Paper 4 , Section II, E

Numbers and Sets | Part IA, 2010

What does it mean for a set to be countable ?

Show that Q\mathbb{Q} is countable, but R\mathbb{R} is not. Show also that the union of two countable sets is countable.

A subset AA of R\mathbb{R} has the property that, given ϵ>0\epsilon>0 and xRx \in \mathbb{R}, there exist reals a,ba, b with aAa \in A and bAb \notin A with xa<ϵ|x-a|<\epsilon and xb<ϵ|x-b|<\epsilon. Can AA be countable ? Can AA be uncountable ? Justify your answers.

A subset BB of R\mathbb{R} has the property that given bBb \in B there exists ϵ>0\epsilon>0 such that if 0<bx<ϵ0<|b-x|<\epsilon for some xRx \in \mathbb{R}, then xBx \notin B. Is BB countable ? Justify your answer.

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