Answer :

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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