Mathematics 242 – Principles of Analysis Discussion 2 – The Cantor Set October 24-26, 2005

Background

In this discussion, we will study some of the surprising properties of a subset of the real number system called the Cantor set. This was first constructed by the same mathematician Georg Cantor who we met earlier when we discussed the uncountability of R. The Cantor set is produced by the following process:

1) Start with the closed interval C_{0} = [0, 1].

2) Discard the open “middle third” (1/3, 2/3) interval to make

C_{1} = [0, 1/3] ∪ [2/3, 1].

3) Discard the open “middle third” of each half of C_{1} to make to make

C_{2} = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1].

4) Discard the open “middle third” of each of the four pieces of C_{2} to make C_{3}, and so on.

5) Repeat this process of removing the middle thirds to produce C_{n} for all n ≥ 1.

6) The Cantor set C is then

C = ⋂_{n=1}^{∞} C_{n}.

Discussion Questions

A) Show that the sum of the lengths of all of the intervals removed in the process described above is 1. (Note: there are infinitely many of them, so this is an infinite series!)

B) Since the whole interval [0, 1] has length 1, you might think that means that C = ∅. But in fact that is far from true. In fact, C is still an uncountably infinite set of real numbers. Here is one way to see this:

1) We usually use base 10 expansions to represent real numbers, but any other integer base b > 1 would work just as well. Let’s consider base b = 3 or “ternary” expansions. The ternary expansion of a whole number like n = 24 would be found like this: 24 = 2 · 3^{2} + 2 · 3 + 0 · 1. So we would say 24 = (220)_{3}. If a number has a fractional part, then that is represented by negative powers of 3. The ternary digits of a number x in [0, 1] are integers d_{i} = 0, 1, or 2 appearing in a series

x = \frac{d_{1}}{3} + \frac{d_{2}}{3^{2}} + \frac{d_{3}}{3^{3}} + ⋯

Show that every such series (that is, for every possible choice of d_{i} ∈ {0, 1, 2}, i ∈ N) converges to some real number in [0, 1].

2) What is true about the ternary digit d_{1} for the numbers removed in the middle half of [0, 1]? What is true about d_{1} for the numbers in the Cantor set C? Similarly, what is true about the ternary digit d_{2} for the number removed from C_{1} to get C_{2}? What is true about d_{2} for the numbers in C?

3) Let’s consider the set of all sequences (x_{n}) where x_{n} = 0 or 2 for each n ∈ N. Show that this set of sequences is an uncountable set by adapting the Cantor Diagonalization proof we used to show [0, 1] is uncountable.

4) Put together parts 1,2,3 to explain why C is uncountably infinite.

C) Here is another surprising property of C. We claim that every number in the closed interval [0, 2] can be obtained as x + y for some x, y ∈ C. (This is true even though C actually contains no intervals itself!). Follow this plan:

1) First show by induction that for each n ≥ 1, every number in [0, 2] can be obtained as x_{n} + y_{n} for some x_{n}, y_{n} ∈ C_{n}.

2) The (x_{n}) and (y_{n}) are bounded sequences. Deduce the desired statement by applying Bolzano-Weierstrass (and other results we have seen, as needed!)

Assignment

Writeups due Friday, October 28.