Is ℝ or ℝ2 Bigger in the Upper Right Quadrant of the Unit Square?

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The discussion concludes that the sets ℝ (the real numbers) and ℝ² (the two-dimensional real number space) are of the same cardinality, despite both being infinite. A one-to-one mapping can be established between ℝ and ℝ², demonstrating that every point on the real line corresponds to a point in ℝ². Specifically, the method involves taking an infinite decimal representation and splitting it into two separate infinite decimals, effectively creating a surjection from ℝ to ℝ² within the upper right quadrant of the unit square.

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ℝ or ℝ2? Both are infinite, but is one greater then the other?
 
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johann1301 said:
ℝ or ℝ2? Both are infinite, but is one greater then the other?
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.
 
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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