# How many equivalence classes does this relation have?

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?

• By "amount," do you mean the cardinality? – Thomas Andrews Nov 12 '14 at 18:12
• Hint: a complete set of representatives would be $[0,1)$. – DKal Nov 12 '14 at 18:13
• $[a] = \{b\in \mathbb{R} \mid b-a\in \mathbb{Z}\} \subseteq \mathbb{R}$ – Frank Vel Nov 12 '14 at 18:14
• Even more understandable, I think: $[a] = a+\mathbb Z$ – MPW Nov 12 '14 at 18:19

Every equivalence class has a unique representant $r\in[0,1)$:
$$[r]=\{x\in\mathbb R\mid x=\lfloor x\rfloor+r\}$$