Paper 4 , Section I, E

Numbers and Sets | Part IA, 2021

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

(i) If gfg \circ f is surjective then ff is surjective.

(ii) If gfg \circ f is injective then ff is injective.

(iii) If gfg \circ f is injective then gg is injective.

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

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