Consider the set A = {1,3,5}.
Assertion (A): The number of reflexive relations on set A is `2^(9)` .
Reason (R): A relation is said to be reflexive if `A in A`.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the problem, we need to determine whether the assertion and reason provided are true or false regarding the number of reflexive relations on the set \( A = \{1, 3, 5\} \).

### Step-by-Step Solution:

1. **Understanding Reflexive Relations**:
A relation on a set is called reflexive if every element is related to itself. For a set \( A \) with \( n \) elements, a reflexive relation must include all pairs of the form \( (a, a) \) for each \( a \in A \).

2. **Identifying the Set**:
The set \( A \) has 3 elements: \( 1, 3, 5 \). Therefore, the reflexive pairs that must be included in any reflexive relation on this set are:
- \( (1, 1) \)
- \( (3, 3) \)
- \( (5, 5) \)

3. **Counting Total Pairs**:
The total number of possible pairs (relations) that can be formed from the set \( A \) is \( n^2 \), where \( n \) is the number of elements in the set. For \( A \):
\[ n = 3 \implies n^2 = 3^2 = 9 \]
Thus, there are 9 possible pairs.

4. **Determining Reflexive Relations**:
Since a reflexive relation must include the 3 reflexive pairs mentioned above, we are left with \( 9 - 3 = 6 \) pairs that can either be included or excluded from the relation. These pairs are:
- \( (1, 3) \)
- \( (1, 5) \)
- \( (3, 1) \)
- \( (3, 5) \)
- \( (5, 1) \)
- \( (5, 3) \)

5. **Calculating the Number of Reflexive Relations**:
Each of the 6 remaining pairs can either be included in the relation or not, giving us \( 2 \) choices (include or exclude) for each pair. Therefore, the total number of reflexive relations is:
\[ 2^6 = 64 \]

6. **Conclusion**:
The assertion states that the number of reflexive relations is \( 2^9 \). We found that it is \( 2^6 \), so the assertion is **false**. The reason states that a relation is reflexive if \( A \in A \), which is a misinterpretation of reflexivity. The reason is also **false**.

### Final Answer:
- Assertion (A): False
- Reason (R): False