A note on countability

• Philipp Festerling
• Published 2012

Abstract

This note contains solutions to all exercises from the Exercise Book dealing with countability (Exercise Book: Problems 9 to 12). You are welcome to report any error. Before beginning solving the exercises, two introductory remarks are necessary. Firstly, in Problem 9 we will use the fact that any subset of a finite set is finite. You will probably be surprised that proving this apparently quite intuitive result requires some amount of work. According to agreement with Niels Jørgen, we do not need to go through this formal proof. If you are interested, however, you find a sketch of the proof in the appendix to this note. Secondly, Definition 0.8 of the lecture notes requires the existence of a bijection f : N → A for a set A to be countable. In some cases, however, you may end up with showing the existence of a bijection g : A → N. Since g is a bijection, the inverse g : N→ A exists, and this inverse g is a bijection of N with A, showing that A is countable. You find a proof of this result in the appendix to this note. Similarly, you may end up with showing the existence of a bijection h : A → B where A is countable. Hence, there exists a bijection g : N → A. The composition h ◦ g : N→ B such that h ◦ g(n) = h(g(n)) for n ∈ N is then a bijection of N with B, showing that B is countable. The straightforward proof of this result is left as a voluntary exercise. Now we are ready to have a look on the first of the exercises about countability.

Cite this paper

@inproceedings{Festerling2012ANO, title={A note on countability}, author={Philipp Festerling}, year={2012} }