Prove that for all sets A, B, and C, Ax(B-C) = (AxB)-(AxC)

[kimberlyd1020]

"Prove that for all sets A, B, and C, Ax(B-C) = (AxB)-(AxC)"

I know that I need to prove that the left side is a subset of the right side and the right side is a subset of the left side in order for this to be equal. I also know that B-C means the relative complement, so x is an element of B but not an element of C for the left side. I also know that the cross product of, say, AxB in the left side means it is the set {(a,b)|a is an element of A and b is an element of B}.

I have no idea how to apply any of this to write a correct proof. I am going to continue working on it and continue throwing out my scrap paper until I figure this one out....

 

[daon]

You have the right idea.

If x is in the LHS, then it is of the form (a,b) where a is an element of A and b is an element of B that is not in C. Therefore we know (a,b) belongs to AxB, but it can't be in AxC since b is not in C. In other words, (a,b), that is, x, is in the difference.Try the other way.