Paper 4, Section II, D

Numbers and Sets | Part IA, 2012

Show that there is no injection from the power-set of R\mathbb{R} to R\mathbb{R}. Show also that there is an injection from R2\mathbb{R}^{2} to R\mathbb{R}.

Let XX be the set of all functions ff from R\mathbb{R} to R\mathbb{R} such that f(x)=xf(x)=x for all but finitely many xx. Determine whether or not there exists an injection from XX to R\mathbb{R}.

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