how do you prove that A ∩ ( B ∪ C) = (A ∩ B) ∪ ( A ∩ C) for any sets A, B,C?
thanks.
You prove that both sets are contained in each other as follows:
Let $\displaystyle x\in A\cap (B\cup C)$. Then by definition $\displaystyle x\in A$ and $\displaystyle x\in B\cup C$
So $\displaystyle x\in A$ and $\displaystyle (x\in B\text{ or }x\in C)$
If $\displaystyle x\in B$ then $\displaystyle x\in A\cap B$ and so $\displaystyle x\in (A\cap B)\cup (A\cap C)$
If $\displaystyle x\in C$ then $\displaystyle x\in A\cap C$ and so $\displaystyle x\in (A\cap B)\cup (A\cap C)$
This proves $\displaystyle A\cap (B\cup C)\subseteq (A\cap B)\cup (A\cap C)$
can you show the other direction?
Well it won't help you prove anything. I mean you can of course declare whatever you want, but someone could ask but what if x is not in B and you wouldn't be able to answer that person
To prove that 2 sets, say X and Y are equal, we need to show that they contain the same elements right? So one way of doing this is by picking an arbitrary element of X, call it p.
Let $\displaystyle p\in X$. Then we use the given information to conclude that p is also in Y. Since we picked p arbitrarily, it holds for all $\displaystyle p\in X$ and so every element of X is in Y
Then we pick an arbitrary element of Y and show it must be in X. Then since every element of X is in Y and every element of Y is in X, X=Y
So to prove the other direction of your question, you want to start out by saying
Let $\displaystyle x\in (A\cap B)\cup (A\cap C)$
now use your knowledge of unions and intersections do determine that x must also be in $\displaystyle A\cap (B\cup C)$
So try to take it from here before I help you further