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Homework Help: Left and right inverses

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data
    a) Prove or disprove: If f:X---->Yhas at least one left inverse g:Y---->X but has no right inverse, then f has more than one such left inverse.

    b) Prove or disprove: If f and g are maps from a set X to X and fog is injective, then f an g are both injective. (fog being function composition).
    2. Relevant equations



    3. The attempt at a solution
    a) I think it is true.

    Assume f only has one such inverse, i.e. g is unique.
    If f has no right inverse, there exists no map h such that f(h(a))=a for all a in X.
    g(f(a))=a for all a in X.

    b) False, found a counterexample. the inner function need not be injective. Still stuck on a though.
     
    Last edited: Sep 29, 2010
  2. jcsd
  3. Sep 29, 2010 #2
    No one? :(
     
  4. Sep 30, 2010 #3
    I hate bumping threads but I'm getting desperate. I found that my counterexample for b does not work because I forgot that f and g need to map X into itself, and now I actually think b may be true.
     
    Last edited: Sep 30, 2010
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