Which set is greater?

1. Nov 12, 2014

johann1301

ℝ or ℝ2? Both are infinite, but is one greater then the other?

Last edited: Nov 12, 2014
2. Nov 12, 2014

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.

3. Nov 12, 2014

mathwonk

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?