Group Theory Problem 1

There is a canonical surjection $\mathbb Q \rightarrow \mathbb Q/\mathbb Z$ given by $q\mapsto q+\mathbb Z$, i.e. $|\mathbb Q|\geq |\mathbb Q/\mathbb Z|$. The sets $r+\mathbb Q\in\mathbb R /\mathbb Q$ for $r\in\mathbb R$ are countable. Additionally, we know that $\bigcup_{r\in\mathbb R} r+\mathbb Q = \mathbb R$. Because $\mathbb R$ is not countable, the set $\mathbb R/\mathbb Q$ must be uncountably infinite. Otherwise, the countable union of countable sets would be countable. Thus, $|\mathbb R/\mathbb Q| = |\mathbb R|$. In summary: $$|\mathbb R/\mathbb Q| = |\mathbb R| > |\mathbb Q| \geq |\mathbb Q/\mathbb Z|.$$ This means that there does not exists a bijective function between the two groups.