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Math Help - Function Proof

  1. #1
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    Function Proof

    Suppose f: A--> B and let C be a subset of A.

    Prove: if f is injective, then f^(-1)[f(C)] = C

    I know from a previous theorem that C is a subset of f^(-1)[f(C)] so need only to prove f^(-1)[f(C)] is a subset of C.

    So far I have:

    Let f be injective. Let x be an element of f^(-1)[f(C)] . Then f(x) is an element of f(C).
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  2. #2
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    Then by definition of f(C) we have \left( {\exists z \in C} \right)\left[ {f(z) = f(x)} \right].
    Now what does injectivity tell us?
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  3. #3
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    Since f is injective x = y and it then follows that x as an element of f^(-1)[f(C)] makes f^(-1)[f(C)] a subset of C.

    Thanks, Plato!
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