# Show that R is an equivalence relation on the set Z of integers given by R = {(a, b) : 2 divides a – b}. Write the equivalence class [0].

Given R = {(a, b) : 2 divides a – b}

For equivance relation we have to check three parameters:

(i) 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.

Now equivalence class [0] is the set of all those elements in A which are related to 0 under relation.

Now,

(a,0) R

a – 0 is divisible by 2 and a A.

a A such that 2 divides a.

a = 0, 2,4

Thus [0] = {0,2,4}

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