Paper 4 , Section I, E

Consider functions $f: X \rightarrow Y$ and $g: Y \rightarrow X$. Which of the following statements are always true, and which can be false? Give proofs or counterexamples as appropriate.

(i) If $g \circ f$ is surjective then $f$ is surjective.

(ii) If $g \circ f$ is injective then $f$ is injective.

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

If $X=\{1, \ldots, m\}$ and $Y=\{1, \ldots, n\}$ with $m<n$, and $g \circ f$ is the identity on $X$, then how many possibilities are there for the pair of functions $f$ and $g$ ?

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