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Math Help - Diagonalization troubles.

  1. #1
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    Diagonalization troubles.

    Let T be the union of the Tks from k=1 to infinity. Use a diagonalization process like that one used to show the set of positive rational numbers is denumerable to show that T is denumerable.

    I looked at the argument for the positive rationals, but can not decide what to make the entries that I enumerate here. I am a bit confused.

    Thanks.
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  2. #2
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    Quote Originally Posted by zhupolongjoe View Post
    Let T be the union of the Tks from k=1 to infinity. Use a diagonalization process like that one used to show the set of positive rational numbers is denumerable to show that T is denumerable.
    You will have to supply a great deal more information to make the question intelligible.
    Like what are the T_k?

    By “diagonalization process” do you mean the so-called zigzag argument?
    I think of Cantor’s ‘diagonal argument’ as being used to show a set is non-denumerable.
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  3. #3
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    My apologies, I forgot the first part of the problem...I got this part, but it is probably needed to answer the part I didn't get:

    For all natural numbers n, let Tn be the set of all sequences of exactly n 1's. I showed this is denumerable for all n.

    Now I need to use the diagonalization, which my book uses to show the positive rationals are denumerable.
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  4. #4
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    I can show you a proof for that statement, but it is not a diagonalzation argument.
    It involves the prime factorization theorem, mapping \{T_n\} to the positive integers one-to-one.
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  5. #5
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    Thanks, but I already did that part and now I just need a diagonalization argument.
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  6. #6
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    Wink

    From my understanding, the diagonal argument is used to prove a set is uncountable, not to prove it is countable.
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