1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

|P(N)| = |Reals|

  1. Feb 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that |P(N)|=|R|. (R=reals, |X| is the cardinality of X).

    2. Relevant equations



    3. The attempt at a solution

    #1.) P(N) --> R

    Given any element, A, of P(N), construct a decimal expansion .x1x2x3x4... by the rule that x_i=1 (if i is in A) and x_i=0 (if i is not in A).

    So the element {1,7} would give .1000001

    This map is 1-1 but not onto the Reals.

    #2.) R --> P(N)

    If I can show this direction then #3 follows. I know a little about the mapping of [0,1] onto the Reals. I cannot determine if that helps here.

    #3.) Using the Cantor-Schroeder-Bernstein Theorem, |P(N)|=|R|.
     
  2. jcsd
  3. Feb 13, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    For #1, instead of a decimal expansion make it a binary expansion. In fact I think you can show this to give a bijection between [0, 1] and P(N). Then use (or prove) the "famous" result that | [0, 1] | = |R|.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: |P(N)| = |Reals|
  1. Evaluate ord_p(p^N!) (Replies: 2)

  2. Lim of P(n)/(a^n) = 0 (Replies: 4)

Loading...