I was given a question that says Prove that $(A-B) \cap (A-C) = A \cap (B \cup C)^c$ for any three sets A, B, C.
I'm completely lost with this question. In a previous question that says $A \cap C \subseteq A- (B-C)$. I used this proof
Let $x \in A \cap C$. Then $x \in A$ and $x \in C$. Also note that $x \notin B-C$. As $x \in A$ and $x \notin B-C$, we see that $x \in A – (B-C)$. Therefore $A \cap C \subseteq A- (B-C)$.
With the question I just did i tried to apply that method with the question i struggled with but I couldn't see how it would work.
Best Answer
The set $(A-B)\cap(A-C)$ is "everything in $A$ that is not in $B$ and not in $C$", and the set $A\cap(B\cup C)^{\mathsf{c}}$ is "everything that is in $A$ and not in ($B$ or $C$)".
Can you see why these are the same thing?