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Math Help - Proof: Intersection, Inverse Function

  1. #1
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    Proof: Intersection, Inverse Function

    For this proof, we are to show that each containment is a subset of the other.

    Part 1: f^{-1}(\bigcap_{\lambda\in\Lambda}B_{\lambda}) = \bigcap_{\lambda\in\Lambda}f^{-1}(B_{\lambda}).

    Part 2: f^{-1}(\bigcup_{\lambda\in\Lambda}B_{\lambda}) = \bigcup_{\lambda\in\Lambda}f^{-1}(B_{\lambda}).

    I honestly have no idea how to get started on this...any insight is appreciated!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    It is almost routine knowing the definition of f^{-1}(B) . For example:

    x\in f^{-1}(\bigcap_{\lambda\in\Lambda}B_{\lambda})\Rightar  row f(x)\in \bigcap_{\lambda\in\Lambda}B_{\lambda}\Rightarrow f(x)\in B_{\lambda}\;\forall \lambda\in\Lambda\Rightarrow

    x\in f^{-1}(B_{\lambda})\;\forall \lambda\in\Lambda \Rightarrow x\in \bigcap_{\lambda\in\Lambda}f^{-1}(B_{\lambda})

    etc.
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  3. #3
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    This question was also discussed in this thread.
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