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Some set proofs

  1. Jan 21, 2007 #1
    Disclaimer: I might have some problems getting my LaTeX code to work properly so please bear with me while I figure out how to properly use the forum software.

    1. The problem statement, all variables and given/known data
    The exercise is to prove the following statements.

    Suppose that [itex]f:X \rightarrow Y[/itex], the following statement is true.
    If [itex]\{G_{\alpha} : \alpha \in A\}[/itex] is an indexed family of subsets of Y, then [itex]f^{-1}(\bigcup_{\alpha \in A} G_\alpha) =\bigcup_{\alpha \in A} f^{-1}( G_\alpha) [/itex].


    2. Relevant equations

    The relevant information in this case is the definitions. In this case I need to know what the definition of an inverse function is.

    DEF: Suppose [itex]f:X \rightarrow Y[/itex] and [itex] A \subset Y[/itex]. [itex]f^{-1}(A) = \{x \in X: f(x) \in A\}[/itex]


    3. The attempt at a solution

    The solution I've been looking thus far is a point wise argument.

    Choose [itex]t \in f^{-1}(\bigcup_{\alpha \in A} G_\alpha)[/itex]. So by definition we know
    [itex]t \in \{x \in X: f(x) \in (\bigcup_{\alpha \in A} G_\alpha)\}[/itex]. And here is where I'm kind of stuck. I need to some how get my chosen element to be in the other set, therefore making the one set a subset of the other. Then complete the converse of the argument to finish the proof.

    Any ideas on where I should be looking, or what I should be thinking about here?
     
    Last edited: Jan 21, 2007
  2. jcsd
  3. Jan 21, 2007 #2

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    [itex]f(x) \in \cup_{\alpha \in A} G_\alpha[/itex] is equivalent to "there is some [itex]\alpha \in A[/itex] such that [itex]f(x) \in G_\alpha[/itex]". Also, you don't have to worry about doing the converse if every step you take is an equivalence.
     
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