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Thread: Cartesian product of universal set.

  1. #1
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    Cartesian product of universal set.

    Hi, I don't know how to prove this.

    $\displaystyle UXA^C = (UXA)^C$

    What is the result of UxA? I know Ax= but have no idea about universal set. Thanks for any help.
    Last edited by truevein; Nov 6th 2010 at 12:37 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by truevein View Post
    Hi, I don't know how to prove this.

    $\displaystyle UXA^C = (UXA^C)$

    What is the result of UxA? I know Ax= but have no idea about universal set. Thanks for any help.
    I believe you're trying to prove that $\displaystyle U\times\left(U-A\right)=U-\left(U\times A\right)$. Is that right?
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    Really sorry about the displacement of the parenthesis, I corrected my question.
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    We know that $\displaystyle \left( {\forall x} \right)\left[ {x \in U} \right]$, the universe.

    So if $\displaystyle (x,y)\in U\times A^c $ then because $\displaystyle x\in U$ we know $\displaystyle y\notin A$ so $\displaystyle (x,y)\in (U\times A)^c$.

    If $\displaystyle (u,w)\in (U\times A)^c$ then $\displaystyle u \notin U \vee w \notin A$.
    Again we know that $\displaystyle u\in U$ so $\displaystyle w\notin A$.
    So $\displaystyle (u,w)\in U\times A^c$.
    Last edited by Plato; Nov 6th 2010 at 02:12 PM.
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