We have,

R = {(a,b) : a–b is divisible by 3; a, b ∈ Z}

To prove : R is an equivalence relation

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Let a ∈ Z

⇒ a – a = 0

⇒ a – a is divisible by 3

(∵ 0 is divisible by 3).

⇒ (a, a) ∈ R

⇒ R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let a, b ∈ Z and (a, b) ∈ R

⇒ a – b is divisible by 3

⇒ a – b = 3p(say) For some p ∈ Z

⇒ –( a – b) = –3p

⇒ b – a = 3 × (–p)

⇒ b – a ∈ R

⇒ R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Let a, b, c ∈ Z and such that (a, b) ∈ R and (b, c) ∈ R

⇒ a – b = 3p(say) and b – c = 3q(say) For some p, q ∈ Z

⇒ a – c = 3 (p + q)

⇒ a – c = 3 (p + q)

⇒ (a, c) ∈ R

⇒ R is transitive

Since, R is reflexive, symmetric and transitive

⇒ R is an equivalence relation.