Proofs of Sets and Subsets

Question

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

Answer 1

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 $B=C$.