# Show that P(A) is a subset of P(B) if A is a subset of B

**Proof**: Suppose A is a subset of B.
Let x be any element of $$\mathcal{P}(A)$$, the power set of A.
Then x is a subset of A by the definition of the power set.
But because A is a subset of B, x must also be a subset of B.
Hence x is an element of $$\mathcal{P}(B)$$ by the definition of power set.
Because this condition holds for any element of $$\mathcal{P}(A)$$,
$$\mathcal{P}(A)$$ is a subset of $$\mathcal{P}(B)$$.
Q.E.D.

*Does this proof work for uncountably infinite sets A and B?*
It is intuitively clear that this proof works for any finite set A.
But if A was countably infinite, or uncountably infinite,
then I’m not sure if this argument makes any sense at all.