# Left and right inverses

1. Sep 29, 2010

### prettymidget

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. Sep 29, 2010

### prettymidget

No one? :(

3. Sep 30, 2010

### prettymidget

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