How to describe the equivalence class for this equivalence relation

Question

For $x, y \in R$, define $x \sim y$ as the equivalence relation meaning $x-y \in Z$.

Am I right in thinking that the relation on $R$ splits it up into different equivalence classes? And that in this case there should be an infinite amount of $[x]=\{x+n$ | $n \in Z, x \in [0, 1)\}$? I am not too familiar with equivalence classes so any guidance is appreciated.

Thanks!

Answer 1

There would be an infinite (uncountable) number of equivalence classes. Your set description of it is correct.