Results 1 to 4 of 4

Math Help - Cantor set problem

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    177
    Thanks
    1

    Cantor set problem

    Could someone give me a hint as to how I could go about proving the following?

    Let C be the cantor set. Show that x=0.a_1a_2a_3..., where RHS is the base 3 expansion of x, is in C iff, for all natural numbers n, a_n\epsilon\{0, 2\}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    I'm not saying I understand what this is saying, but it might point you in the right direction.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2009
    Posts
    177
    Thanks
    1
    Thanks, I think I figured out how to prove it. It turns out that I should have mentioned something earlier. This something is the actual definition of the cantor set. Here it is. Let T=[0,1]. Then, remove the midle 3rd open segment to form the set
    A_1= [0,1/3]U[2/3,1]. Then, remove the middle 3rd open segement in each of those intervals whose union forms A_1 to get A_2. Continue this k times to get A_k. Then, define the cantor set as \bigcap\{A_k: k\epsilon N\}.

    You have to proceed by induction. Writing x=1/3 as 0.022222222222222222222222222222222..., to handle the endpoints. You suppose that x is in the cantor set, and that the nth digit of x is either 0 or 2. Then, x is in A_n. Then, the kth dight of x sort of tells you which 3rd of a segment from A_{k-1} x is in. It dosen't matter which one, as long as it's either the first or the 2nd. On the k+1th iteration, the k+1th digit must be either 0 or 2 since it's either in the first segment or the last segment of whatever segment it was in before. Hence, x only has 0s or 2s in its base 3 expansion. Then, you do something similar to prove the converse
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2009
    Posts
    177
    Thanks
    1
    I got the problem from a book that I'm going through. It's the type of problem where the author guides you through someone's research
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cantor set
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 10th 2008, 12:42 PM
  2. A Cantor set
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 29th 2008, 02:12 PM
  3. help with Cantor-Schroder-Berstein problem please!
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: May 24th 2008, 05:08 AM
  4. Cantor Set
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 29th 2007, 04:11 AM
  5. Cantor Set
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: November 7th 2006, 07:18 AM

Search Tags


/mathhelpforum @mathhelpforum