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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


    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


    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?
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