Proving That {x ∈ R | x = t1 + t2, t1, t2 ∈ C} = [0,2]

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In summary, to prove that the set of all numbers that are the sum of two Cantor set elements is precisely [0,2], we can use the fact that the elements of the Cantor set can be written in base 3 with only 0's and 2's. However, when constructing two Cantor set elements that sum to a generic base 3 element, we must consider carrying over from infinitely far away. To overcome this, we can use the fact that if a_n and b_n are sequences in the Cantor set that approach a number c in [0,2], then the sequences have a cluster point in the Cantor set due to its compactness. This shows that the set of sums is
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Homework Statement


If C is the cantor set, prove that [tex]{x \in R | x=t_1+t_2, t_1, t_2 \in C} = [0,2] [/tex]. In english if that wasn't clear, show the set of all numbers that are the sum of two cantor set elements is precisely [0,2]


Homework Equations


The Cantor set of course being constructed on the interval [0,1]


The Attempt at a Solution


I know that the elements of the Cantor set are precisely those that can be written in base 3 with only 0's and 2's, so I thought maybe for a generic base 3 element, I could construct two cantor set elements that sum to it. This didn't work because I needed to worry about carrying over from infinitely far away (since when constructing the two cantor set elements, I obviously need to start at the first decimal place, but addition 'starts' at the infinitieth or whatever you want to call it) so that didn't pan through so well. What's the best way to start this? It's obvious the set of sum of two cantor elements is a subset of [0,2], but any attempt to go the other way just ends in failure.
 
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  • #2
You seem to be saying that you wouldn't have any trouble constructing a such a sum if the number of digits were finite. Isn't that good enough? If you have a_n+b_n approaching some number c in [0,2] with a_n and b_n all in the Cantor set then the sequences a_n and b_n have a cluster point in the Cantor set, since it's compact.
 
  • #3
Dick said:
You seem to be saying that you wouldn't have any trouble constructing a such a sum if the number of digits were finite. Isn't that good enough? If you have a_n+b_n approaching some number c in [0,2] with a_n and b_n all in the Cantor set then the sequences a_n and b_n have a cluster point in the Cantor set, since it's compact.

Ah, of course. Thanks
 

What is the definition of "Proving That {x ∈ R | x = t1 + t2, t1, t2 ∈ C} = [0,2]"?

This statement is essentially a mathematical proof that a specific set of numbers (x) that belong to the set of real numbers (R) can be expressed as the sum of two complex numbers (t1 and t2) and fall within the interval of [0,2].

How is this statement proven?

This statement can be proven using mathematical techniques such as substitution, algebraic manipulation, and logical reasoning.

What is the significance of proving this statement?

This statement is significant because it provides a mathematical understanding of the relationship between real and complex numbers, and how they can be combined to create a specific set of numbers within a certain range. It also demonstrates the power and versatility of mathematical proofs in solving complex problems.

Are there any real-world applications of this statement?

Yes, this statement can be applied in various fields such as physics, engineering, and economics where the use of real and complex numbers is necessary for calculations and analysis. For example, in electrical engineering, this statement could be used to analyze the behavior of circuits with both real and complex components.

Is this statement universally true?

Yes, this statement is universally true as it is a mathematical proof that is based on logical reasoning and mathematical principles. However, it is important to note that it only applies to the specific set of numbers stated in the statement and may not necessarily hold true for all sets of numbers.

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