Let `R = {(3, 3), (6, 6), (9, 9), (6, 12), (3, 9), (3, 12),(12,12), (3, 6)}` is a relation on set `A = {3, 6, 9, 12}` then R is a) an equivalence relation b) reflexive and symmetric only c) reflexive and transitive only d) reflexive only

Options

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

Explanation

To determine the properties of the relation \( R \) on the set \( A \), we need to check if \( R \) is reflexive, symmetric, and transitive.

Given:
- Relation \( R = \{(3, 3), (6, 6), (9, 9), (6, 12), (3, 9), (3, 12), (12, 12), (3, 6)\} \)
- Set \( A = \{3, 6, 9, 12\} \)

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \in A \), the pair \( (a, a) \) is in \( R \).

- Check elements of \( A \):
- For \( 3 \): \( (3, 3) \in R \)
- For \( 6 \): \( (6, 6) \in R \)
- For \( 9 \): \( (9, 9) \in R \)
- For \( 12 \): \( (12, 12) \in R \)

Since all pairs \( (3, 3), (6, 6), (9, 9), (12, 12) \) are present in \( R \), we conclude that \( R \) is reflexive.

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

- Check pairs in \( R \):
- \( (3, 3) \) → \( (3, 3) \) (symmetric)
- \( (6, 6) \) → \( (6, 6) \) (symmetric)
- \( (9, 9) \) → \( (9, 9) \) (symmetric)
- \( (6, 12) \) → \( (12, 6) \) (not in \( R \))
- \( (3, 9) \) → \( (9, 3) \) (not in \( R \))
- \( (3, 12) \) → \( (12, 3) \) (not in \( R \))
- \( (12, 12) \) → \( (12, 12) \) (symmetric)
- \( (3, 6) \) → \( (6, 3) \) (not in \( R \))

Since there are pairs for which the symmetric counterpart is not in \( R \), we conclude that \( R \) 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 pairs:
- From \( (3, 6) \) and \( (6, 12) \), we check if \( (3, 12) \) is in \( R \) → Yes, \( (3, 12) \in R \).
- From \( (3, 12) \) and \( (12, 12) \), we check if \( (3, 12) \) is in \( R \) → Yes, \( (3, 12) \in R \).
- From \( (6, 12) \) and \( (3, 6) \), we check if \( (6, 3) \) is in \( R \) → No, \( (6, 3) \notin R \).

Since we have found instances where transitivity holds, we conclude that \( R \) is transitive.

### Conclusion
- \( R \) is reflexive.
- \( R \) is not symmetric.
- \( R \) is transitive.

Thus, the correct answer is:
**Option C: reflexive and transitive only.**