If I is the set of integers and if the relation R is defined over I by aRb$\iff$a-b is an even integer. a,b$\in$I the relation R is

$\left(\mathrm{a}\right)\forall \mathrm{a}\in \mathrm{I},\mathrm{aRa}\iff a-a=0 is an even integer$

So it is  reflexive
(b)

 Let  aRb$\iff$a-b is an even integer$\Rightarrow b-a is also an even integer\Rightarrow bRa$

So it is symmetric
(c)

 $$\begin{gathered} Let (a,b)\in R, (b,c)\in R\Rightarrow a-b and b-c are even integers \\ \Rightarrow a-c is also an even integer \\ \Rightarrow (a,c)\in R \end{gathered}$$

So it is  transitive.
(d)since the relation is reflexive,symmetric and trasitive then the relation R is an  equivalence relation.