a R b ⬄ a – b is a even integer

Given R = {(a, b) : a – b is an even integer(i.e divisible by 2)}

For equivance relation we have to check three parameters:

(iii) Reflexive:

If (a-b) is divisible by 2 then,

⇒ (a-a)=0 is also divisible by 2

⇒ (a,a) ∈ R

Hence R is Reflexive ∀ (a,b) ∈ Z

(ii)Symmetric:

If (a-b) is divisible by 2 then,

⇒ (b-a)=-(a-b) is also divisible by 2

⇒ (a,b) ∈ R and (b,a) ∈ R

Hence R is Symmetric ∀ (a,b) ∈ Z

(iii)Transitive:

If (a-b) and (b-c) are divisible by 2 then,

⇒ a-c=(a-b)+(b-c) is also divisible by 2

⇒ (a,b) ∈ R, (b,c) ∈ R and (a,c) ∈ R

Hence R is Transitive ∀ (a,b) ∈ Z

→ As Relation R is satisfying all the three parameters, hence R is an equivalence relation.