Cardinality of equivalence classes

Question

The relation ~ is defined on P(N): A~B if |A| = |B|.

I need to prove that the cardinality of the equivalence classes is countable.

Any ideas??

Answer 1

Hint: The cardinalities of the subsets of $\mathbb{N}$ are the natural numbers themselves and $\aleph_0$ ($=|\mathbb{N}|$), so this amounts to proving that the set $\mathbb{N} \cup \{ \aleph_0 \}$ is countable.