The fractal C is constructed as follows. (i) Begin with the line segment Ia = [0, 1). (ii) Remove the open middle third (, }), to get I = [0, }) U [3, 1). (iii) Remove the open middle third from every remaining line segment, to get (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 ilhustrated below. (m) Find the total length of the intervals removed from (0, 1] after three iterations, ie. 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?
The fractal C is constructed as follows. (i) Begin with the line segment Ia = [0, 1). (ii) Remove the open middle third (, }), to get I = [0, }) U [3, 1). (iii) Remove the open middle third from every remaining line segment, to get (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 ilhustrated below. (m) Find the total length of the intervals removed from (0, 1] after three iterations, ie. 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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![The fractal C is constructed as follows.
(i) Begin with the line segment Ig = [0, 1).
(ii) Remove the open middle third (, }), to get I = [0, ]U , 1].
(iii) Remove the open middle third from every remaining line segment, to get
(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 ilhustrated below.
(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%2Fe2d84131-0120-4813-b2c4-66678a0ce9a7%2F8f2bf403-8480-4f3d-9192-992b2e233af9%2F8mjmvz_processed.png&w=3840&q=75)
Transcribed Image Text:The fractal C is constructed as follows.
(i) Begin with the line segment Ig = [0, 1).
(ii) Remove the open middle third (, }), to get I = [0, ]U , 1].
(iii) Remove the open middle third from every remaining line segment, to get
(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 ilhustrated below.
(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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