1.I am trying to prove that two circles has the same cardinality I declared two intervals
$[0,2\pi R]$ and $[0,2\pi \widetilde{R}]$ I build I bijection between the two interval
It's the correct proof ?
$2$.The equation for a closed disk $(x-a)^{2}+(y-b)^{2}\leqslant R^{2}$.
Can I prove it by doing in the same way that I prove the circles cardinality but now with the area $\pi R^{2}$ instead of using the circumference.
Thanks
[Math] how to prove that any two circles and any two disks have the same cardinality
elementary-set-theory
Best Answer
Place the circles so that both of their centres are at origin. Now draw straight lines from origin to the points on the circumference of the bigger circle, which will intersect the circumference of the smaller circle at exactly one point. This will give a bijection between the two circles.
For discs, the map $$f(r,\theta )=(\frac{r R_2}{R_1},\theta)$$ will give a bijection from the disc of radius $R_1$ to the disc of radius $R_2.$