Given two concentric circles (seen to the right), where could P be located so that there is a one-to-one correspondence between the two circles? A. anywhere outside of the two circles B. anywhere between the two circles or on either circle C. anywhere between the two circles D. anywhere outside of the two circles or on the larger circle E. anywhere within the smaller circle F. anywhere within the smaller circle or on the smaller circle

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UpStudy AI · Accepted Answer

To determine where point $$ P $$ could be located so that there is a one-to-one correspondence between the two concentric circles, we need to analyze the properties of one-to-one correspondence (bijective function) in the context of the circles.

1. **Understanding the Circles**: 
   - We have two concentric circles, meaning they share the same center but have different radii. Let's denote the smaller circle with radius $$ r_1 $$ and the larger circle with radius $$ r_2 $$.

2. **One-to-One Correspondence**: 
   - A one-to-one correspondence between two sets means that each element in one set can be paired with exactly one element in the other set, and vice versa. In this case, we want to map points from the smaller circle to points on the larger circle.

3. **Possible Locations for $$ P $$**:
   - **Outside the Circles**: If $$ P $$ is outside both circles, it can correspond to points on both circles, but it does not create a unique mapping since there are infinitely many points on both circles.
   - **Between the Circles**: If $$ P $$ is located between the two circles, it can correspond to points on both circles, but again, it does not create a unique mapping.
   - **On the Circles**: If $$ P $$ is on the larger circle, it can correspond to points on the smaller circle, but this does not guarantee a one-to-one mapping.
   - **Within the Smaller Circle**: If $$ P $$ is within the smaller circle, it can correspond to points on the larger circle, but again, this does not create a unique mapping.

4. **Conclusion**: 
   - The only way to ensure a one-to-one correspondence is if $$ P $$ is located outside of both circles or on the larger circle, as this allows for a unique mapping from points on the smaller circle to points on the larger circle.

Thus, the correct answer is:

**D. anywhere outside of the two circles or on the larger circle**.
$$ P $$ can be anywhere outside both circles or on the larger circle to have a one-to-one correspondence with the two concentric circles.

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