Q. 35.0( 2 Votes )

Let R = {(a, b) :

Answer :

In order to show R is an equivalence relation, we need to show R is Reflexive, Symmetric and Transitive.


Given that, a, b Z, R = {(a, b) : (a + b) is even }.


Now,


R is Reflexive if (a,a) R a Z


For any a A, we have


a+a = 2a, which is even.


(a,a) R


Thus, R is reflexive.


R is Symmetric if (a,b) R (b,a) R a,b Z


(a,b) R


a+b is even.


b+a is even.


(b,a) R


Thus, R is symmetric .


R is Transitive if (a,b) R and (b,c) R (a,c) R a,b,c Z


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


a+b = 2P and b+c = 2Q


Adding both, we get


a+c+2b = 2(P+Q)


a+c = 2(P+Q)-2b


a+c is an even number


(a, c) R


Thus, R is transitive on Z.


Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.


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