I proved that $x\sim y$ iff $x-y\in \mathbb Z$ is an equivalence relation on $\mathbb R$. I'd like to know if $[x]=\{x+n:n\in\mathbb Z\}$ is an equivalence class for every $x\in \mathbb R$ (if it is well written).
Now, I proved that $(x_1,y_1)\sim(x_2,y_2)$ iff $ x_1=x_2$ is an equivalence relation on $\mathbb R^2$ and know that $[(x,y)]$ is the set of points of the form $(x_0,y)$ for some fixed $x_0$, but don't know how to write it down. I appreciate your help.
For the first question, yes you are correct. If $x\sim y$ then $x-y=n$ for some $n\in\Bbb Z$, so $y=x-n$; and of course if $y=x+n$ then $x-y\in\Bbb Z$. So $[x]$ is exactly the set $\{x+n\mid n\in\Bbb Z\}$.
For the second question, note that if $(x,y)\sim(x',y')$ then $x=x'$. So the equivalents of $(x,y)$ are those where $x$ is fixed, and $y$ varies over $\Bbb R$. Namely, $$[(x,y)]=\{(x,y')\in\Bbb R^2\mid y'\in\Bbb R\}.$$