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Math Help - Basic proof of onto functions

  1. #1
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    Basic proof of onto functions

    "Suppose $f: A \to B$ is onto. Show that $\forall \, Y \subset B, f(f^{-1}(Y)) = Y$.

    Here's what I did:

    1. Observe that $\forall x \in f^{-1}(Y), f(x) \in Y$. Then $f(f^{-1}(Y)) \subset Y$.
    2. Since f is onto, $Y \subset f(A)$.

    This is where I'm stuck. How do I continue step 2 to show that $f(f^{-1}(Y)) \supset Y$? Obviously $f^{-1}(Y) \subset A$, but I don't see how that helps...
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  2. #2
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    Re: Basic proof of onto functions

    Quote Originally Posted by phys251 View Post
    "Suppose $f: A \to B$ is onto. Show that $\forall \, Y \subset B, f(f^{-1}(Y)) = Y$.

    1. Observe that $\forall x \in f^{-1}(Y), f(x) \in Y$. Then $f(f^{-1}(Y)) \subset Y$.
    2. Since f is onto, $Y \subset f(A)$.
    Suppose $t\in Y$ then $\exists b\in A $ such that $f(b)=t$. Does that mean $t\in f(A)~?$
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  3. #3
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    Re: Basic proof of onto functions

    Let $ y \in Y $. Since $f$ is onto, there exists $ a \in A $ such that  f(a) = y . Hence, a\in f^{-1}(Y) implies y = f(a) \in f(f^{-1}(Y)). This shows Y\subseteq f(f^{-1}(Y)).
    Last edited by Plato; May 15th 2014 at 01:18 PM.
    Thanks from phys251
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