Paper 4, Section II, E

Numbers and Sets | Part IA, 2019

(a) What is a countable set? Let X,A,BX, A, B be sets with A,BA, B countable. Show that if f:XA×Bf: X \rightarrow A \times B is an injection then XX is countable. Deduce that Z\mathbb{Z} and Q\mathbb{Q} are countable. Show too that a countable union of countable sets is countable.

(b) Show that, in the plane, any collection of pairwise disjoint circles with rational radius is countable.

(c) A lollipop is any subset of the plane obtained by translating, rotating and scaling (by any factor λ>0\lambda>0 ) the set

{(x,y)R2x2+y2=1}{(0,y)R23y1}.\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} \cup\left\{(0, y) \in \mathbb{R}^{2} \mid-3 \leqslant y \leqslant-1\right\} .

What happens if in part (b) we replace 'circles with rational radius' by 'lollipops'?

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