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Math Help - Inverse of the cartesian product

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    Inverse of the cartesian product

    Supppose that  \{ X_ \alpha \} _ { \alpha \in A } is an indexed family of sets, and define their Cartesian product  \pi_ { \alpha \in A } X_ \alpha to be the set of all maps  f: A \rightarrow \bigcup _{ \alpha \in A } such that  f( \alpha ) \in X_ \alpha \ \ \ \ \ \forall \alpha \in A

    Find the inverse function of the Cartestian product,  \pi ^ {-1} _ { \alpha } (E) for  E _ \alpha \subset Z_ \alpha

    My solution:

    The inverse function of the cartestian product should be all the inverse maps  f^{-1} : E \rightarrow Z_ \alpha which is defined by  f^{-1} = \{ \alpha \in A : f( \alpha ) \in E _ { \alpha } \subset Z _ \alpha \}

    But is that the best we can do? Thanks.
    Last edited by tttcomrader; August 29th 2009 at 07:01 AM.
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