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

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