How do you prove the following? Let $\displaystyle f: A \to B $.

(1) $\displaystyle f $ is injective if it has a left inverse.

(2) $\displaystyle f $ is surjective if it has a right inverse.

(3) $\displaystyle f $ is bijective if it has both left and right inverse.

(4) if $\displaystyle |A| = |B| $ then $\displaystyle f: A \to B $ is bijective if and only if $\displaystyle f $ is injective if and only if $\displaystyle f $ is surjective.

Thanks