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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

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Notice that any continuous curve has the cardinality of $\mathbb{R}$ being an injective image of $[0, 1]$ and also every set containing an open set can be shown to contain a continuous curve and thus has the same cardinality. –  Karolis Juodelė Aug 20 '12 at 11:15
    
$(1)$. Need to be careful, you want to prove a geometric fact. Note want to use half-open intervals, like $[0,2\pi R)$. One cannot assess what you did without being given some detail. $(2)$. Again, need explicit geometric bijection. –  André Nicolas Aug 20 '12 at 11:32
    
Hello Nicolas:Why did you write half open interval $[0,2\pi R)$ if the point $2\pi R$ is a memeber it is the last point of the circle. –  Hernan Aug 20 '12 at 12:18
    
@Hernan: Circles don't have endpoints. –  Hurkyl Oct 29 '12 at 8:13
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1 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 $R_2.$

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Thanks, Is my proof for the circles are correct? What is the significance of r? –  Hernan Aug 20 '12 at 10:22
    
Yeah, your proof is also correct, but I found it more easy to visualize. Here $(r,\theta )$ is the polar coordinates of a point in the plane –  pritam Aug 20 '12 at 10:34
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