Let R= {(3, 3),(6,6), (9,9), (12, 12),(6, 12),(3,9), (3, 12), (3, 6)} be a relation on the set A={3,6,9,12}. The relation is:

Options

A. an equivalence relation.
B. reflexive and symmetric
C. reflexive and transitive
D. reflexive only

Answer

To determine the properties of the relation \( R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\} \) on the set \( A = \{3, 6, 9, 12\} \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element in the set \( A \) is related to itself. This means that for every \( a \in A \), the pair \( (a, a) \) must be in \( R \).

- The elements of \( A \) are \( 3, 6, 9, 12 \).
- The pairs we need to check are \( (3, 3), (6, 6), (9, 9), (12, 12) \).

**Check:**
- \( (3, 3) \in R \)
- \( (6, 6) \in R \)
- \( (9, 9) \in R \)
- \( (12, 12) \in R \)

Since all required pairs are in \( R \), the relation is **reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every \( (a, b) \in R \), the pair \( (b, a) \) must also be in \( R \).

**Check:**
- For \( (6, 12) \in R \), we check if \( (12, 6) \in R \). It is not.
- For \( (3, 9) \in R \), we check if \( (9, 3) \in R \). It is not.
- For \( (3, 12) \in R \), we check if \( (12, 3) \in R \). It is not.
- For \( (3, 6) \in R \), we check if \( (6, 3) \in R \). It is not.

Since there are pairs for which the reverse is not present in \( R \), the relation is **not symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \).

**Check:**
1. From \( (3, 6) \) and \( (6, 12) \), we check if \( (3, 12) \in R \). It is.
2. From \( (3, 9) \) and \( (9, 9) \), we check if \( (3, 9) \in R \). It is.
3. From \( (3, 12) \) and \( (12, 12) \), we check if \( (3, 12) \in R \). It is.
4. From \( (6, 12) \) and \( (12, 12) \), we check if \( (6, 12) \in R \). It is.

Since all combinations satisfy the transitive property, the relation is **transitive**.

### Conclusion
- The relation \( R \) is **reflexive** and **transitive**, but **not symmetric**. Therefore, it is not an equivalence relation.