Expert: Sherry Wallin - 3/21/2009

Question
For any sets A, B, C, D prove (A x B) U (C X D)= (A U C) x (B U D).  I am not too sure where to start with this one.  I am sure I will need to assume x is in (A x B) U (C X D) then show it is in (A U C) x (B U D) and then vice versa but am not sure how to do this?

Any guidance would be appreciated.

Thanks, Steve

Answer
Hi Steve, you are right in your assumption. The next thing for you to decide is what an element in
(A x B) U (C x D) looks like. Elements in (A x B) are just ordered pairs (a,b) where a comes from A and b comes from B. So the element in (C x D) can be (c,d) likewise. Now you have the union of (a,b) and (c,d). What can you do with ordered pairs as a Cartesian cross product? (a+c, b+d) but isn't this really just
(A U C) x (B U D)? Now you do the other direction.