Bijective matching between the points of a disk and the points in the plain

**doug** · September 26th 2010, 06:36 AM

There are two sets:
- A set: is the points of a disk (of radius 1), so x^2+y^2<1
- B set is: RxR

We should prove that two sets have the same cardinality, so we should find a bijection from A to B.

Thanks for helping me!

**HappyJoe** · September 26th 2010, 07:49 AM

You can take the following function $f\colon D^2\rightarrow \mathbb{R}^2$ , where $D^2$ is the open unit disc:

$f(x,y) = \frac{(x,y)}{1-|(x,y)|^2},$

where $(x,y)$ is a point of $D^2$ , and where $|(x,y)|$ is the norm of the point, i.e.

$|(x,y)|^2 = x^2+y^2.$

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