Prove Sums of Cantor Sets in [0,2]

  • Thread starter Thread starter kathrynag
  • Start date Start date
  • Tags Tags
    Cantor Sets Sums
Click For Summary

Homework Help Overview

The discussion revolves around proving properties of sums of Cantor sets within the interval [0,2]. The original poster attempts to show that the sum of elements from the Cantor set results in the entire interval [0,2], specifically focusing on sequences derived from these sets.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the existence of specific elements in the Cantor sets that sum to a given value, questioning how to demonstrate this for arbitrary natural numbers. There are attempts to express elements of the Cantor set in terms of sequences, and some participants express uncertainty about the definitions and properties of the Cantor sets involved.

Discussion Status

The discussion is ongoing, with participants sharing their interpretations and definitions of the Cantor sets. Some have provided specific examples of the sets, while others are seeking clarification on the problem's requirements and the nature of the sequences involved.

Contextual Notes

There is ambiguity regarding the definitions of C, C1, C2, and Cn, as well as the specific properties of the Cantor sets being referenced. Participants are working under the assumption that C refers to the Cantor set, but this is not explicitly confirmed in the problem statement.

kathrynag
Messages
595
Reaction score
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.
 
Physics news on Phys.org
I also considered writing x as a^i/3^i where a is in {0,1} but not sure what else to do
 
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.
 
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.
 
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])
 
Cantor set C=intersection(C_n)
 

Similar threads

Partial derivative of radial basis function
  • · Replies 2 ·
Replies
2
Views
5K
How Do You Calculate Predicted Values in Least Squares Regression?
  • · Replies 10 ·
Replies
10
Views
2K
Continuous function and limits
  • · Replies 5 ·
Replies
5
Views
2K
How can I show that the sup(S)=lim{Xn} and the inf(S)=lim{Yn} as n goes to infinity
  • · Replies 1 ·
Replies
1
Views
3K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Contraction Mapping Theorem: Proving Continuity and Convergence of a Sequence
  • · Replies 1 ·
Replies
1
Views
3K
Proving Sum (r^ncos(nt))=rcos(t)-r^2/(1-2rcos(t)+r^2
  • · Replies 2 ·
Replies
2
Views
3K
Infinite sum of squares converges
  • · Replies 1 ·
Replies
1
Views
4K
Prove ##a + b = b + a## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Proof in Set Theory: Subset of ]0,2]
  • · Replies 3 ·
Replies
3
Views
2K