Let the relation $\sim$ on $\mathbb{Q}$ be defined as follows: $x\sim y \Leftrightarrow x- \lfloor x \rfloor= y-\lfloor y \rfloor$ Here $\lfloor x \rfloor$ is equal to the largest integer $n$ such that $n\le x$.
(a) Let $M=\{x\in\mathbb{Q} \mid 0\le x \lt 1\}$. Show that ${Q}/_{\sim} \cong M$.
The obvious choice for me became that if I map $x \to x - \lfloor x \rfloor$. But then how can I prove that this is a bijection? Can anyone please help me understanding the process? I've tried a lot but the best that I could come up with is the mapping. Also I can't even see how the equivalence classes look like here.