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Math Help - left and right inverses of functions

  1. #1
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    left and right inverses of functions

    Prove or disprove: If f:X---->Yhas 1 or more left inverses g:Y---->X but possesses no right inverse, then f has more than one left inverse.

    I am inclined to 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.

    stuck.
    Last edited by prettymidget; October 1st 2010 at 07:14 PM.
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  2. #2
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    Quote Originally Posted by prettymidget View Post
    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.

    I am inclined to 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.

    stuck.
    I believe the function f: A -> B where A = {1}, B = {1,2} and f(1)=1 is a counterexample.
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