Let $Z$ be the set of integers. Show that the relation $R=\left\{\left(a,b\right):a,b\in Z\text{and a+b is even }\right\}$ is an equivalence relation on $Z$.

Again the relation is symmetric as if $(x,y)\in R\Rightarrow (y,x)\in R$ since $x+y$ is even gives $y+x$ is also even for all $x,y\in Z$.

Again this relation $R$ is transitive as if $(x,y)\in R$ and $(y,z)\in R$ this gives $(x,z)\in R$ as $x+y=2k_1$.....(1) [ Since $x+y$ is even ] and $y+z=2k_2$.....(2) [ Since $y+z$ is even] [ For $k_1$ and $k_2$ being integers]

Now adding (1) and (2) we get 

or, $x+z=2(k_1+k_2-y)$

This gives $x+z$ is even.

Hence this relation is also transitive.

So the relation is equivalance relation.