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Thread: proving a set is countable

  1. #1
    Super Member
    Sep 2008

    proving a set is countable


    Can anyone please help me with this question, I am finding it very hard. I am not sure where to begin
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  2. #2
    MHF Contributor
    Nov 2010

    Re: proving a set is countable

    Let's write S(E) a slightly different way:

    S(E) = \left\{A \subseteq E \mid \text{card}(A) = n, n\in \mathbb{N}\right\}

    Note that this is similar to the power set, but the power set includes subsets with infinite cardinality and the empty set.

    Let P be the set of primes (as suggested in the hint). Define f:E \to P to be any injection. Such an injection exists since both E and P are countable. Then, define g:S(E) \to \mathbb{N} by g(A) = \prod_{e \in A}f(e). Prove that g is an injection.

    Another proof of this: If E is finite, then S(E) = \mathcal{P}(E) \setminus \{\emptyset\}. Hence, \text{card}(S(E)) = 2^{\text{card}(E)}-1 where \mathcal{P}(E) is the power set of E, so S(E) is finite and therefore countable. Suppose E is infinite and let E = \{e_0,e_1,\ldots\} be an enumeration of the elements of E. Then, define h: S(E) \to \mathbb{N} by h(A) = \sum_{e_k \in A}2^k. Show that h is a bijection.
    Last edited by SlipEternal; Nov 5th 2013 at 08:00 AM.
    Thanks from Tweety
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