$\Bbb{R} / \Bbb{Q}$ is a quotient set of $\Bbb{R}$ with the following equivalence relation $\sim$ :
$$r \sim s \Longleftrightarrow r-s \in \Bbb{Q}$$
Then is there a bijection between $\Bbb{R}$ and $\Bbb{R} / \Bbb{Q}$?
I know that, with Axiom of Choice, there exists an injection from $\Bbb{R} / \Bbb{Q}$ to $\Bbb{R}$.
Thus $|\Bbb{R} / \Bbb{Q}| \leq |\Bbb{R}|$.
But I'm not certain that there exists an injection from $\Bbb{R}$ to $\Bbb{R} / \Bbb{Q}$.