Let A, B, C be sets.

Here's my attempt:

Suppose $\displaystyle x \in A - (B \cap C)$ then $\displaystyle x \in A$ and $\displaystyle x \notin B$ and $\displaystyle x \notin C$
So, x is in A not B or C, therefore (is this explanation sufficient?):

$\displaystyle x \in (A-B) \cup (A-C)$

$\displaystyle

A - (B \cap C) \subseteq (A-B) \cup (A-C)

$

Conversely, let $\displaystyle x \in (A-B) \cup (A-C)$, which means $\displaystyle x \in A$ and $\displaystyle x \notin B$ or $\displaystyle x \in A$ and $\displaystyle x \notin C$

(I'm not sure what to explain here!)

Therefore, $\displaystyle x \in A - (B \cap C) $

Hence: $\displaystyle (A-B) \cup (A-C) \subseteq A - (B \cap C)$

I appreciate it if anyone could help me with this. Thanks.