Paper 4, Section I, E

What does it mean to say that a function $f: X \rightarrow Y$ has an inverse? Show that a function has an inverse if and only if it is a bijection.

Let $f$ and $g$ be functions from a set $X$ to itself. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.

(i) If $f$ and $g$ are bijections then $f \circ g$ is a bijection.

(ii) If $f \circ g$ is a bijection then $f$ and $g$ are bijections.

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