1. ## Cartesian product

My teatcher wasn't able to explain this for me in a clear way. See if you guys can help.
Let A and B be two random sets. Prove that A $\displaystyle \times$ B (B $\displaystyle \cap$ C) = (A $\displaystyle \times$ B) $\displaystyle \cap$ (A $\displaystyle \cap$ C) by putting in the element (x,y)

2. Sort of difficult to see what you want to prove like this...

Did you mean: $\displaystyle A \times (B \times C) = (A \times B) \times (A \times C)$ ?

3. Sorry about that. Didn't know how to handle the LaTex-codes right. But I think know what's going on now.

But yes, that was what I meant. The book wants me to prove the equivalence by putting in the elemet (x,y) into that.

4. LaTex Help:
$$A \times \left( {B \times C} \right) \ne \left( {A \times B} \right) \times \left( {A \times C} \right)$$ gives $\displaystyle A \times \left( {B \times C} \right) \ne \left( {A \times B} \right) \times \left( {A \times C} \right)$.

Why? $\displaystyle A \times \left( {B \times C} \right)$ is a set of triples.
Whereas, $\displaystyle \left( {A \times B} \right) \times \left( {A \times C} \right)$ is a set of pairs made up of pairs.

5. Ok, this is how the question looks like and what the key says.

$\displaystyle A \times (B \cap C) = (A \times B) \cap (A \times C)$

Let (x,y) be a random element in $\displaystyle A \times (B \cap C)$
Then we have that X $\displaystyle \epsilon$ A and y $\displaystyle \epsilon$ B$\displaystyle \cap$C which give y $\displaystyle \epsilon$ B and y $\displaystyle \epsilon$C.

(x,y) $\displaystyle \epsilon$ A x B and (x,y) $\displaystyle \epsilon$ A x C so (x,y) $\displaystyle \epsilon$ (A x B) $\displaystyle \cap$ (A x C)
and now we can prove that (A x B) $\displaystyle \cap$ (A x C) $\displaystyle \subseteq$ A x (B$\displaystyle \cap$ C).

But I don't get a grip over this. For example why do you have to put the x in A and the y in (B$\displaystyle \cap$C) and not the other way around?

I would like to see this in a graphic 3D model so I could see what I was doing.

6. Originally Posted by The Lama
Ok, this is how the question looks like and what the key says.

$\displaystyle A \times (B \cap C) = (A \times B) \cap (A \times C)$

Let (x,y) be a random element in $\displaystyle A \times (B \cap C)$
Then we have that X $\displaystyle \epsilon$ A and y $\displaystyle \epsilon$ B$\displaystyle \cap$C which give y $\displaystyle \epsilon$ B and y $\displaystyle \epsilon$C.

(x,y) $\displaystyle \epsilon$ A x B and (x,y) $\displaystyle \epsilon$ A x C so (x,y) $\displaystyle \epsilon$ (A x B) $\displaystyle \cap$ (A x C)
and now we can prove that (A x B) $\displaystyle \cap$ (A x C) $\displaystyle \subseteq$ A x (B$\displaystyle \cap$ C).

But I don't get a grip over this. For example why do you have to put the x in A and the y in (B$\displaystyle \cap$C) and not the other way around?

I would like to see this in a graphic 3D model so I could see what I was doing.
You need to show $\displaystyle x \in A$ and $\displaystyle y \in {B \cap C}$ because that's simply the order of the cartesian product -- for example:

$\displaystyle X \times Y = \left\{ (x,y) : x \in X ; y \in Y\right\}$ While
$\displaystyle Y \times X = \left\{ (y,x) : y \in Y; x \in X\right\}$

And those are ordered pairs, so:
$\displaystyle (x,y) \neq (y,x) \Rightarrow X \times Y \neq Y \times X$

Also, your wording is a bit off... You should say "Let $\displaystyle (x,y) \in {A \times B}$", or "Let $\displaystyle (x,y)$ be some element in $\displaystyle A \times B$"; random is not quite the right word here!

7. Originally Posted by The Lama
(x,y) $\displaystyle \epsilon$ A x B and (x,y) $\displaystyle \epsilon$ A x C so (x,y) $\displaystyle \epsilon$ (A x B) $\displaystyle \cap$ (A x C)
and now we can prove that (A x B) $\displaystyle \cap$ (A x C) $\displaystyle \subseteq$ A x (B$\displaystyle \cap$ C).
But I don't get a grip over this. For example why do you have to put the x in A and the y in (B$\displaystyle \cap$C) and not the other way around?
More LaTex help.
$$(x,y)\in A\times (B\cap C)$$ gives $\displaystyle (x,y)\in A\times (B\cap C)$.

The is simply the way a cross product is defined.

$\displaystyle (x,y)\in W\times Z$ means $\displaystyle x\in W\text{ and }y\in Z$

8. Originally Posted by Defunkt
Also, your wording is a bit off... You should say "Let $\displaystyle (x,y) \in {A \times B}$", or "Let $\displaystyle (x,y)$ be some element in $\displaystyle A \times B$"; random is not quite the right word here!
Alright. Thanks. I just tried to translate it as best I could from a swedish compendium.

yea. x and $\displaystyle \times$. Got a bit lazy there I guess.