I have the following equivalence relation:
$$ \begin{equation} R=\{(x,y) \in \mathbb{Z} \times \mathbb{Z} \mid |x| = |y|\} \end{equation} $$
I have already verified its properties, but, as the set is infinite, I have no idea about how can I describe this relation's equivalence classes. How should I do it?
Let $x \in \mathbb{Z}$. Then, we describe the equivalence class of $x$ as the following set:
$$[x] = \{y \in \mathbb{Z}: (y,x) \in R\}$$
Let $x \neq 0$. Then, we can see that:
$$|x| = |-x|$$
In other words, $x \in [x]$ and $-x \in [x]$. These are going to be the only two elements of $[x]$. There is no other element that will satisfy $|x| = |y|$.
Now, that is going to hold for almost every equivalence class. Now, let $x = 0$. Then, we can see that:
$$|0| = |-0|$$
So, it follows that $0 \in [0]$ and that's it. The set $[0]$ contains no other elements of $\mathbb{Z}$. That completely describes the equivalence classes for this relation.