The Cantor Set: find total length of all segments removed

[Jamers328]

The Cantor set is a subset of the unit interval [0,1]. To construct the Cantor set, first remove the middle third (1/2, 2/3) of the interval, leaving two line segments. For the second step, remove the middle third of each of the two remaining segments, leaving for line segments. Continue this procedure indefinitely, as shown in the figure. The Cantor set consists of all numbers in the unit interval [0,1] that still remain.

a) Find the total length of all the line segments that are removed.

b) Write down three numbers that are in the Cantor set.

c) Let Cn denote the total length of the remaining line segments after n steps. Find lim(n to infinity) Cn.

part (a) it would be the SUM of (2^n)/(3^n+1) = 1.

part (b) could be a number of things: 0, 1/3, 2/3...

part (c) is 0... I just did the limit of n to infinity (2^n)/(3^n+1) and I got 0.

I've been working a lot on this problem. My biggest question is concerning part (c). I understand the limit is equal to 0, but what does this mean?
For instance, in part (a) I think the point of the question is that when you see that the answer is 1, you think "how is that possible? Does that mean the Cantor set is empty?" which is not true.
My question now is, what is the significance of the limit equaling 0?

 

[pka]

Re: The Cantor Set

The Cantor Set is one of the most studied examples is all mathematics.
It is uncountable subset of [0,1] which has no content (i.e its measure is zero).
That is the answer to your question about zero. It is a very ‘large’ subset that takes up no room because as you have shown its complement has length one the length of the superset [0,10].

Go to Mathworld.com for more information.

 

[Jamers328]

Re: The Cantor Set

Thanks for the reply, pka. Always very appreciated.

Why do you say it has no content though?
I do understand that its complement, the segments removed, is 1 (which is the whole segment that we began with), but we also know there are values in the Cantor Set.
I'm sorry, I'm sure it's not that difficult to understand, but I don't get it.

I do thank you for your reply!

Edit: I did my presentation today on the Cantor set and it went well. pka, thank you again for your help.