I am stuck on this question and would greatly appreciate any help:
Recall, for an arbitrary set $S$ and equivalence relation $\equiv$ on $S$, $S/\equiv$ denotes the set of equivalence classes in $S$. Consider the relation $\sim$ on the plane $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ defined in the following way: for $\vec{x}, \vec{y} \in \mathbb{R}^2$, say $\vec{x} \sim \vec{y}$ if $\vec{x}-\vec{y} \in \mathbb{Z}^2$. Show that $\sim$ is an equivalence relation, and describe $\mathbb{R}^2 / \sim$.
You've defined
$$\binom ab\sim\binom xy\iff\begin{cases}x-a\in\Bbb Z\\{}\\\text{and also}\\{}\\y-b\in\Bbb Z\end{cases}$$
For transitivity we have:
$$\binom ab\sim\binom cd\sim\binom xy\implies \begin{cases}c-a\\d-b\end{cases}\in\Bbb Z\;\;\text{and also}\;\;\begin{cases}x-c\\y-d\end{cases}\in\Bbb Z$$
and for example
$$x-a=(x-c)+(c-a)\;\ldots\;\;\text{complete and end the exercise}$$