Cardinality of a set of subsets obtained from different power sets

Question

the question is: $|A|=m,\; |B|=n,\; |A \cap B|=k.$ Find: $|P(A\cup B)-(P(A)\cup P(B))|$

How do I even start? I thought going with real numbers will help but the answers didn't make any sense.. $P(A)=2^m,\; P(b)=2^n$ but how to use it in case of $(P(A)\cup P(B))?$ It can be a lot of options.

Any help will be appreciated.

Answer 1

First prove for any sets K,L that
|K $\cap$ L| + |K $\cup$ L| = |K| + |L|.

Next show P(A $\cap$ B) = P(A) $\cap$ P(B).
and P(A) $\cup$ P(B) subset P(A $\cup$ B).

Apply as needed.