Results 1 to 3 of 3

Thread: Cardinality of R x R

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    Cardinality of R x R

    Is it true that for any uncountable set S, S \times S is uncountable, and therefore this leads to the claim that \mathbb{R \times R} is uncountable?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Quote Originally Posted by Pinkk View Post
    Is it true that for any uncountable set S, S \times S is uncountable, and therefore this leads to the claim that \mathbb{R \times R} is uncountable?
    Yes, this is correct. If we were to assume \mathbb{R} \times \mathbb{R} is countable, it would be easy to conclude that \mathbb{R} is also countable (in contradiction, of course), since:

     f:\mathbb{R \to R \times R} defined by f(x) = (x,0) is an injection, and therefore \mathbb{|R| \leq |R \times R|}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,573
    Thanks
    789
    I would not be able to post anything on this forum without Wikipedia...

    Namely, the section about cardinal arithmetic defines cardinal multiplication as the cardinality of the Cartesian product of the given sets. It also says that if the axiom of choice is accepted (which it usually is in regular mathematics), then for two non-zero cardinals \kappa and \mu, if at least one of them is infinite, then \kappa\cdot\mu=\max(\kappa, \mu), which, interestingly, equals \kappa+\mu.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cardinality
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 11th 2010, 07:08 AM
  2. Cardinality of R, [0,1], R^2, [0,1] x [0,1]
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: Mar 18th 2010, 02:20 PM
  3. Cardinality
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 31st 2009, 03:56 PM
  4. Cardinality
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Feb 11th 2009, 05:36 PM
  5. Cardinality
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Feb 8th 2009, 02:23 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum