Understanding Cantor set C in Ternary form with 1/n factor in front C

In summary, at the first iteration, one removes the middle third of the set. At the second iteration, one removes the entire set. At the nth iteration, one removes the entire set except for the last n/3 elements.
  • #1
cbarker1
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I am trying to comprehend the Cantor set C with a 1/n factor in base 3
Dear Everybody,

I am confused by ##1/n C##, where C is a cantor set in base 3 and ##n\geq2##. I can understand the construction of the normal Cantor set.

How do I comprehend this set with this extra condition. Do I multiply the set with ##1/n## or not?

Thanks,
Cbarker1

mentor note: adjusted latex to use double # instead of single #
 
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  • #2
cbarker1 said:
Understanding Cantor set C in Tetany form with 1/n factor in front C
I have never seen the word "Tetany" before: do you mean "ternery"? Even with this correction I'm afraid the rest of the post doesn't make much sense to me.

Do you have a reference for the ideas you are talking about?

If not, can you provide a more complete description of the set avoiding ambiguous notation like ##1/n C## which whether in ## \LaTeX ## or plain text can mean either ## \frac{1}{nC} ## or ## \frac{1}{n}C ##. Perhaps you could start by rephrasing "The Cantor ternary set is created by iteratively deleting parts of a set of line segments. One starts by deleting the open middle third ## \left ( \frac{1}{3} , \frac{2}{3} \right) ## from the interval ## [ 0 , 1 ] ##."
 
  • #3
pbuk said:
have never seen the word "Tetany" before: do you mean "ternery"?
I changed "tetany" in the thread title to "ternary," as my best guess as to what the OP was trying to convey.

Also, perhaps the "1/nC" (with same complaint about what 1/nC actually means) is meant to convey the level of middle third deletions. Again, that's a guess. If so, with n = 1, we would have the two subintervals [0, .1] and [.2, 1], using base-3 fractions. With n =2, we remove the middle third from each of the two previously listed subintervals. This would produce four subintervals: [0, .01], [.02, .1], [.2, .21], and [.22, 1], again using base-3 fractions.

And so on.
 
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  • #4
your guesses are right. I want that word Ternary and ##(1/n)*C##.
 
  • #5
OK, so how much of the set do you remove at the first iteration?

How much at the second?
...
How much at the ## n ##th?
...

How much in total as ## n \to \infty ##?
 

1. What is the Cantor set C in ternary form with 1/n factor in front?

The Cantor set C is a set of real numbers between 0 and 1 that is constructed by removing the middle third of each interval in the previous iteration. The 1/n factor in front means that each interval is divided into n equal parts before the middle third is removed.

2. How is the Cantor set C constructed?

The Cantor set C is constructed by starting with the interval [0,1] and removing the middle third, resulting in two intervals [0,1/3] and [2/3,1]. This process is repeated infinitely, with the middle third of each remaining interval being removed in each iteration.

3. What is the significance of representing the Cantor set C in ternary form?

The ternary form of the Cantor set C allows for a clearer understanding of its structure, as it shows the pattern of removing the middle third of each interval in a visual way. It also highlights the self-similarity of the set, as each iteration results in smaller copies of the original set.

4. How does the 1/n factor affect the construction of the Cantor set C?

The 1/n factor in front of the Cantor set C means that each interval is divided into n equal parts before the middle third is removed. This results in a finer division of intervals and a more complex structure of the set. As n approaches infinity, the Cantor set C becomes a perfect Cantor set, with a fractal structure.

5. What are some real-world applications of the Cantor set C in ternary form with 1/n factor?

The Cantor set C has applications in various fields such as mathematics, physics, and computer science. It is used in the study of fractals and self-similarity, and it has been applied in signal processing, data compression, and image analysis. It also has connections to chaos theory and the behavior of dynamical systems.

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