Originally Posted by

**The Lama** Ok, this is how the question looks like and what the key says.

$\displaystyle A \times (B \cap C) = (A \times B) \cap (A \times C)$

Let (x,y) be a random element in $\displaystyle A \times (B \cap C) $

Then we have that X $\displaystyle \epsilon$ A and y $\displaystyle \epsilon$ B$\displaystyle \cap$C which give y $\displaystyle \epsilon$ B and y $\displaystyle \epsilon$C.

(x,y) $\displaystyle \epsilon$ A x B and (x,y) $\displaystyle \epsilon$ A x C so (x,y) $\displaystyle \epsilon$ (A x B) $\displaystyle \cap$ (A x C)

and now we can prove that (A x B) $\displaystyle \cap$ (A x C) $\displaystyle \subseteq$ A x (B$\displaystyle \cap$ C).

But I don't get a grip over this. For example why do you have to put the x in A and the y in (B$\displaystyle \cap$C) and not the other way around?

I would like to see this in a graphic 3D model so I could see what I was doing.