Paper 4, Section I, E

Numbers and Sets | Part IA, 2011

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

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

(i) If ff and gg are bijections then fgf \circ g is a bijection.

(ii) If fgf \circ g is a bijection then ff and gg are bijections.

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