Find equivalence classes of x ~ y : <=> x-y ∈ Z

Question

The equivalence relation is:

$$X=\mathbb{R}$$ $$x∼y:⇔x−y∈\mathbb{Z}$$ I proved the relation properties but how can I find the equivalence classes?

Also I was wondering whether the equivalence relation above is the same as the following?

$$n∈\mathbb{N}$$ $$x∼y:⇔n|x−y$$

Does this help? Is it related to congruence? Also, I think x and y must have the same decimal expansion, does this help?

EDIT: Thank you all for your help.

Answer 1

I think someone posted an answer before but it seems it is deleted now. Anyway consider $[x]$ the equivalence class for some $x\in \mathbb{R}$.

If $y\in [x]$ then $x\sim y$ so $y-x\in \mathbb{Z}$. Then $y\in x+\mathbb{Z}$.

Conversely if $y\in x+\mathbb{Z}$ then $y-x\in \mathbb{Z}$ so $y\sim x$ and $y\in [x]$

So $[x]\subseteq x+\mathbb{Z} \subseteq [x] \implies [x]=x+\mathbb{Z}$.