Show that if A and B are sets with the same cardinality, then the power set of A and the power set of B have the same cardinality.

Since A and B have the same cardinality there is a bijection between A and B.

Therefore each element of A can be paired with each element of B.

It then follows that every subset of A can be paired with every subset of B.

This means that there is a bijection between the power set of A and the power set of B.

Which proves that the power set of A has the same cardinality as the power set of B.

Can anyone validate or show if and where this proof is wrong? I have a quiz tomorrow.

Thanks