Question

solidifying ideas

1. a circle is a circle (h). prove that a small circle has the same number of points as a large circle. stated precisely, prove that the cardinality of points on a small circle is the same as the cardinality of points on a large circle. describe a one - to - one correspondence between these two sets.

Explanation:

Step1: Define the circles

Let the small - circle be \(C_1\) with center \(O_1\) and radius \(r_1\), and the large - circle be \(C_2\) with center \(O_2\) and radius \(r_2\) (\(r_1

Step2: Establish one - to - one correspondence

For any point \(P_1\) on the small circle \(C_1\), draw a ray starting from the common center \(O\) (in the concentric case) passing through \(P_1\). This ray will intersect the large circle \(C_2\) at exactly one point \(P_2\). Conversely, for any point \(P_2\) on the large circle \(C_2\), the ray starting from the center \(O\) passing through \(P_2\) will intersect the small circle \(C_1\) at exactly one point \(P_1\). This defines a one - to - one correspondence \(f:C_1
ightarrow C_2\) where \(f(P_1)=P_2\) and \(f^{-1}(P_2)=P_1\). Since there is a one - to - one correspondence between the points of the small circle and the points of the large circle, the cardinality of the set of points on the small circle is the same as the cardinality of the set of points on the large circle.

Answer:

The cardinality of points on a small circle is the same as that on a large circle, and the one - to - one correspondence is established by drawing rays from the common center (in the concentric case) or a corresponding center (in the non - concentric case) through the points of one circle to get the corresponding points on the other circle.