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Math Help - Cantor set question

  1. #1
    Senior Member Danneedshelp's Avatar
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    Cantor set question

    Let C represent the Cantor set.

    Q: Show that there exists x_{1},y_{1}\in{C} satisfying x_{1}+y_{1}=s\in[0,2].

    I found some work that goes like this...

    C_{1}=[0,1/3]\cup[2/3,1]. Then we have, [0,1/3]+[0,1/3]=[0,1/3]\cup[0,2/3]+[2/3,1]=[2/3,4/3]\cup[2/3,1]<br />
+[2/3,1]. So, C_{1}+C_{1}=[0, 2/3]\cup[2/3, 4/3]\cup[4/3, 2]=[0, 2]. Thus, for all s\in[0, 2], we can find an x_{1} , y_{1} C_{1} satisfying x_{1} + y_{1} = s

    I am not sure that I follow exactly what is being done. Can someone help me to understand the work above. Do I think of these intervals as only lengths?

    thanks
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Danneedshelp View Post
    Let C represent the Cantor set.

    Q: Show that there exists x_{1},y_{1}\in{C} satisfying x_{1}+y_{1}=s\in[0,2].

    I found some work that goes like this...

    C_{1}=[0,1/3]\cup[2/3,1]. Then we have, \color{blue}[0,1/3]+[0,1/3]=[0,1/3]\cup[0,2/3]+[2/3,1]=[2/3,4/3]\cup[2/3,1]<br />
+[2/3,1]. So, C_{1}+C_{1}=[0, 2/3]\cup[2/3, 4/3]\cup[4/3, 2]=[0, 2]. Thus, for all s\in[0, 2], we can find an x_{1} , y_{1} C_{1} satisfying x_{1} + y_{1} = s

    I am not sure that I follow exactly what is being done. Can someone help me to understand the work above. Do I think of these intervals as only lengths?
    C_1+C_1 means the set of all numbers that are equal to an element of C_1 plus an element of C_1. But C_1 = [0,1/3]\cup[2/3,1]. By adding two numbers from the interval [0,1/3] you can get anything in the interval [0,2/3]. By adding a number from the interval [0,1/3] and a number from the interval [2/3,1] you can get anything in the interval [2/3,4/3]. By adding two numbers from the interval [2/3,1] you can get anything in the interval [4/3,2]. Putting those facts together, you see that by adding two numbers from the set C_1 you can get anything in the interval [0,2]. That is what is meant by the line C_{1}+C_{1}=[0, 2/3]\cup[2/3, 4/3]\cup[4/3, 2]=[0, 2]. (The previous line, which I have highlighted in blue, seems totally confused.) Therefore, for any s\in[0,2], we can find x_1,\ y_1 \in C_1 with x_1+y_1=s.

    That is just the first stage of this proof. You need to go on to show that C_n+C_n = [0,2] for each of the sets C_n used in the construction of the Cantor set. So there exist x_n,\ y_n \in C_n with x_n+y_n=s. Finally, you need to show that a subsequence of the sequence (x_n) converges to a point x in the Cantor set. The corresponding subsequence of the y_ns will then converge to a point y in the Cantor set, with x+y=s.
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