Which one of the following is a null set?

Options

A. A)`(0)`
B. B)`{(11)}`
C. C)`{11}`
D. D)`[|x|x^(2)+1=0, x in R]`

Answer

To determine which of the given options is a null set, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Null Set**: A null set, also known as an empty set, is a set that contains no elements. It is denoted by the symbol ∅ or {}.

2. **Analyzing Option A**:
- Option A contains the set {0}.
- Since this set contains one element (0), it is not a null set.

3. **Analyzing Option B**:
- Option B contains the set {11}.
- This set contains one element (11), so it is also not a null set.

4. **Analyzing Option C**:
- Option C contains the set {11} again.
- Similar to option B, this set contains one element (11), thus it is not a null set.

5. **Analyzing Option D**:
- Option D presents the set defined by the equation: {x | x² + 1 = 0, x ∈ ℝ}.
- To solve for x, we rearrange the equation: x² = -1.
- The solutions to this equation are x = ±√(-1), which gives us x = ±i (where i is the imaginary unit).
- However, the condition states that x must belong to the real numbers (ℝ). Since there are no real solutions to the equation x² + 1 = 0, this means that there are no elements in this set.

6. **Conclusion**:
- Since option D has no elements, it is a null set (or empty set).
- Therefore, the correct answer is **Option D**.