Paper 4, Section II, E

(a) Let $A$ and $B$ be non-empty sets and let $f: A \rightarrow B$.

Prove that $f$ is an injection if and only if $f$ has a left inverse.

Prove that $f$ is a surjection if and only if $f$ has a right inverse.

(b) Let $A, B$ and $C$ be sets and let $f: B \rightarrow A$ and $g: B \rightarrow C$ be functions. Suppose that $f$ is a surjection. Prove that there is a function $h: A \rightarrow C$ such that for every $a \in A$ there exists $b \in B$ with $f(b)=a$ and $g(b)=h(a)$.

Prove that $h$ is unique if and only if $g(b)=g\left(b^{\prime}\right)$ whenever $f(b)=f\left(b^{\prime}\right)$.

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