The cardinality of $\mathbb{R}/\mathbb Q$

How to prove the cardinality of $\mathbb{R}/ \mathbb Q$ is equal to the cardinality of $\mathbb{R}$

I don't see what I would have expected to be the "standard answer" to this question, so let me leave it in the hopes it will be helpful to someone.

Proposition: Let $G$ be an infinite group, and let $H$ be a subgroup with $\# H < \# G$. Then $\# G/H = \# G$.

Proof: Let $\{g_i\}_{i \in G/H}$ be a system of coset representatives for $H$ in $G$: then every element $x$ in $G$ can be written as $x = g_{i_x} h_x$ for unique $h_x \in H$ and $i_x \in G/H$. (Note that there is no canonical system of coset representatives: getting one is an archetypical use of the Axiom of Choice.) Thus we have defined a bijection from $G$ to $G/H \times H$, so $\# G = \# G/H \cdot \# H$. Since $\# G$ is infinite, so must be at least one of $\# G/H$, $\# H$, and then standard cardinal arithmetic (again AC gets used...) gives that

$\# G = \# G/H \cdot \# H = \max(\#G/H, \# H)$.

Since we've assumed $\# H < \# G$, we conclude $\# G = \#G/H$.

This applies in particular with $G = \mathbb{R}$, $H = \mathbb{Q}$ to give $\# \mathbb{R}/\mathbb{Q} = \# \mathbb{R} = 2^{\aleph_0}$.

To prove equality we need to either find a bijection between the sets, or two injections between them.

As noted in the comments, this cannot be proved without the axiom of choice. So I am going to use it freely.

Assuming the axiom of choice, if so, we have a function $f\colon\mathbb{R/Q\to R}$ which chooses $f(A)\in A$ for every $A\in\mathbb{R/Q}$. This is an injection because if $A\neq A'$ then $f(A)\in A$ and $f(A)\notin A'$, and vice versa, therefore $f(A)\neq f(A')$.

On the other hand, let $V=\operatorname{rng}(f)$, then $\mathbb R$ is a countable union of copies of $V$, namely $\bigcup_{q\in\mathbb Q}q+V$. Therefore $|V|\cdot\aleph_0=2^{\aleph_0}$. Again, using the axiom of choice, we have that $2^{\aleph_0}=|V|\cdot\aleph_0=\max\{|V|,\aleph_0\}=|V|$.

Remark: This answers the original version of the question.

The following bijection uses Hilbert's infinite hotel.

The rationals can be enumerated as $q_0,q_1,q_2,\dots$ in various explicit ways.

Define $f: \mathbb{R}\to \mathbb{R}\setminus \mathbb{Q}$ as follows.

If $x$ does not have shape $q$, or $\sqrt{3}+q\sqrt{2}$, where $q$ is rational, let $f(x)=x$.

If $x$ is the rational $q_i$, let $f(x)=\sqrt{3}+q_{2i}\sqrt{2}$.

If $x=\sqrt{3}+q_i\sqrt{2}$, let $f(x)=\sqrt{3}+q_{2i+1}\sqrt{2}$.