Given A = {1,2,3,4} and R = {(a, b) : a−b is even}

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

(i) Reflexive:

Let a ϵ A

then a−a = 0 is an even

∴ (a, a) ϵ R, ∀ a ϵ A

∴ R is reflexive.

(ii) Symmetric

Let a, b ϵ A

s.t., (a, b) ϵ R ⇒ a−b is even

⇒ −(b−a) is even

⇒ (b−a) is also even

⇒ (b−a) ϵ R

∴R is symmetric

(iii) Transitive

Let a, b, c ϵ A

s.t., (a, b) ϵ R and (b, c) ϵ R

thus, we have (a-b) and (b-c) both are even

⇒ (a−b) + (b−c) is also even

⇒ a−c is even

⇒ (a, c) ϵ R

∴R is transitive

Hence, R is an equivalence relation.