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Homework Help: Prove onto

  1. Sep 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if f: [tex]X \rightarrow Y[/tex] is onto, then [tex]f(f^{-1}(B))=B[/tex] [tex]\forall B \in Y[/tex]


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 15, 2010
  2. jcsd
  3. Sep 15, 2010 #2

    radou

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    Homework Helper

    What does it mean for a function to be onto? What kind of inverse does f possess iff it is onto?
     
  4. Sep 15, 2010 #3
    Onto means that for a function [tex]f:A \rightarrow B [/tex] if [tex]\forall b \in B[/tex] there is an [tex]a \in A: f(a)=b[/tex]

    The inverse means that if you take the [tex]f^{-1}(b)[/tex] that it should map back to a?
     
  5. Sep 15, 2010 #4

    radou

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    Correct. But note that a right inverse exists if the function is onto. I.e., if g is a right inverse of f, then f(g(y)) = y, for every y in Y. What you need to prove is a direct consequence of this fact. (I used "g" rather than "f^-1" for the right inverse to avoid confusion leading to a conclusion that f^-1 is an inverse, i.e. both left and right).
     
  6. Sep 16, 2010 #5
    So I need to prove that if [tex]f(y)=Y[/tex] and [tex]f^{1}(Y)=y[/tex], that [tex]f(f^{1}(Y))=Y[/tex]?
     
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