Prove Sums of Cantor Sets in [0,2]

In summary, the problem is dealing with showing that the sum C+C={[x+y] | x,y in C} is equal to [0,2]. This can be shown by finding x1,y1 in C1 such that x1+y1=s and for any n in the naturals, there exists xn, yn in Cn for which xn+yn=s. Even though the sequences xn and yn may not converge, they can still be used to produce the desired x and y in C that satisfy x+y=s. The Cantor set, denoted as C, is defined as the intersection of Cn for all n in the naturals.
  • #1
kathrynag
598
0
I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2]
a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s.
b) Keeping in mind that the sequences xn and yn do not necessarily converge show show that they never the less be used to produce the desired x and y in C satisfying x+y=s.


a) Let's be in [0,2]
C1=[0,1/3]U[2/3,1]
That's about as far as I get and then I get stuck.
 
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  • #2
I also considered writing x as a^i/3^i where a is in {0,1} but not sure what else to do
 
  • #3
I would guess from your title that C is the Cantor trinary set but it would have been better if you had said so. And I have no idea what C1, C2, or Cn is.
 
  • #4
I'm assuming C is the Cantor set since the problem is dealing with Cantor sets, but the problem doesn't explicitly say what C is.
 
  • #5
I say earlier in my book some talk about cantor sets with C1=C0\(1/3,2/3)=[0,1/3]U[2/3,1]
C2=([0,1/9)U[2/3,1/3])U([2/3,7/9]U[8/9,1])
 
  • #6
Cantor set C=intersection(C_n)
 

1. What are Cantor sets?

Cantor sets are a type of fractal set that is created by repeatedly removing the middle third of a line segment, and then removing the middle third of each remaining line segment, and so on. This process results in a set of points that are dense in some parts and sparse in others, with a fractal-like structure.

2. What is the significance of proving sums of Cantor sets in [0,2]?

Proving sums of Cantor sets in [0,2] is significant because it allows us to understand and analyze the behavior of these sets in a specific interval. This can lead to insights and applications in various areas of mathematics, such as measure theory and dynamical systems.

3. How can we prove sums of Cantor sets in [0,2]?

There are various methods for proving sums of Cantor sets in [0,2]. One approach is to use the concept of ternary expansions, where each number in the interval [0,2] is represented as a base-3 number. Another approach is to use the concept of binary expansions, where each number in the interval [0,2] is represented as a base-2 number. Both of these methods involve breaking down the interval into smaller subintervals and analyzing the sums of the Cantor sets in each subinterval.

4. Are sums of Cantor sets in [0,2] always finite?

No, sums of Cantor sets in [0,2] can be either finite or infinite. It depends on the specific Cantor set being considered and the method used for proving the sum. Some Cantor sets have a finite sum, while others have an infinite sum.

5. What are some real-world applications of proving sums of Cantor sets in [0,2]?

One real-world application of proving sums of Cantor sets in [0,2] is in the study of chaotic systems. These systems often exhibit fractal-like behavior and can be described using Cantor sets. By understanding the sums of these sets, we can gain insights into the behavior and dynamics of chaotic systems. Additionally, the concept of Cantor sets and their sums has applications in signal processing, data compression, and cryptography.

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