Cantor set defined via sums, whaaaaa?

In summary, the Cantor set is defined as the set of real numbers between 0 and 1 that can be represented in trinary form with only 0s and 2s, and is constructed recursively. This set is proven to be the Cantor middle-thirds set.
  • #1
SiddharthM
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Cantor set defined via sums, whaaaaa?!?

problem 19 chapter 3 of Rudin. I'm totally lost, I've even done a project on the Cantor set before but I just don't know where to start here.

Associate to each sequence a=(p_n) in which p_n is either 0 or 2, the real number

x(a) = sum from 1 to infinity of (p_n)/(3^n).

Prove that the set of all x(a) is the cantor middle-thirds set.

Cheerio
 
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  • #2
What's the problem? Just consider the base three expansion of points in [0,1].
 
  • #3
Represent the real numbers between 0 and 1 inclusive in trinary. It will help to represent 1 as 0.222...3. If you use this representation, when you construct Cantor's set recursively you will arrive at the set {x(a)}.
 

What is the Cantor set defined via sums?

The Cantor set defined via sums is a mathematical set that is constructed by repeatedly removing the middle third of a line segment and its subsequents segments. This set is named after the German mathematician Georg Cantor, who first described it in 1883.

How is the Cantor set defined via sums constructed?

The Cantor set defined via sums is constructed by starting with a line segment and dividing it into three equal segments. The middle segment is then removed, leaving two line segments. This process is repeated infinitely, resulting in a set of line segments with no middle points.

What are some properties of the Cantor set defined via sums?

The Cantor set defined via sums is a self-similar set, meaning that it contains smaller copies of itself. It is also an uncountable set, which means it has an infinite number of elements. Additionally, the Cantor set defined via sums has a fractal dimension of ln(2)/ln(3), which is approximately 0.631.

What is the significance of the Cantor set defined via sums?

The Cantor set defined via sums has many interesting properties and applications in mathematics, such as being a counterexample to the common belief that a set must have a non-zero length in order to be uncountable. It also has connections to chaos theory and dynamical systems.

How is the Cantor set defined via sums related to other mathematical concepts?

The Cantor set defined via sums is closely related to other fractal sets, such as the Sierpinski triangle and the Koch curve. It is also related to other constructions in mathematics, such as the Cantor function and the ternary numeral system.

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