Chapter 11.2, Problem 5E

Interpretation Introduction

Interpretation:

• To find the base ^{$\text{-3}$ expansion of $\frac{1}{2}$}

• To find a one-to-one correspondence between the Cantor set C and the interval $[0,1]$

• To show Cantor set is not an “all endpoints” by explicitly identifying a point in C that is not an endpoint.

Concept Introduction:

• The Cantor set is created by starting with closed interval ${\text{S}}_{\text{0}}=[0,1]$ and removing its open middle third, i.e. interval $\left(\frac{1}{3},\frac{2}{3}\right)$ and the endpoints are kept as they are. This procedure creates the pair of the closed interval ${\text{S}}_{\text{1}}$, then removing the open middle third of the intervals ${\text{S}}_{\text{0}}{\text{ and S}}_{\text{1}}$ creates ${\text{S}}_2$ and so on. The limiting set ${\text{C = S}}_{\infty }$ is the Cantor set.

• The fractal properties of the Cantor set are

  1. C has structure at arbitrarily small scales.

  2. C is self-similar

  3. The dimension of C is not an integer

  4. C has measure zero

  5. C consists of unaccountably many points