Determine the equivalence classes of an equivalence relation R.

Question

Let $\mathbf{R}$ be the relation on $Z \times (Z \setminus \{0\})$ given by $m \mathrel{\mathbf{R}} n$ iff $m - n =2k$ for some $k \in\mathbb Z$. I have proven that this is indeed an equivalence relation by meeting the three required properties, yet I am having trouble understanding the meaning of and determining the equivalence classes of $\mathbf{R}$. Any pointers would be a great help.

Answer 1

I'm emitting a doubt about the symetrical property of $\mathbf R$ because $0$ is forbidden as a second argument. So $0$ is in relation to the left with some numbers, but there is no description of what happen to the right. I would have defined it on $\mathbb Z^2$ without restriction.

Anyway, two numbers $m$ and $n$ are in the same equivalence class if they differ by and even number.

So $\mathbf R$ is splitting $\mathbb Z$ into odd and even numbers, and there are two classes $\tilde 1$ (odd numbers) and $\tilde 2$ (even numbers). [and I still don't know what to think about $\tilde 0$, which without the restriction should be equal to $\tilde 2$].

Answer 2

I don't think your relation is symmetric since $0R0$ is not true. Remark that $R$ is defined over $\mathbb{Z}\times (\mathbb{Z}\setminus\{0\})$.