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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