Q. 143.5( 4 Votes )

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

Answer :

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


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 01Functions - 01Functions - 0152 mins
Range of FunctionsRange of FunctionsRange of Functions58 mins
Quick Revision of Types of RelationsQuick Revision of Types of RelationsQuick Revision of Types of Relations59 mins
Some standard real functionsSome standard real functionsSome standard real functions61 mins
Functions - 06Functions - 06Functions - 0648 mins
Functions - 11Functions - 11Functions - 1156 mins
Battle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their Graphs48 mins
Functions - 09Functions - 09Functions - 0947 mins
Quick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functions58 mins
Range of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functions45 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses