Q. 2

# Which of the following is not an equivalence relation on Z?A. a R b ⬄ a + b is an even integerB. a R b ⬄ a – b is a even integerC. a R b ⬄ a < bD. a R b ⬄ a = b

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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