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Math Help - Set Theory.

  1. #1
    Junior Member
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    Set Theory.

    Hi, could anyone help me start off this proof:

    Notation: For a set X we define \bigcup\ X \;=\;\{x \mid x\in y\ for\   some\   y\in X \}

    Then show if X_{ij} for i,j\in\mathbb{N} are sets then:

    \bigcap\limits^\infty_{i=0}(\bigcup\limits^\infty_  {j=0}X_{ij})\;=\;\bigcup\{( \bigcap\limits^\infty_{i=0}X_{ih(i)})\mid h\in\mathbb{N^{N}}\}
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  2. #2
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    If x\in LHS then \left( {\forall i} \right)\left( {\exists j} \right)\left[ {x \in X_{i~j} } \right].
    We are going to use that to define h:\mathBB{N}\to\mathBB{N} by h:i\mapsto j.
    So x\in X_{i~h(i)}.
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  3. #3
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    Thanks Plato, I think understand what you're saying but doesn't this only give \bigcap\limits^\infty_{i=0}(\bigcup\limits^\infty_  {j=0}X_{ij})\;\subseteq\;\bigcup\{( \bigcap\limits^\infty_{i=0}X_{ih(i)})\mid h\in\mathbb{N^{N}}\}?
    If so, how would I prove \supseteq
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