Q. 17

# 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.

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