# Thread: [SOLVED] Set Proof Help

1. ## [SOLVED] Set Proof Help

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

2. Hi

Usually A x B is the set of couples (x,y) where x is in A and y in B

3. So instead of letting x be an element when you deal with anything that has Cartesian products you use let (x,y) be an element? Seems reasonable. I can solve the right side easily I think, but tips on how to start the left to right?

4. 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?

5. ## Cartesian Product Proof

Hello BSC hiBi
Originally Posted by BSC hiBi
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 the proof that $A \times (B-C) \subseteq (A\times B)-(A\times C)$.

$(x, y) \in A \times (B-C) \Rightarrow x \in A \wedge y \in (B-C)$

$\Rightarrow x \in A \wedge y \in B \wedge y \notin C$

$\Rightarrow (x,y) \in A \times B \wedge (x,y) \notin A \times C$

$\Rightarrow (x, y) \in (A \times B) - (A \times C)$

$\Rightarrow A \times (B-C) \subseteq (A\times B)-(A\times C)$

Now you need to prove that
$(A\times B)-(A\times C) \subseteq A \times (B-C)$ in order to complete the proof.