If A={4, 6, 10, 12} and R is a relation defined on A as ''two elements are related iff they have exactly one common factor other than 1''. Then the relation R is

Options

A. antisymmetric
B. only transitive
C. only symmetric
D. equivalence

Answer

To determine the nature of the relation \( R \) defined on the set \( A = \{4, 6, 10, 12\} \) based on the condition that two elements are related if and only if they have exactly one common factor other than 1, we will follow these steps:

### Step 1: Identify the Cartesian Product \( A \times A \)

The Cartesian product \( A \times A \) consists of all possible ordered pairs of elements from set \( A \).

\[ A \times A = \{(4, 4), (4, 6), (4, 10), (4, 12), (6, 4), (6, 6), (6, 10), (6, 12), (10, 4), (10, 6), (10, 10), (10, 12), (12, 4), (12, 6), (12, 10), (12, 12)\} \]

### Step 2: Determine the Relation \( R \)

We need to find pairs \( (a, b) \) such that \( a \) and \( b \) have exactly one common factor other than 1. Let's analyze the pairs:

- **Common factors of 4**: {1, 2, 4}
- **Common factors of 6**: {1, 2, 3, 6}
- **Common factors of 10**: {1, 2, 5, 10}
- **Common factors of 12**: {1, 2, 3, 4, 6, 12}

Now let's check pairs:

1. \( (4, 6) \): Common factors = {2} → Related
2. \( (4, 10) \): Common factors = {2} → Related
3. \( (4, 12) \): Common factors = {2, 4} → Not related
4. \( (6, 10) \): Common factors = {2} → Related
5. \( (6, 12) \): Common factors = {2, 3, 6} → Not related
6. \( (10, 12) \): Common factors = {2} → Related

Thus, the relation \( R \) consists of the following pairs:

\[ R = \{(4, 6), (4, 10), (6, 4), (6, 10), (10, 4), (10, 6), (10, 12), (12, 10)\} \]

### Step 3: Check Properties of Relation \( R \)

1. **Symmetric**: A relation is symmetric if for every \( (a, b) \in R \), \( (b, a) \in R \).
- Here, \( (4, 6) \) and \( (6, 4) \) are both in \( R \), \( (4, 10) \) and \( (10, 4) \) are both in \( R \), etc.
- Thus, \( R \) is symmetric.

2. **Antisymmetric**: A relation is antisymmetric if for every \( (a, b) \in R \) and \( (b, a) \in R \), \( a = b \).
- Since \( (4, 6) \) and \( (6, 4) \) are in \( R \) but \( 4 \neq 6 \), \( R \) is not antisymmetric.

3. **Transitive**: A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
- Checking pairs like \( (4, 6) \) and \( (6, 10) \): \( (4, 10) \) is in \( R \), but we can find cases where it fails (e.g., \( (6, 10) \) and \( (10, 12) \) does not imply \( (6, 12) \)).
- Thus, \( R \) is not transitive.

### Conclusion

The relation \( R \) is symmetric but not antisymmetric and not transitive. Therefore, the correct answer is that the relation \( R \) is **only symmetric**.