Dear Friends ,
The following points can be proved easily :-
a) A function has a left inverse if and only if it is one-one.
b) A function has a right inverse if and only if it is onto.
A function is said to have an inverse if and only if its left and right inverses exist and are equal to each other.
We also know that a function has an inverse only if it is bijective. However the if part is never said. That is , it is never said that a function has an inverse if and only if it is bijective. We remain silent about the if part.
The question is that is it possible for a bijective function to have left and right inverses that are unequal ? Putting it into another way , the question is that is it possible for a function to be bijective and still not have an inverse ?
Please reply and point out any error in the argument if any .