Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

Question

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?

Answer 1

In this case, the proof is rather trivial: since the $\cap$ operator is both associative and commutative, all the following are equivalent: $$ (A \cap B) \cap (A \cap C) $$ $$ A \cap B \cap A \cap C $$ $$ A \cap A \cap B \cap C $$ $$ A \cap B \cap C $$ $$ A \cap (B \cap C) $$

As others said, whether or not you actually have to include the proof is mostly a matter of context.

Answer 2

Statement: $A \cap (B' \cap C') = (A \cap B') \cap (A \cap C')$

• $A \cap (B' \cap C') \subset (A \cap B') \cap (A \cap C')$

$x \in A \cap (B' \cap C') \implies x \in A$ and $x \in B' \cap C' \implies x \in A, x \in B', x \in C' \implies x \in A \cap B'$ and $x \in A \cap C' \implies x \in (A \cap B') \cap (A \cap C')$

• $(A \cap B') \cap (A \cap C') \subset A \cap (B' \cap C')$

You can do it similarly.