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Homework Help: If g o f is surjective, then is f surjective?

  1. Aug 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Assume f:A[tex]\rightarrow[/tex]B
    g:B[tex]\rightarrow[/tex]C
    h=g(f(a))=c
    Give a counterexample to the following statement. If h is surjective, then f is surjective.



    2. Relevant equations
    Definition ofSurjection: Assume f:A[tex]\rightarrow[/tex]B, For all b in B there is an a in A such that f(a)=b


    3. The attempt at a solution
    f(a)=1/a from [tex]\Re[/tex] to [tex]\Re[/tex]
    g(b)=1/b from [tex]\Re[/tex] to [tex]\Re[/tex]
    h(a)=a from [tex]\Re[/tex] to [tex]\Re[/tex]

    h is a surjection and f is not.

    Does this work?
     
  2. jcsd
  3. Aug 5, 2010 #2

    vela

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    Re: Surjection

    No, h(0) isn't defined since f(0) isn't defined. You can only say h(a)=a for a≠0.
     
  4. Aug 5, 2010 #3

    Dick

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    Re: Surjection

    Don't try and make life complicated. You can find a counterexample with finite sets. Take A={1,2}, B={1,2} and C={1}. Now define f and g.
     
  5. Aug 5, 2010 #4
    Re: Surjection

    It is always little things like that which I don't see with these problems. Does this one work?

    f(a)=a2 [tex]\Re[/tex] [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
    g(b)=b3/2 [0,[tex]\infty[/tex]) [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
    h(a)=a3 [tex]\Re[/tex] [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
     
  6. Aug 5, 2010 #5
    Re: Surjection

    oh ok thanks dick I will try that
     
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