Q. 2

Which of the following is not an equivalence relation on Z?
A. a R b a + b is an even integer

B. a R b a – b is a even integer

C. a R b a < b

D. a R b a = b

Answer :

a R b a – b is a even integer

Given R = {(a, b) : a – b is an even integer(i.e divisible by 2)}


For equivance relation we have to check three parameters:


(iii) Reflexive:


If (a-b) is divisible by 2 then,


(a-a)=0 is also divisible by 2


(a,a) R


Hence R is Reflexive (a,b) Z


(ii)Symmetric:


If (a-b) is divisible by 2 then,


(b-a)=-(a-b) is also divisible by 2


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


Hence R is Symmetric (a,b) Z


(iii)Transitive:


If (a-b) and (b-c) are divisible by 2 then,


a-c=(a-b)+(b-c) is also divisible by 2


(a,b) R, (b,c) R and (a,c) R


Hence R is Transitive (a,b) Z


As Relation R is satisfying all the three parameters, hence R is an equivalence relation.

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