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Math Help - Inverse of sets

  1. #1
    Mel
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    Inverse of sets

    I have attached a question that I am having trouble with. Thanks to anyone who can help.DOC1.DOC
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  2. #2
    MHF Contributor
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    Sets and Inverse functions

    Hello Mel

    Here is the problem.

    Suppose X and Y are nonempty sets and f: X \rightarrow Y is a function. Let A and B be subsets of Y. Prove that f^{-1} (A \cup B) = f^{-1}(A) \cup f^{-1}(B)

    We do this by showing that f^{-1} (A \cup B) \subseteq f^{-1}(A) \cup f^{-1}(B) and f^{-1}(A) \cup f^{-1}(B) \subseteq f^{-1} (A \cup B)


    So, for the first part:

    Suppose x \in f^{-1}(A \cup B). Then for some y \in Y, f(x) = y, and y \in A \cup B.

    \Rightarrow y \in A or y \in B [Note: throughout this proof 'or' means 'inclusive or'. So y \in A or y \in B means y \in A or y \in B or both.

    \Rightarrow f(x) \in A or f(x) \in B

    \Rightarrow x \in f^{-1}(A) or x \in f^{-1}(B)

    \Rightarrow x \in f^{-1}(A) \cup f^{-1}(B)

    So x \in f^{-1}(A \cup B) \Rightarrow x \in f^{-1}(A) \cup f^{-1}(B)

    \Rightarrow f^{-1}(A \cup B) \subseteq f^{-1}(A) \cup f^{-1}(B) (1)


    For the second part:

    Suppose x \in f^{-1}(A) \cup f^{-1}(B)

    Then x \in f^{-1}(A) or x \in f^{-1}(B)

    Then for some y \in Y, f(x) = y and y \in A or y \in B

    \Rightarrow y \in A \cup B

    \Rightarrow x \in f^{-1}(A \cup B)

    So x \in f^{-1}(A) \cup f^{-1}(B) \Rightarrow x \in f^{-1}(A \cup B)

    \Rightarrow f^{-1}(A) \cup f^{-1}(B) \subseteq f^{-1} (A \cup B) (2)


    So, from (1) and (2): f^{-1} (A \cup B) = f^{-1}(A) \cup f^{-1}(B)


    Grandad
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