162 4 Functions
Exercise 4.4.14. [Used in Section 4.4.] Let A, B and C be sets, and let $f$ : A$\to$ B and
g: B$\to$ C be functions.
(1) Prove that if $g \circ f$ is injective, then $f$ is injective.
(2) Prove that if $g \circ f$ is surjective, then $g$ is surjective.
(3) Prove that if $g \circ f$ is bijective, then $f$ is injective, and $g$ is surjective.
(4) Find an example of functions $f$ : A$\to$ B and $g$ : B$\to$ C such that $g \circ f$ is
bijective, but $f$ is not surjective, and $g$ is not injective. Hence Parts (1)-(3) of
this exercise are the best possible results.