(iii) Let (a, b) ∈ R, (b, c) ∈ R

Then a – b = 2m, b – c = 2n,

∴ a – c = 2(m + n), where m, n are integers.

∴ 2 divides a – c

∴ (a, c) ∈ R

∴ R is transitive.

Thus, R is an equivalence relation.