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 × Z0


Prove 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


⇒ ad = bc


⇒ 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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