Results 1 to 3 of 3

Math Help - any denumerable set has uncountably many subsets

  1. #1
    Senior Member
    Joined
    Dec 2008
    Posts
    288

    any denumerable set has uncountably many subsets

    How do I go about proving:

    any denumerable set has uncountably many subsets.

    I have a proposition saying that for any set A |A|<|\mathcal{P}(A)  does this help in anyway?

    It is just stated as a corollary but it has not been proven.

    Thanks for any help

    As |\mathcal{P}(A)|>|A| then no bijection exists between the two and so no bijection exits between \mathcal{P}(A) and \mathcal{N}

    (as there is a bijection between \mathcal{N} and A)

    so \mathcal{P}(A) is uncountable
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: any denumerable set has uncountably many subsets

    You could slightly reorder some of those statements to make is 'flow' better. However, you do have the correct ideas there.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2010
    Posts
    466
    Thanks
    4

    Re: any denumerable set has uncountably many subsets

    You don't even have to involve the cardinality operator '| |'.

    We'd have already proven that there is no bijection between A and PA (by Cantor's argument that there is no surjection from A onto PA).

    So if A is denumerable (countably infinite), then PA is not denumerable. And PA is infinite.

    Therefore, PA is uncountable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove that a set is denumerable iff...
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 2nd 2010, 11:26 AM
  2. Replies: 1
    Last Post: October 2nd 2010, 11:21 AM
  3. Denumerable Set
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 9th 2010, 11:54 AM
  4. Denumerable sets...
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 5th 2009, 10:26 PM
  5. Replies: 3
    Last Post: February 3rd 2009, 02:33 PM

Search Tags


/mathhelpforum @mathhelpforum