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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?

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

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

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