Paper 4, Section II, E
Part IA, 2015
What does it mean for a set to be countable? Prove that
(a) if is countable and is injective, then is countable;
(b) if is countable and is surjective, then is countable.
Prove that is countable, and deduce that
(i) if and are countable, then so is ;
(ii) is countable.
Let be a collection of circles in the plane such that for each point on the -axis, there is a circle in passing through the point which has the -axis tangent to the circle at . Show that contains a pair of circles that intersect.