Results 1 to 4 of 4

Math Help - Countable set?

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    19

    Countable set?

    Is this set countable or uncountable?

    The set I of all two-element subsets of Z+
    (Z+ being positive integers)

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,969
    Thanks
    1788
    Awards
    1
    Quote Originally Posted by iamthemanyes View Post
    Is this set countable or uncountable?
    The set I of all two-element subsets of Z+ (Z+ being positive integers
    Would you agree that \mathbb{I}=\left\{ {\{ j,k\}  \subset \mathbb{Z}^ +  :j < k} \right\} is your set?

    Now define \Phi: \mathbb{I} \to \mathbb{Z}^+ by \{j,k\}\mapsto 2^j\cdot 3^k.

    Can you prove that \Phi is injective?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    19
    Fantastic! Thank you so much.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,914
    Thanks
    779
    Hello, iamthemanyes!

    Is this set countable or uncountable?

    The set I of all two-element subsets of Z^+\;\;(Z^+ being positive integers)

    Since we have a one-to-one correspondance: . (a,b) \;\leftrightarrow\;\dfrac{a}{b}
    . . set I has the same cardinality as the set of rational numbers,
    . . which is countable.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: January 9th 2013, 07:14 AM
  2. Replies: 2
    Last Post: November 11th 2010, 05:56 AM
  3. [SOLVED] Countable union of closed sets/countable interesection of open sets
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 8th 2010, 02:59 PM
  4. Replies: 1
    Last Post: February 9th 2010, 02:51 PM
  5. Replies: 11
    Last Post: October 11th 2008, 07:49 PM

Search Tags


/mathhelpforum @mathhelpforum