The set `A={x: a^(x)=1, a gt 0, xin R}` can never be

Options

A. Null set
B. Singleton set
C. Finite set
D. Infinite set

Answer

To solve the problem, we need to analyze the set \( A = \{ x : a^x = 1, a > 0, x \in \mathbb{R} \} \) and determine the nature of this set.

### Step-by-step Solution:

1. **Understanding the Equation**:
The equation \( a^x = 1 \) holds true under certain conditions. Specifically, for any positive \( a \), the equation \( a^x = 1 \) is true if \( x = 0 \). This is because any non-zero number raised to the power of 0 equals 1.

**Hint**: Recall the properties of exponents, particularly that \( a^0 = 1 \) for any \( a > 0 \).

2. **Identifying Possible Values of \( x \)**:
Since \( a^x = 1 \) when \( x = 0 \), we can conclude that \( x = 0 \) is a solution. However, we must consider if there are other solutions.

**Hint**: Think about whether there are other values of \( x \) that can satisfy \( a^x = 1 \) for \( a > 0 \).

3. **Exploring Different Cases for \( a \)**:
- If \( a = 1 \), then \( 1^x = 1 \) for all \( x \). Thus, the set \( A \) would contain all real numbers, making it an infinite set.
- If \( a > 1 \) or \( 0 < a < 1 \), the only solution remains \( x = 0 \). Thus, the set \( A \) would be \( \{0\} \), which is a singleton set.

**Hint**: Analyze the behavior of the function \( a^x \) for different values of \( a \).

4. **Conclusion on the Nature of Set \( A \)**:
From the above analysis, we can conclude:
- If \( a = 1 \), \( A \) is infinite.
- If \( a \neq 1 \), \( A \) is a singleton set containing just \( \{0\} \).

Therefore, the set \( A \) can never be a null set (empty set) because there will always be at least one value (either all real numbers or just \( 0 \)).

**Hint**: Remember that a null set is defined as a set that contains no elements at all.

### Final Answer:
The set \( A \) can never be a null set.