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Thread: Algebra - Please help

  1. #1
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    Algebra - Please help

    $\displaystyle X_1$, $\displaystyle X_2$ are nonempty sets. Let $\displaystyle \mathcal{F}_1 \subset 2^{X_1}$ and $\displaystyle \mathcal{F}_2 \subset 2^{X_2}$ be $\displaystyle \sigma$-algebras.

    Let $\displaystyle \mathcal{R} = \{A \times B: A \in \mathcal{F}_1, B \in \mathcal{F}_2 \}$

    And $\displaystyle \mathcal{R}_1 = \left\{\bigcup_{i=1}^n R_i: n \in \mathbb{N}, \ \{R_i\}_{i=1}^n \subset \mathcal{R} \right\}$

    Prove that $\displaystyle \mathcal{R}_1$ is an algebra.

    It was not hard to prove that it is colsed with respect to finite unions. However I failed to prove it is closed with respect to complement!
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    Forum Admin topsquark's Avatar
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    And this relates to Probability and Statistics, how?

    Thread moved.

    -Dan

    Edit: My bad. Sorry about that!
    Last edited by topsquark; Jun 16th 2008 at 03:18 PM.
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  3. #3
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    Quote Originally Posted by topsquark View Post
    And this relates to Probability and Statistics, how?

    Thread moved.

    -Dan
    Because axiomatic probability theory involves measures and $\displaystyle \sigma$-algebra.
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  4. #4
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    Frankly speaking I was working on some probability theory theorem when this problem occured. I'm 99% sure $\displaystyle R_1$ is algebra but the full proof seems bit too long.... If somebody knows a short proof, please post it.

    PS I think I can prove that $\displaystyle R_1$ is closed under finite intersections. Than I take complement of set that belongs to $\displaystyle R_1$ and I get finite intersection - the only problem is to find a simple proof that the "intersected" sets belong to $\displaystyle R_1$ .
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  5. #5
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    Quote Originally Posted by albi View Post
    Frankly speaking I was working on some probability theory theorem when this problem occured. I'm 99% sure $\displaystyle R_1$ is algebra but the full proof seems bit too long.... If somebody knows a short proof, please post it.

    PS I think I can prove that $\displaystyle R_1$ is closed under finite intersections. Than I take complement of set that belongs to $\displaystyle R_1$ and I get finite intersection - the only problem is to find a simple proof that the "intersected" sets belong to $\displaystyle R_1$ .
    these two trivial identities should be enough for you to complete the proof:

    $\displaystyle 1) \ \ (A_1 \times B_1) \cap (A_2 \times B_2) = (A_1 \cap A_2) \times (B_1 \cap B_2).$

    $\displaystyle 2) \ \ (A \times B)^c = (A^c \times B) \cup (A \times B^c) \cup (A^c \times B^c),$ where $\displaystyle Y^c$ is the complement of $\displaystyle Y.$
    Last edited by NonCommAlg; Jun 16th 2008 at 01:07 PM.
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  6. #6
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    The last one is not true. I belive it should be:
    $\displaystyle
    (A \times B)^c = (A^c \times B) \cup (A \times B^c) \cup (A^c \times B^c),
    $

    However that will help.
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  7. #7
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    Quote Originally Posted by albi View Post
    The last one is not true. I belive it should be:
    $\displaystyle
    (A \times B)^c = (A^c \times B) \cup (A \times B^c) \cup (A^c \times B^c),
    $
    of course! thanks! i fixed it. i've already done too much latex for today ... i guess i'm getting really tired!
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