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Thread: [SOLVED] Finally in LaTex: Pre-images and indexed set.

  1. #1
    Member ilikedmath's Avatar
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    Exclamation [SOLVED] Finally in LaTex: Pre-images and indexed set.

    Let f be a function from X into Y.
    If is a family of subsets of Y, prove that = .

    My thinking so far (which isn't a lot because I still don't have a grasp on this concept of indexed sets):

    So how I'm understanding this is that the question is asking me to prove that the pre-image (and not the inverse function!) of the intersection of a certain family of subsets of Y (the set {
    $\displaystyle B_\alpha$}) is the intersection of the pre-images of $\displaystyle B_\alpha$ .

    The pre-image is in the domain. In this case the domain is X. So I have to show that the intersection of the pre-images of
    $\displaystyle B_\alpha$ is the same as the pre-image of the intersection of all sets $\displaystyle B_\alpha$.

    Is that even close?
    Thanks for your time.
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  2. #2
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    $\displaystyle u \in \left( { \cap _\alpha f^{ - 1} (B_\alpha )} \right) \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {u \in f^{ - 1} (B_\alpha )} \right]$
    $\displaystyle \begin{gathered}
    \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {f(u) \in (B_\alpha )} \right] \hfill \\
    \Leftrightarrow \left[ {f(u) \in \cap _\alpha (B_\alpha )} \right] \hfill \\
    \Leftrightarrow u \in f^{ - 1} \left( { \cap _\alpha (B_\alpha )} \right) \hfill \\
    \end{gathered} $

    That is a start. Can you do the details?
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  3. #3
    Member ilikedmath's Avatar
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    Quote Originally Posted by Plato View Post
    $\displaystyle u \in \left( { \cap _\alpha f^{ - 1} (B_\alpha )} \right) \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {u \in f^{ - 1} (B_\alpha )} \right]$
    $\displaystyle \begin{gathered}
    \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {f(u) \in (B_\alpha )} \right] \hfill \\
    \Leftrightarrow \left[ {f(u) \in \cap _\alpha (B_\alpha )} \right] \hfill \\
    \Leftrightarrow u \in f^{ - 1} \left( { \cap _\alpha (B_\alpha )} \right) \hfill \\
    \end{gathered} $

    That is a start. Can you do the details?
    I think I can see it now. Thanks!
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