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Homework Help: Number Theory (Finite and Infinite Sets)

  1. Nov 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Why is R\Q not countably infinate or denumerable? Given R (Real Number) is not countably infinate or denumerable and Q (rational number) is denumerable.

    2. Relevant equations

    A set is said to be denumberable or countably infinate if there exists a bijestion of N (natural Number) onto S.

    3. The attempt at a solution

    Let Q be a subset of R and Let S be R-Q, which is denumerable. Via the defn there exists a bijection but R-Q is not bijective R-Q is the set of Real Numbers which is already not denumerable. I showed this to my prof and he said its not correct. He said to use Q u (R-Q)=R, we know Q is denumerable and R not to be denumerable use this to show R-Q is not denumerable? HOW!
  2. jcsd
  3. Nov 19, 2009 #2
    What you need to show is that the union of two countably infinite sets is countable. So take two sets {A_n} and {B_n}. Can you find a bijection from their union to the natural numbers? (hint: it's really easy)
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