Proofs of Sets and Subsets I have these proof problems that I need some help on, any direction would be great. Thanks
Let A, B, and C be subsets of some universal set U
(a) Prove the following:
IF $A \cap B$ $\subseteq$ C, and $'A \cap B$ $\subseteq$ C, THEN $B \subseteq C$
(b) Either prove the following or provide a counterexample:
IF $A \cap B$ = $A \cap C$ and $'A \cap B$ = $'A \cap B$ = $'A \cap C$, THEN B = C 

Hint.  For (a) it is given that
$$A\cap B\subseteq C\ ,\quad A'\cap B\subseteq C$$
and you have to prove $B\subseteq C$.  You should know the basic way of proving a subset statement like this: assume $x$ is in the LHS, and use this assumption (and the given facts) to prove that $x$ is in the RHS.
So, let $x\in B$.  Consider two cases: either $x\in A$ or $x\in A'$.


*

*Case 1, $x\in A$.  Then $x\in A\cap B$, so $x\in C$.

*Case 2, [fill in the details yourself].


In both cases, $x\in C$.  Therefore $B\subseteq C$.
You can use (a) to answer (b).  We have
$$A\cap B=A\cap C\subseteq C\ ,\quad A'\cap B=A'\cap C\subseteq C\ ,$$
so by (a) we get $B\subseteq C$.  See if you can write out a similar argument to show $C\subseteq B$ and thereby prove $B=C$.
