Given A = {1,2,3,4,5} and R = {(a, b) : ∣a−b∣ is divisible by 4}

Relation is an equivalent relation if R is reflexive, symmetric and transitive.

(i) Reflexive:

Let a ϵ A

then ∣a−a∣=0 is divisible by 4

∴ (a, a) ϵ R, ∀ a ϵ A

∴ R is reflexive.

(ii) Symmetric

Let a, b ϵ A

∀ (a, b) ϵ R ⇒ ∣a−b∣ is divisible by 4

⇒ ∣−(b−a)∣ is divisible by 4

⇒ ∣b−a∣ is divisible by 4

⇒ ∣b−a∣ ϵ R

or (b, a) ϵ R

∴R is symmetric

(iii) Transitive

Let a, b, c ϵ A

∀ (a, b) ϵ R and (b, c) ϵ R

we have ∣a−b∣ is divisible by 4 and ∣b−c∣ is divisible by 4

⇒ a−b is divisible by 4 and b−c is divisible by 4

⇒ (a−b) + (b−c) is divisible by 4

⇒ a−c is divisible by 4

⇒ ∣a−c∣ is divisible by 4 ⇒ (a, c) ϵ R

∴R is transitive

Hence, R is an equivalence relation.