cantor set = set of all what in base 3?
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cantor set = set of all what in base 3?
Reading through my notes in my Real Analysis class, I wrote down that the cantor set P = {x \in \R  x has n 1s in its base three expansion}. I guess I forgot to put what n was. Is n any element of Z? Did I mean to write 'inf'? Little help please?

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Re: cantor set = set of all what in base 3?
IIRC n means "a finite number" here.
ETA: I do not recall correctly and did not do any checking of my recollection and it shows.
ETA: I do not recall correctly and did not do any checking of my recollection and it shows.
Last edited by GreedyAlgorithm on Fri Oct 26, 2007 5:17 am UTC, edited 1 time in total.
GENERATION 1i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation.
Re: cantor set = set of all what in base 3?
I don't think you forgot to define n. I think you dropped an 'o'.
Every number in the Cantor set can be written with no 1s in its ternary expansion.
(Note that 1/3 = 0.1_{3} = 0.02222222..._{3}, so 1/3 is in the Cantor set even though one of the two ways to write it violates the above.)
Every number in the Cantor set can be written with no 1s in its ternary expansion.
(Note that 1/3 = 0.1_{3} = 0.02222222..._{3}, so 1/3 is in the Cantor set even though one of the two ways to write it violates the above.)
Re: cantor set = set of all what in base 3?
Buttons is right on. If you're used to thinking of the Cantor set as "Start with [0,1] and repeatedly remove the middle thirds of all segments", then notice that the first iteration removes everything that must be written in ternary with a 1 right after the radix point, the second iteration removes everything that must be written with a 1 in the _next_ position, etc.
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