6. The Cantor Set is one of the most famous sets in mathematics. To construct the Cantor set, start with the interval [0, 1]. Now remove the middle third (,). This leaves you with the set [0, ]u , 1]. For each of the two subintervals, remove the middle third; in this case, remove the intervals (G.) and (7. ). Continue in this way, removing the middle thirds of each remaining interval. The Cantor set is all the points in [0, 1] that are NOT removed. Definition of Cantor's Set - Step 0: we begin with the interval [0, 1]. 1 Step 1: we divide [0, 1] into 3 subintervals and delete the open middle subinterval (). - Step 2: we divide each of the 2 resulting intervals above into 3 subintervals and delete the open middle subintervals (,) and (3. ). 2/9 23 7/9 S/9 We continue this procedure indefinitely. At each step, we delete the open middle third subinterval of each interval obtained in the previous step. (a) Explain why 0, 1, and are in the Cantor set. Identify at least 4 more points that lie in the Cantor set. (In fact, the set contains an infinite number of points.) (b) At each step, determine the total length of the subintervals that have been removed. For example, in step 1 an interval of length S1 = is removed. In step two, an additional two intervals each of length are removed, so the total length removed in S2 =}+ 23. Determine S, and show that this is a geometric series. (c) Determine the limit of this series. (d) Given that you started with an interval of length 1, how much "length" does the Cantor set have?
6. The Cantor Set is one of the most famous sets in mathematics. To construct the Cantor set, start with the interval [0, 1]. Now remove the middle third (,). This leaves you with the set [0, ]u , 1]. For each of the two subintervals, remove the middle third; in this case, remove the intervals (G.) and (7. ). Continue in this way, removing the middle thirds of each remaining interval. The Cantor set is all the points in [0, 1] that are NOT removed. Definition of Cantor's Set - Step 0: we begin with the interval [0, 1]. 1 Step 1: we divide [0, 1] into 3 subintervals and delete the open middle subinterval (). - Step 2: we divide each of the 2 resulting intervals above into 3 subintervals and delete the open middle subintervals (,) and (3. ). 2/9 23 7/9 S/9 We continue this procedure indefinitely. At each step, we delete the open middle third subinterval of each interval obtained in the previous step. (a) Explain why 0, 1, and are in the Cantor set. Identify at least 4 more points that lie in the Cantor set. (In fact, the set contains an infinite number of points.) (b) At each step, determine the total length of the subintervals that have been removed. For example, in step 1 an interval of length S1 = is removed. In step two, an additional two intervals each of length are removed, so the total length removed in S2 =}+ 23. Determine S, and show that this is a geometric series. (c) Determine the limit of this series. (d) Given that you started with an interval of length 1, how much "length" does the Cantor set have?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![6. The Cantor Set is one of the most famous sets in mathematics. To construct the Cantor set, start with
the interval [0, 1]. Now remove the middle third (,). This leaves you with the set [0, ]u , 1].
For each of the two subintervals, remove the middle third; in this case, remove the intervals (G.) and
(7. ). Continue in this way, removing the middle thirds of each remaining interval. The Cantor set
is all the points in [0, 1] that are NOT removed.
Definition of Cantor's Set
- Step 0: we begin with the interval [0, 1].
1 Step 1: we divide [0, 1] into 3 subintervals and delete the
open middle subinterval ().
- Step 2: we divide each of the 2 resulting intervals above into
3 subintervals and delete the open middle subintervals (,)
and (3. ).
2/9
23 7/9
S/9
We continue this procedure indefinitely. At each step, we delete
the open middle third subinterval of each interval obtained in the
previous step.
(a) Explain why 0, 1, and are in the Cantor set. Identify at least 4 more points that lie in the
Cantor set. (In fact, the set contains an infinite number of points.)
(b) At each step, determine the total length of the subintervals that have been removed. For example,
in step 1 an interval of length S1 = is removed. In step two, an additional two intervals each of
length are removed, so the total length removed in S2 =}+ 23. Determine S, and show that
this is a geometric series.
(c) Determine the limit of this series.
(d) Given that you started with an interval of length 1, how much "length" does the Cantor set have?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0a2c60a-65ef-4963-839a-bfc03c6ccd72%2Fa520e377-20d3-469a-a83d-f25cf5a00c33%2Fgv05d4.png&w=3840&q=75)
Transcribed Image Text:6. The Cantor Set is one of the most famous sets in mathematics. To construct the Cantor set, start with
the interval [0, 1]. Now remove the middle third (,). This leaves you with the set [0, ]u , 1].
For each of the two subintervals, remove the middle third; in this case, remove the intervals (G.) and
(7. ). Continue in this way, removing the middle thirds of each remaining interval. The Cantor set
is all the points in [0, 1] that are NOT removed.
Definition of Cantor's Set
- Step 0: we begin with the interval [0, 1].
1 Step 1: we divide [0, 1] into 3 subintervals and delete the
open middle subinterval ().
- Step 2: we divide each of the 2 resulting intervals above into
3 subintervals and delete the open middle subintervals (,)
and (3. ).
2/9
23 7/9
S/9
We continue this procedure indefinitely. At each step, we delete
the open middle third subinterval of each interval obtained in the
previous step.
(a) Explain why 0, 1, and are in the Cantor set. Identify at least 4 more points that lie in the
Cantor set. (In fact, the set contains an infinite number of points.)
(b) At each step, determine the total length of the subintervals that have been removed. For example,
in step 1 an interval of length S1 = is removed. In step two, an additional two intervals each of
length are removed, so the total length removed in S2 =}+ 23. Determine S, and show that
this is a geometric series.
(c) Determine the limit of this series.
(d) Given that you started with an interval of length 1, how much "length" does the Cantor set have?
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