A relation `R` is defined on the set of circles such that "`C_(1)RC_(2)implies` circle `C_(1)` and circle `C_(2)` touch each other externally", then relation `R` is

Options

A. Reflexive and symmetric but not transitive
B. Symmetric only
C. Symmetric and transitive but not reflexive
D. Equivalence

Explanation

To determine the nature of the relation \( R \) defined on the set of circles such that \( C_1 R C_2 \) implies that circle \( C_1 \) and circle \( C_2 \) touch each other externally, we will analyze the properties of this relation: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Reflexivity**:
- A relation \( R \) is reflexive if every element is related to itself. This means for every circle \( C \), it should hold that \( C R C \).
- In our case, a circle cannot touch itself externally; it can only touch itself internally or not touch at all.
- Therefore, \( C R C \) does not hold for any circle \( C \).
- **Conclusion**: The relation \( R \) is **not reflexive**.

2. **Symmetry**:
- A relation \( R \) is symmetric if whenever \( C_1 R C_2 \) holds, then \( C_2 R C_1 \) must also hold.
- If circle \( C_1 \) touches circle \( C_2 \) externally, then circle \( C_2 \) also touches circle \( C_1 \) externally at the same point.
- Therefore, if \( C_1 R C_2 \) is true, then \( C_2 R C_1 \) is also true.
- **Conclusion**: The relation \( R \) is **symmetric**.

3. **Transitivity**:
- A relation \( R \) is transitive if whenever \( C_1 R C_2 \) and \( C_2 R C_3 \) hold, then \( C_1 R C_3 \) must also hold.
- Consider three circles \( C_1, C_2, \) and \( C_3 \). If \( C_1 \) touches \( C_2 \) externally and \( C_2 \) touches \( C_3 \) externally, it does not necessarily imply that \( C_1 \) touches \( C_3 \) externally.
- For example, if \( C_1 \) and \( C_2 \) are touching at one point, and \( C_2 \) and \( C_3 \) are touching at another point, \( C_1 \) and \( C_3 \) may not touch at all.
- **Conclusion**: The relation \( R \) is **not transitive**.

### Final Conclusion:
The relation \( R \) defined on the set of circles such that \( C_1 R C_2 \) implies that circles \( C_1 \) and \( C_2 \) touch each other externally is **symmetric but neither reflexive nor transitive**.