# Thread: Proof involving sets, crossproducts, and intersections

1. ## Proof involving sets, crossproducts, and intersections

I have this problem, i'm not quite sure how to go about proving it. If anyone could point something helpful out or ideas, that would be great.

Prove: (AxB)∩(CxD)=(A∩C)x(B∩D)

Thanks.

2. ## Re: Proof involving sets, crossproducts, and intersections

Prove two inclusions.
1. $(A \times B) \cap (C \times D) \subset (A \cap C) \times (B \cap D)$
Let $x \in (A \times B) \cap (C \times D)$ then $x \in A \times B \wedge x \in C \times D$. Let $x=(u,v)$ then we have $(u,v) \in A \times B \wedge (u,v) \in C \times D$, thus $u \in A \cap C$ and $v \in B \cap D$. Hence $x = (u,v) \in (A \cap C) \times (B \cap D)$.

2. Prove the other inclusion ...