Difference between revisions of "Cantor set"
(New page: The '''Cantor set''' is equal to <math>C(0,1)</math>, where <math>C</math> is a recursively defined function: <math>C(a,b)=C\left(a, \frac{2a+b}{3}\right)\cup C\left(\frac{a+...) |
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| − | The '''Cantor set''' | + | The '''Cantor set''' <math>\mathcal{C}</math> is a [[subset]] of the [[real number]]s that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a [[fractal]]. |
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| + | The Cantor set can be described [[recursion|recursively]] as follows: begin with the set [0,1] and then remove the ([[open interval | open]]) middle third, dividing the [[interval]] into two intervals of length <math>\frac{1}{3}</math>. Then remove the middle third of the two remaining segments, and remove the middle third of the four remaining segments, and so on ''ad infinitum''. | ||
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| + | Equivalently, we may define <math>\mathcal{C}</math> to be the set of real numbers between <math>0</math> and <math>1</math> with a [[base number | base]] three expansion that contains only the digits <math>0</math> and <math>1</math>. | ||
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Revision as of 10:42, 22 July 2009
The Cantor set 
is a subset of the real numbers that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a fractal.
The Cantor set can be described recursively as follows: begin with the set [0,1] and then remove the ( open) middle third, dividing the interval into two intervals of length 
. Then remove the middle third of the two remaining segments, and remove the middle third of the four remaining segments, and so on ad infinitum.
Equivalently, we may define to be the set of real numbers between 
and 
with a base three expansion that contains only the digits and .
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