equiv. class if aRb means a+b is a+b even

Question

let s be set of integers. and say that aRb=a+b only if a+b is even.

i've already shown that this is indeed a equivalance relation, but how to show its equivalance classes?

Answer 1

Lets's start by picking some integer and finding its equivalence class. I pick $0$. Now what is the equivalence class of $0$? It's the set of all integers $a$ such that $aR0$, in other words the set of all integers $a$ such that $a+0$ is even, in other words, the set of all even integers!

So one equivalence class is the set $E$ of all even integers.

Now let's find another equivalence class by picking another integer. If we pick an integer that belongs to $E$, we will just get the same equivalence class $E$ again. So let's pick an odd number. I pick $1$.

You can take it from there.