# Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be defined as follows :(a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0Prove that R is an equivalence relation on Z × Z0

We have, Z be set of integers and Z0 be the set of non-zero integers.

R = {(a, b) (c, d) : ad = bc} be a relation on Z and Z0.

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

(a, b) Z × Z0

ab = ba

((a, b), (a, b)) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let ((a, b), (c, d) R

cd = da

((c, d), (a, b)) R

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), (c, d) R and (c, d), (e, f) R

ad = bc and cf = de

af = be

(a, c) (e, f) R

R is transitive

Hence, R is an equivalence relation on Z × Z0

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