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Math Help - another mapping question

  1. #1
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    another mapping question

    Let f be a function from X onto Y and let B \subset Y. Show that f(f^{-1}(B))=B.

    I was thinking of something along the lines of

    let b \in B \ \& \ x \in X then f^{-1}(b) = x \longrightarrow f(f^{-1}(b)) = f(x) = b, since it's onto and it's pretty much by definition. I'm not quite sure that this is correct.
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  2. #2
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    z \in f\left( {f^{ - 1} (B)} \right)\; \Rightarrow \;\left( {\exists t \in f^{ - 1} (B)} \right)\left[ {z = f(t) \in B} \right]\;\; \Rightarrow \;f\left( {f^{ - 1} (B)} \right) \subseteq B

    By surjectivity b \in B\; \Rightarrow \;\left( {\exists s \in X} \right)\left[ {f(s) = b} \right].
    It follows: \left[ {f(s) = b} \right]\; \Rightarrow \;s \in f^{ - 1} (B)\; \Rightarrow \;b = f(s) \in f\left( {f^{ - 1} (B)} \right)
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  3. #3
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    Plato:

    I am sorry to bother you again but can you do this problem in predicate calculus,i would like to see the face of my lecturer to show him a proof in predicate calculus.

    Thanks
    Last edited by archidi; October 13th 2008 at 04:06 PM. Reason: forgot the name Plato
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