Counting

Abstract

This chapter contains a brief introduction to cardinality. The focus is on countable sets. Of course, Cantor’s results that an interval is not countable and that the set of irrational numbers is uncountable are also included. In fact, most of the results in this chapter are due to Cantor.

Appendices

Problems

1.1 Problems for Sect. 4.1

• 1. Is the set of all enumerations of $\mathbb {N}$ countable?

• 2. The set of all subsets of $\mathbb {N}$ is not countable.

  A real number $A$ is algebraic if it solves $p(x)=0$ for some polynomial with integer coefficients. A real number that is not algebraic is transcendental.

• 3. The set of polynomials with integer coefficients is countable.

• 4. The set of algebraic numbers is countable.

• 5. The set of transcendental numbers is uncountable. In particular, some real numbers are transcendental.

• 6. Let $R$ be the set of numbers of the form

  $$ 0.a_{1}b_{1}0a_{2}b_{2}00a_{3}b_{3}000a_{4}b_{4}\ldots, $$

  where $\left \{ a_{k},b_{k}\right \} =\left \{ 0,1\right \}$ for all $k\in \mathbb {N}.$

  (a) Show each number in $R$ is irrational.

  (b) Show the set $R$ is uncountable.

• 7. Let $A$ be a set, for example, $A$ could be an interval. Consider a function $g:A\to \mathbb {R}.$ Suppose there is a real number $M,$ such that

  $$ -M\leq\sum_{b\in B}g(b)\leq M $$

  for all finite subsets $B$ of $A.$ Prove

  $$ \left\{ a\in A\mid g(a)\neq0\right\} $$

  is countable. [Hint: for each integer $n\geq 1,$ the set $G_{n}:=\left \{ a\in A\mid g(a)>\frac {1}{n}\right \}$ is finite, in fact has at most $Mn$ elements.]

• 8. Let $f:\left ]0,1\right [\to \mathbb {R}$ be determined by $f(x):=\frac {1}{n}$ for $\frac {1}{n+1}\leq x<\frac {1}{n}$ and all $n\in \mathbb {N}.$ Show that the set of points where $f$ is discontinuous is an infinite countable set.

1.2 Problems for Sect. 4.2

• 1. Find a bijection $f:[0,1]\to [0,1[.$ [Hint: verify

  $$ f(x):=\begin{cases} 1/(n+1) & \text{when }x=1/n,\text{ for some }n\in\mathbb{N}\\ x & \text{when }x\neq1/n,\text{ for all }n\in\mathbb{N} \end{cases} $$

  works.]

• 2. Find a bijection $g:[0,1[\to ]0,1[.$

• 3. Find a bijection $h:]0,1[\to \mathbb {R}.$

Combining the three problems we conclude the map $h\circ g\circ f$ is a bijection $[0,1]\to \mathbb {R}.$

Solutions and Hints for the Exercises

Exercise 4.1.6 Let A_k be the rational numbers with denominator k.

Exercise 4.1.8 Mimic the proof that the interval [0,1] is uncountable.

Exercise 4.1.9 Suppose A ⊆ B and B is countable. If A is finite we are done.Suppose A is infinite. Enumerate the elements of B. Deleting the elements from thelist that are not in A gives an enumeration of the elements of A.

Exercise 4.1.10 If A∪B is countable, then A is countable by Exercise 4.1.9.

Exercise 4.1.11 Use Cantor’s Theorem and Exercise 4.1.10.

Exercise 4.1.12 Mimic the proof of Theorem 4.1.5