How many equivalence classes does this relation have?

Question

I have this relation:

$$A = \mathbb {R} \\ \quad\;\; x\sim y \iff x-y \in \mathbb {Z} $$

I have already proved if it is an equivalence relation. Now I am just searching for the equivalence classes of this relation.

How many equivalence classes does this relation have?

Answer 1

Every equivalence class has a unique representant $r\in[0,1)$:

$$[r]=\{x\in\mathbb R\mid x=\lfloor x\rfloor+r\}$$