Let C be the set

We have,

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

Now,

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 ∈ C₀

And, 0 is real

∴ (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

⇒ p is real.

And ∵ p is real

⇒ -p is also a real no.

⇒ (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

⇒ p is real no.

….(1)

⇒ q is real.

…..(2)

Dividing (1) by (2), we get-

Where, Q is a rational number.

⇒ Q is real number

Now, by componendo dividendo-

⇒ (a, c) ∈ R.

⇒ R is transitive

Thus, R is reflexive, symmetric and, transitive on C₀.

Hence, R is an equivalence relation on C₀.