The Cantor set To construct this set, we begin with the closed interval [0, 1]. From that interval, remove the middle open interval (1/3, 2/3), leaving the two closed intervals [0, 1/3] and [2/3, 1]. At the second step we remove the open middle third interval from each of those remaining. From [0, 1/3] we remove the open interval (1/9, 2/9), and from [2/3, 1] we remove (7/9, 8/9), leaving behind the four closed intervals [0, 1/9], [ 2/9,1/3], [ 2/3, 7/9], and [ 8/9, 1]. At the next step, we remove the middle open third interval from each closed interval left behind, so (1/27, 2/27) is removed from [0, 1/9 ], leaving the closed intervals [0, 1/27 ] and [2/27, 1/9]; (7/27, 8/27 ) is removed from [2/9, 1/3 ], leaving behind [ 2/9, 7/27 ] and [8/27, 1/3 ], and so forth. We continue this process repeatedly without stopping, at each step removing the open third interval from every closed interval remaining behind from the preceding step. The numbers remain- ing in the interval [ 0, 1 ], after all open middle third intervals have been removed, are the points in the Cantor set (named after Georg Cantor, 1845–1918). The set has some interesting properties. a. The Cantor set contains infinitely many numbers in [0, 1 ]. List 12 numbers that belong to the Cantor set. b. Show, by summing an appropriate geometric series, that the total length of all the open middle third intervals that have been removed from [ 0, 1 ] is equal to 1.
The Cantor set To construct this set, we begin with the closed interval [0, 1]. From that interval, remove the middle open interval (1/3, 2/3), leaving the two closed intervals [0, 1/3] and [2/3, 1]. At the second step we remove the open middle third interval from each of those remaining. From [0, 1/3] we remove the open interval (1/9, 2/9), and from [2/3, 1] we remove (7/9, 8/9), leaving behind the four closed intervals [0, 1/9], [ 2/9,1/3], [ 2/3, 7/9], and [ 8/9, 1]. At the next step, we remove the middle open third interval from each closed interval left behind, so (1/27, 2/27) is removed from [0, 1/9 ], leaving the closed intervals [0, 1/27 ] and [2/27, 1/9]; (7/27, 8/27 ) is removed from [2/9, 1/3 ], leaving behind [ 2/9, 7/27 ] and [8/27, 1/3 ], and so forth. We continue this process repeatedly without stopping, at each step removing the open third interval from every closed interval remaining behind from the preceding step. The numbers remain- ing in the interval [ 0, 1 ], after all open middle third intervals have been removed, are the points in the Cantor set (named after Georg Cantor, 1845–1918). The set has some interesting properties. a. The Cantor set contains infinitely many numbers in [0, 1 ]. List 12 numbers that belong to the Cantor set. b. Show, by summing an appropriate geometric series, that the total length of all the open middle third intervals that have been removed from [ 0, 1 ] is equal to 1.
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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![The Cantor set To construct this set, we begin with the closed
interval [0, 1]. From that interval, remove the middle open
interval (1/3, 2/3), leaving the two closed intervals [0, 1/3]
and [2/3, 1]. At the second step we remove the open middle
third interval from each of those remaining. From [0, 1/3]
we remove the open interval (1/9, 2/9), and from [2/3, 1]
we remove (7/9, 8/9), leaving behind the four closed intervals
[0, 1/9], [ 2/9,1/3], [ 2/3, 7/9], and [ 8/9, 1]. At the
next step, we remove the middle open third interval from each
closed interval left behind, so (1/27, 2/27) is removed from
[0, 1/9 ], leaving the closed intervals [0, 1/27 ] and
[2/27, 1/9]; (7/27, 8/27 ) is removed from [2/9, 1/3 ],
leaving behind [ 2/9, 7/27 ] and [8/27, 1/3 ], and so forth.
We continue this process repeatedly without stopping, at each
step removing the open third interval from every closed interval
remaining behind from the preceding step. The numbers remain-
ing in the interval [ 0, 1 ], after all open middle third intervals have
been removed, are the points in the Cantor set (named after Georg
Cantor, 1845–1918). The set has some interesting properties.
a. The Cantor set contains infinitely many numbers in [0, 1 ].
List 12 numbers that belong to the Cantor set.
b. Show, by summing an appropriate geometric series, that the
total length of all the open middle third intervals that have
been removed from [ 0, 1 ] is equal to 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc9c00a1-bb37-45fd-bbeb-7c437096da08%2Ff4f0c92f-493d-405a-8d87-9c614881ee2f%2Fxnzls8q.png&w=3840&q=75)
Transcribed Image Text:The Cantor set To construct this set, we begin with the closed
interval [0, 1]. From that interval, remove the middle open
interval (1/3, 2/3), leaving the two closed intervals [0, 1/3]
and [2/3, 1]. At the second step we remove the open middle
third interval from each of those remaining. From [0, 1/3]
we remove the open interval (1/9, 2/9), and from [2/3, 1]
we remove (7/9, 8/9), leaving behind the four closed intervals
[0, 1/9], [ 2/9,1/3], [ 2/3, 7/9], and [ 8/9, 1]. At the
next step, we remove the middle open third interval from each
closed interval left behind, so (1/27, 2/27) is removed from
[0, 1/9 ], leaving the closed intervals [0, 1/27 ] and
[2/27, 1/9]; (7/27, 8/27 ) is removed from [2/9, 1/3 ],
leaving behind [ 2/9, 7/27 ] and [8/27, 1/3 ], and so forth.
We continue this process repeatedly without stopping, at each
step removing the open third interval from every closed interval
remaining behind from the preceding step. The numbers remain-
ing in the interval [ 0, 1 ], after all open middle third intervals have
been removed, are the points in the Cantor set (named after Georg
Cantor, 1845–1918). The set has some interesting properties.
a. The Cantor set contains infinitely many numbers in [0, 1 ].
List 12 numbers that belong to the Cantor set.
b. Show, by summing an appropriate geometric series, that the
total length of all the open middle third intervals that have
been removed from [ 0, 1 ] is equal to 1.
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