Chapter 11.2, Problem 6E

Interpretation Introduction

Interpretation:

1. To show that ${\text{P}}_{\text{0}}\text{(x) =x}$ by considering that ${\text{P}}_{\text{0}}\text{(x)}$ denotes the probability that a randomly chosen point in ${\text{S}}_{\text{0}}$ lies to the left of x, where $\text{0 }\le \text{x}\le \text{ 1}$.

2. Considering $\text{S1}$, define ${\text{P}}_{\text{1}}\text{(x)}$, and draw the graph of ${\text{P}}_{\text{1}}\text{(x)}\text{.}$

3. To draw the graphs of ${\text{P}}_{\text{n}}\text{(x)}$ for $\text{n= 2, 3, 4}$.

4. To tell if the limiting function $\text{P}\infty \text{(x)}$ the devil’s staircase is continuous and to draw the graph of its derivative.

Concept Introduction:

• The fixed point of a differential equation is a point where -$\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$.

• For countable and uncountable sets, the notion created by the cantor will be followed. Different infinite sets can be compared as follows: considering the basis set N as the set of natural numbers, if set A has one - to – one correspondence, i.e. every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

• Start with the closed interval ${\text{S}}_{\text{0}}\text{= }\left[\text{0, 1}\right]$ and removing itsopen middle third. Then, repeat the procedure for the remaining two sets and continue the procedure infinitely. The limiting set $\text{C=S}\infty$ is the Cantor set.

• The probability that the point lies to the left of some point x on the cantor, where $0\le \text{x}\le \text{1}$ is given by a function P(x); this function is called the devil’s staircase.