(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the total length of the intervals removed from [0,1] to get I3. (b) By considering a suitable series, show that the total length of all of the intervals removed is equal to 1. (c) Despite the fact that an infinite number of intervals of total length 1 have been removed to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there are points in C. What is one such point?
(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the total length of the intervals removed from [0,1] to get I3. (b) By considering a suitable series, show that the total length of all of the intervals removed is equal to 1. (c) Despite the fact that an infinite number of intervals of total length 1 have been removed to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there are points in C. What is one such point?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. The fractal C is constructed as follows.
(i) Begin with the line segment Io = [0, 1].
(ii) Remove the open middle third (,), to get I = [0,]U, 1].
3
(iii) Remove the open middle third from every remaining line segment, to get
2.
U
2 7
8
I2
0,
U
3'9
(iv) Repeat the process of removing the open middle thirds of the remaining line segments
infinitely many times.
The first few iterations of this construction are illustrated below.
I||
(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the
total length of the intervals removed from [0, 1] to get I3.
(b) By considering a suitable series, show that the total length of all of the intervals removed
is equal to 1.
(c) Despite the fact that an infinite number of intervals of total length 1 have been removed
to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there
are points in C. What is one such point?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2d2659c-073c-4d9b-b022-db6863ac78eb%2Fd5a5124b-2302-48f8-8efd-2bd068dbf142%2Filewccel_processed.png&w=3840&q=75)
Transcribed Image Text:3. The fractal C is constructed as follows.
(i) Begin with the line segment Io = [0, 1].
(ii) Remove the open middle third (,), to get I = [0,]U, 1].
3
(iii) Remove the open middle third from every remaining line segment, to get
2.
U
2 7
8
I2
0,
U
3'9
(iv) Repeat the process of removing the open middle thirds of the remaining line segments
infinitely many times.
The first few iterations of this construction are illustrated below.
I||
(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the
total length of the intervals removed from [0, 1] to get I3.
(b) By considering a suitable series, show that the total length of all of the intervals removed
is equal to 1.
(c) Despite the fact that an infinite number of intervals of total length 1 have been removed
to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there
are points in C. What is one such point?
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