We have,

R = {(a, b) : (a – b) is divisible by 5} on Z.

We want to prove that R is an equivalence relation on Z.

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 5.

∴ (a, a) ∈ R so R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let (a, b) ∈ R

⇒ a – b = 5p(say) For some p ∈ Z

⇒ b – a = 5 × (–p)

⇒ b – a is divisible by 5

⇒ (b, a) ∈ R, so 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) ∈ R and (b, c) ∈ R

⇒ a – b = 5p(say) and b – c = 5q(say), For some p, q ∈ Z

⇒ a – c = 5 (p + q)

⇒ a – c is divisible by 5.

⇒ R is transitive

∴ R being reflexive, symmetric and transitive on Z.

⇒ R is equivalence relation on Z.