Paper 4, Section II, D

Numbers and Sets | Part IA, 2012

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

(i) If fgf g is the identity map then gfg f is the identity map.

(ii) If fg=gf g=g then ff is the identity map.

(iii) If fg=ff g=f then gg is the identity map.

How (if at all) do your answers change if we are given that XX is finite?

Determine which sets XX have the following property: if ff is a function from XX to XX such that for every xXx \in X there exists a positive integer nn with fn(x)=xf^{n}(x)=x, then there exists a positive integer nn such that fnf^{n} is the identity map. [Here fnf^{n} denotes the nn-fold composition of ff with itself.]

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