Prove the following for all sets A, B, and C:
$\displaystyle A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$
and
$\displaystyle A\cap B \subset A \subset A \cup C$
The first is known as Demorgan's law. Try thinking about it logically, it says that $\displaystyle x\in A\text{ and }(x\notin B\text{ or }x\notin C)\leftrightarrow (x\in A\text{ and }x\notin B)\text{ and }(x\in A\text{ and }x\notin C)$. Can you take it from there?
The second part is fairly standard. Where are you having trouble?
Actually, that's as far as I got on the first one... I did the $\displaystyle \rightarrow$ but I got stumped trying to do the $\displaystyle \leftarrow$ after stating what you did.
On the second one, I guess I just can't see it...
I've never been that good at Set Theory, it slightly confuses me for some reason. I have some time off so I'm trying to study it, and it's not going so well. :/