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Cantor set fun

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data

    I am trying to find if 5/27 and 8/9 are in the Cantor set.

    2. Relevant equations

    [tex]C_2=[0,1/9]\cup[2/9,3/9]\cup...\cup[8/9,1][/tex]

    [tex]C_3=[0,1/27]\cup[2/27,3/27]\cup[4/27,5/27]\cup...\cup[26/27,1][/tex]

    3. The attempt at a solution

    I have: [tex]8/9=(0.22)_3[/tex]
    and it is an endpoint in one of the closed sets of C2, so it is in the Cantor set.

    I also have: [tex]5/27=(0.012)_3[/tex]
    It is an endpoint in C3, but doesn't the 1 in the base 3 expansion mean that it isn't part of the Cantor set?
     
    Last edited: Nov 29, 2011
  2. jcsd
  3. Nov 29, 2011 #2

    micromass

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    Your [itex]C_3[/itex] is wrong. The interval [itex]\left[\frac{4}{27},\frac{5}{27}\right][/itex] is not part of the third stage of the Cantor set. Can you think about what would be the correct interval?
     
  4. Nov 29, 2011 #3
    Oh yes, how stupid of me.

    It's:

    [tex][0,1/27]\cup[2/27,3/27]\cup[6/27,7/27]\cup[8/27,9/27]\cup...[/tex]

    So 5/27 has been removed. Thanks.
     
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