[SOLVED] Set Proof Help

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March 7th 2009, 12:11 PM
BSC hiBi

[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 :)
March 7th 2009, 12:20 PM
running-gag

Hi

Usually A x B is the set of couples (x,y) where x is in A and y in B
March 7th 2009, 12:22 PM
BSC hiBi

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?
March 7th 2009, 12:38 PM
BSC hiBi

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?
March 7th 2009, 12:42 PM
Grandad

Cartesian Product Proof

Hello BSC hiBi

Quote:

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.

Grandad