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Math Help - Set Products Question

  1. #1
    Super Member Aryth's Avatar
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    Set Products Question

    I'm beginning a section on Product Topologies and I have to do a few problems, but I'm not seeing the difference between these two.

    Let \mathcal{A} and \mathcal{B} be two nonempty families of sets. Prove that:

    \bigcap \mathcal{A} \times \bigcap \mathcal{B} = \bigcap\{A \times B : A \in \mathcal{A}, \ B \in \mathcal{B}\}

    Suppose that A_1,\cdots ,A_n and B_1,\cdots ,B_n are sets. Prove that:

    (A_1\times B_1)\cap\cdots\cap (A_n\times B_n) = (A_1\cap\cdots\cap A_n) \times (B_1\cap\cdots\cap B_n)

    Aren't these asking for the same thing??
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  2. #2
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    Re: Set Products Question

    Hey Aryth.

    Does your combination contain all permutations Ai X Bj for all possible i and j or is i = j? (The way you have written it is unclear).
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  3. #3
    Super Member Aryth's Avatar
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    Re: Set Products Question

    To be honest I can't answer that question. The problems are exactly as I have typed them. In the proofs... I imagine that the equations would hold either way (Sets under intersections commute, so I can order them however I like on the right hand side and arrive at any necessary permutation as long as it's A_i \times B_j).
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  4. #4
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    Re: Set Products Question

    Quote Originally Posted by Aryth View Post
    I'm beginning a section on Product Topologies and I have to do a few problems, but I'm not seeing the difference between these two.

    Let \mathcal{A} and \mathcal{B} be two nonempty families of sets. Prove that:

    \bigcap \mathcal{A} \times \bigcap \mathcal{B} = \bigcap\{A \times B : A \in \mathcal{A}, \ B \in \mathcal{B}\}

    Suppose that A_1,\cdots ,A_n and B_1,\cdots ,B_n are sets. Prove that:

    (A_1\times B_1)\cap\cdots\cap (A_n\times B_n) = (A_1\cap\cdots\cap A_n) \times (B_1\cap\cdots\cap B_n)

    Aren't these asking for the same thing??
    No, they are not. The second assumes a finite number of sets (n) while the first does not.
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