Consider $ \ A \cap (B-C)$ and $ \ (A\cap B) - (A \cap C)$. Question:
Consider $ \ A \cap (B-C)$ and $ \ (A\cap B) - (A \cap C)$.
Are the two sets equal or is one a subset of the other?
My attempt: 
We know that if $ \ x\in (A \cap B) - (A \cap C) \implies x \in A \cap B$ and $ \ x \notin A \cap C \implies x \in A$ and $ \ x\in B$ and $ \ x\notin C \implies x \in A $and $ \ x\in B-C \implies x \in A \cap (B-C)$
So, $ \ (A\cap B) - (A \cap C) \subseteq \ A \cap (B-C) $
Is the other way true?

Through logic it should be true using definition $\forall x: x\in (X-A) \iff (x \in X) \land (x \notin A)$  that
$$\forall x: (x \in A) \land ((x \in B) \land (x\notin C)) =\forall x:  ((x \in A) \land (x \in B)) \land ((x \notin C) \lor (x\notin A))$$
because in order to satisfy the formula, $x$ must be in $A$, therefore $(x \notin A)$ is always false and the rightmost term is false, meaning that $(x \notin C) \lor (x\notin A)$ is the same as $(x \notin C)$
