# Cartesian Product in Element Method

I'm trying to do proofs using element method, but a few problems popped up with Cartesian products in them. How do these fit in?

For example, I know I can break

(A $\cap$ B) into $\ [x \in A] \wedge [x \in B]$

How do I work Cartesian products into this?

One example problem I'm working on is:

$$(A\times B)-(A\times C) \subset A\times(B-C)$$

Can someone help step me through this? Thank you!

-

You know that the elements of any Cartesian Products is of form $(x,y)$. Now let $$(x,y)\in(A\times B)-(A\times C)$$ so as $(A-B)=A\cap B'$ so $$(x,y)\in(A\times B),~~(x,y)\notin(A\times C)$$ so $$(x,y)\in(A\times B)\to x\in A,~y\in B$$ and $$(x,y)\notin(A\times C)\to x\in A,~y\notin C$$ These two latter ones mean that $x\in A, y\in B, y\notin C$.
$x\in (A\times B)-(A\times C)$ means that $x\in A\times B$ and $x\notin A\times C$. Now, $x\in A\times B$ means $x=(a,b)$ with $a\in A, b\in B$. Since $x=(a,b)\notin A\times C$ it follows from $a\in A$ that $b\notin C$. Therefore, $x=(a,b)\in A\times (B-C)$.