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Thread: Cartesian product

  1. #1
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    Smile 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)
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  2. #2
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    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) $ ?
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  3. #3
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    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.
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  4. #4
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    LaTex Help:
    [tex]A \times \left( {B \times C} \right) \ne \left( {A \times B} \right) \times \left( {A \times C} \right)[/tex] 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.
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  5. #5
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    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.
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  6. #6
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    Quote Originally Posted by The Lama View Post
    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!
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  7. #7
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    Quote Originally Posted by The Lama View Post
    (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.
    [tex](x,y)\in A\times (B\cap C)[/tex] 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 $\displaystyle x\in W\text{ and }y\in Z$
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  8. #8
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    Quote Originally Posted by Defunkt View Post
    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.
    Last edited by The Lama; Aug 27th 2009 at 08:55 AM. Reason: syntax error
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