Prove that $P(\Bbb R)/\prec$ is countable and show that the class $[A]_\prec$ is an infinite set and not countable

Question

Let $\prec$ be the relation over $P(\Bbb R)$ defined as:

$A \prec B$ if and only if $|A \cap \Bbb Q| = |B \cap \Bbb Q|$.

Prove that the quotient set $P(\Bbb R)/\prec$ is countable and show that the class $[A]_\prec$ is an infinite set and not countable for all $X \in P(\Bbb R)$.

I know this implies that there is a bijection between $A \cap \Bbb Q$ and $B \cap \Bbb Q$. I'm stuck with both proofs.

Any help is greatly appreciated.

Answer 1

Hint
For the first one, the equivalence classes are defined by a unique representative in $\mathcal P(\mathbb Q)$. (why?)
Use a similar argument as for enumerating $\mathbb Q$.

For the second one try to find an uncountable subset of $[A]_\prec$ where $A\subset\mathbb Q$.