Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Which set is greater?

  1. Nov 12, 2014 #1
    ℝ or ℝ2? Both are infinite, but is one greater then the other?
     
    Last edited: Nov 12, 2014
  2. jcsd
  3. Nov 12, 2014 #2

    Mark44

    Staff: Mentor

    They are both the same size. It's not too difficult to find a one-to-one map between the two spaces, which is enough to show that every point on the real line has a counterpart in R2, and vice versa.
     
  4. Nov 12, 2014 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    It suffices to find a surjection from R to R^2. restricting say to the part of the upper right quadrant in the unit square, somethingn like this might work:

    given an infinite decimal .abcdefghijkl......, split it into two infinite decimals (.acegik...., .bdfhjl.....). Can you see that every pair of such decimals arises this way?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Which set is greater?
  1. Which is greater? (Replies: 22)

Loading...