# Let R = {(a, b) : a, b ∈ Z and (a + b) is even}.Show that R is an equivalence relation on Z.

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

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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