The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows: Start with the closed interval [0, 1] and remove the open interval .  That leaves the two intervals  and . Then remove the open middle third of each of those intervals.  Four intervals remain, and again remove the open middle third of each of them.  Continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step.  The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed.   Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infinitely many numbers. Give examples of at least three numbers in the Cantor set.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 29E
icon
Related questions
Question
  1. The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows: Start with the closed interval [0, 1] and remove the open interval .  That leaves the two intervals  and . Then remove the open middle third of each of those intervals.  Four intervals remain, and again remove the open middle third of each of them.  Continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step.  The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed.

 

  1. Show that the total length of all the intervals that are removed is 1.
  2. Despite that, the Cantor set contains infinitely many numbers. Give examples of at least three numbers in the Cantor set.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 3 images

Blurred answer
Knowledge Booster
Sequence
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,