4.I.1C

What does it mean to say that a function $f: A \rightarrow B$ is injective? What does it mean to say that a function $g: A \rightarrow B$ is surjective?

Consider the functions $f: A \rightarrow B, g: B \rightarrow C$ and their composition $g \circ f: A \rightarrow C$ given by $g \circ f(a)=g(f(a))$. Prove the following results.

(i) If $f$ and $g$ are surjective, then so is $g \circ f$.

(ii) If $f$ and $g$ are injective, then so is $g \circ f$.

(iii) If $g \circ f$ is injective, then so is $f$.

(iv) If $g \circ f$ is surjective, then so is $g$.

Give an example where $g \circ f$ is injective and surjective but $f$ is not surjective and $g$ is not injective.

*Typos? Please submit corrections to this page on GitHub.*