# Math Help - left and right inverses of functions

1. ## 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.

2. Originally Posted by prettymidget
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.