True or False: $A\cap (B\times C)=(A\cap B)\times (A\cap C)$ True or false. Prove by showing the statement is true in general or by giving a counter example.
$A\cap (B\times C)=(A\cap B)\times (A\cap C)$
I'm not sure what the question is trying to get at. Maybe there is some way to write the left to look like the right?

The statement doesn't really make much sense if you think about it.  B x C is a set of ordered pairs.  So if $A \cap (B \times C)$ isn't empty A must have some points that are ordered pairs.  The right hand side, however, allows A,B, and C to be "simple" sets that need not have any ordered pairs.  These are ... very different concepts.
Consider $A = B = C = \mathbb R$.
Then $A \cap (B \times C) = \mathbb R \cap (\mathbb R \times \mathbb R) = \emptyset$.
But $(A \cap B) \times (A \cap C) = \mathbb R \times \mathbb R$.
