I'm self studying Ethan Bloch's book Proofs and Fundamentals and I'd like to know how to prove this (this is the book's Exercise 3.3.11).
Let $X$ be a set, and let $A, B, C \subseteq X$ be subsets. Suppose that $A \cap B = A \cap C$, and that $(X-A) \cap B = (X-A) \cap C$. Prove that $B = C$.
To prove that $B = C$ we need to show that $B \subseteq C$ and $C \subseteq B$, but I don't know how to actually begin proving it. I think that this is the first time I'm trying to prove something set theoretic with the aid of given statements, I've only done more "direct" proofs before.
I've managed to gather from $A \cap B = A \cap C$ that the sets $B$ and $C$ are not disjoint (unless one or both of them is the empty set), but that's the farthest I've gotten. Any kind help is appreciated.