Intersection of two families of sets
Solution 1:
First, convince yourself that $$\bigcup_{j\in A\cap B}S_j\subset\left(\bigcup_{j\in A}S_j\right)\cap \left(\bigcup_{j\in B}S_j\right)$$ (you can show this by looking at any element $z\in S_j$ for $j\in A\cap B$, and arguing why it must be an element of the right side.) Let the first of these be $T$ and the second be $U$. Then, it's enough to show $$|T\cap U|\leq |P\cap Q|.$$ However, $P=T\cup(S_1-\{x\})$ and $Q=U\cup(S_2-\{y\})$. So, $T\subset P$ and $U\subset Q$. Can you see how to finish from here?