/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Show that \(\frac{13}{17}\) is n... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

Show that \(\frac{13}{17}\) is not in the Cantor set.

Short Answer

Expert verified
In the ternary representation of \(\frac{13}{17}\), which is \(0.20212\dots\), there is at least one '1'. Since a number is in the Cantor set if and only if its ternary representation does not contain a '1', we can conclude that \(\frac{13}{17}\) is not in the Cantor set.

Step by step solution

01

Write \(\frac{13}{17}\) as a ternary number

We begin by writing \(\frac{13}{17}\) as a ternary (base 3) number. A method to do this is to repeatedly multiply the fraction by 3, taking the integer part as the next digit in the ternary representation, then repeating the process for the remaining fraction portion after subtracting the digit. 1. Multiply the fraction by 3: \(\frac{13}{17} \cdot 3 = \frac{39}{17}\), which is equal to \(2 + \frac{5}{17}\). The first digit is 2. 2. Subtract the digit (2) and multiply the remaining fraction by 3: \(\frac{5}{17} \cdot 3 = \frac{15}{17}\), which is equal to \(0 + \frac{15}{17}\). The second digit is 0. 3. Multiply the remaining fraction by 3: \(\frac{15}{17} \cdot 3 = \frac{45}{17}\), which is equal to \(2 + \frac{11}{17}\). The third digit is 2. 4. Subtract the digit (2) and multiply the remaining fraction by 3: \(\frac{11}{17} \cdot 3 = \frac{33}{17}\), which is equal to \(1 + \frac{16}{17}\). The fourth digit is 1. 5. Subtract the digit (1) and multiply the remaining fraction by 3: \(\frac{16}{17} \cdot 3 = \frac{48}{17}\), which is equal to \(2 + \frac{14}{17}\). The fifth digit is 2. Therefore, the ternary representation of \(\frac{13}{17}\) is \(0.20212\dots\). Note that if at any point in the ternary representation, there is a non-zero digit in the decimal part, then the number does not belong to the Cantor set.
02

Determine if \(\frac{13}{17}\) belongs to the Cantor set

A number is in the Cantor set if and only if its ternary representation does not contain a '1'. In the ternary representation found above, we have \(\frac{13}{17} = 0.20212\dots\). Since there is at least one '1' in the ternary representation, we can conclude that \(\frac{13}{17}\) is not in the Cantor set.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ternary Number System
The ternary number system, also known as base three, is a numeral system that uses three as its base. Every digit in a ternary number is an integer from 0 to 2. This system plays a vital role in some areas of mathematics, including fractals like the Cantor set.

To convert a fraction like \(\frac{13}{17}\) to the ternary system, one can use a method similar to long division for decimal conversion. You multiply the fraction by 3 and take the integer part as the next digit of the result. The remaining fraction after removing the integer forms the basis for the next multiplication. This process is repeated until you reach a repeatable pattern or terminate after a few iterations for practical purposes.

Understanding the ternary system is crucial when studying the Cantor set. In this context, the presence of '1' in the ternary expansion of a number is a deciding factor for its membership in the Cantor set, which doesn't happen with regular decimal numbers.
Mathematical Proof
A mathematical proof is a logical argument that establishes the truth or falsehood of a mathematical statement. Proofs rely on a series of logically sound, interconnected steps that demonstrate that, given certain premises or axioms, a specific conclusion invariably follows.

In the case of proving whether a number like \(\frac{13}{17}\) belongs to the Cantor set, the proof involves showing its ternary representation and checking for the specific criterion that identifies Cantor set members. If any digit in its ternary expansion is '1', the number cannot be part of the Cantor set, as defined by its construction. The proof's conclusiveness rests upon understanding both the ternary number system and the definition of the Cantor set.
Measure Theory
Measure theory is a branch of mathematical analysis that deals with the quantitative assessment of size or 'measure' of mathematical objects, particularly in sets. It allows mathematicians to extend notions like length, area, and volume to more complex and abstract sets than conventional geometric shapes.

The Cantor set is a fascinating object in measure theory because, despite being uncountably infinite, it has a measure of zero. This paradoxical trait underscores the Cantor set's sparseness in the real number line鈥攊t contains no intervals of any positive length. Measure theory helps provide a rigorous framework for understanding such counterintuitive properties of the Cantor set and other abstract mathematical constructs.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that the collection of Borel subsets of \(\mathbf{R}\) is translation invariant. More precisely, prove that if \(B \subset \mathbf{R}\) is a Borel set and \(t \in \mathbf{R},\) then \(t+B\) is a Borel set.

Prove that if \(A \subset \mathbf{R}\) and \(|A|>0,\) then there exists a subset of \(A\) that is not Lebesgue measurable.

Give an example of a measure space \((X, \mathcal{S}, \mu)\) such that $$ \\{\mu(E): E \in \mathcal{S}\\}=[0,1] \cup[3, \infty] $$

Prove that the collection of Borel subsets of \(\mathbf{R}\) is dilation invariant. More precisely, prove that if \(B \subset \mathbf{R}\) is a Borel set and \(t \in \mathbf{R},\) then \(t B\) (which is defined to be \(\\{t b: b \in B\\}\) ) is a Borel set.

Show that $$ \lim _{j \rightarrow \infty}\left(\lim _{k \rightarrow \infty}(\cos (j ! \pi x))^{2 k}\right)=\left\\{\begin{array}{ll} 1 & \text { if } x \text { is rational, } \\ 0 & \text { if } x \text { is irrational } \end{array}\right. $$ for every \(x \in \mathbf{R}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.