Hi
Usually A x B is the set of couples (x,y) where x is in A and y in B
Let, A, B, and C be sets. Prove that A x (B - C) = (A x B) - (A x C)
I started by letting x be an element of A x (B - C). Since I'm starting from the left and going right. Could someone explain to me what the x's mean here and how I apply them? If I understand that I can solve this.
-Thanks
Here's my answer. Anyone mind telling me if there's things wrong with it? Prove A x (B - C) = (A x B) - (A x C)
Pf: A x (B - C)
Let (x,y) be an element of A x (B - C). Based on the Cartesian product we know x is an element of A, and y is an element of B. Thus (x,y) is an element of (AxB). Similarly x is an element of A, but y is not an element of C. In either case x is an element of A, but y is an element of B, and not C. Thus (x,y) is an element of (A x B) - (A x C).
Pf: (A x B) - (A x C)
Let (x,y) be an element of A x (B - C). Based on the Cartesian prudct we know x is an element of A and y is an element of B, or x is an element of A and y is not an element of C. In either case x is an element of A. Thus x is an element of A. If y is an element of B, but not an element of C we get (B - C). Thus (x,y) is an element of A x (B - C).
How'd I do?