In order to show R is an equivalence relation, we need to show R is Reflexive, Symmetric and Transitive.

Given that, ∀ a, b ∈ Z, R = {(a, b) : (a + b) is even }.

Now,

R is Reflexive if (a,a) ∈ R ∀ a ∈ Z

For any a ∈ A, we have

a+a = 2a, which is even.

⇒ (a,a) ∈ R

Thus, R is reflexive.

R is Symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a,b ∈ Z

(a,b) ∈ R

⇒ a+b is even.

⇒ b+a is even.

⇒ (b,a) ∈ R

Thus, R is symmetric .

R is Transitive if (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R ∀ a,b,c ∈ Z

Let (a,b) ∈ R and (b,c) ∈ R ∀ a, b,c ∈ Z

⇒ a+b = 2P and b+c = 2Q

Adding both, we get

a+c+2b = 2(P+Q)

⇒ a+c = 2(P+Q)-2b

⇒ a+c is an even number

⇒ (a, c) ∈ R

Thus, R is transitive on Z.

Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.