In general, let us denote the identity function for a set C by iC. That is, define  to be...

In general, let us denote the identity function for a set C by iC. That is, define  to be the function given by the rule iC(x) = x for all . Given . we say that a function  is a left inverse for f if ; and we say that  is a right inverse for f if .

(a) Show that if f has a left inverse, f is injective; and if f has a right inverse, f is surjective.

(b) Give an example of a function that has a left inverse but no right inverse.

(c) Give an example of a function that has a right inverse but no left inverse.

(d) Can a function have more than one left inverse? More than one right inverse?

(e) Show that if f has both a left inverse g and a light inverse h, then f is bijective and g = h = f−1.