Which of the following in a null set :

Options

A. `{0}`
B. `{x:xgt0orxlt0}`
C. `{x:x^(2)=4orx=3}`
D. `{x:x^(2)+1=0,x inR}`

Answer

To determine which of the given sets is a null set, we need to analyze each option carefully. A null set (or empty set) is defined as a set that contains no elements.

### Step-by-Step Solution:

1. **Analyze the First Set:**
- The first set is {0}.
- This set contains one element, which is 0.
- **Conclusion:** This is not a null set.

2. **Analyze the Second Set:**
- The second set is defined as all x such that x > 0 or x < 0.
- This means it includes all positive and negative real numbers (but not 0).
- **Conclusion:** This set contains infinitely many elements, so it is not a null set.

3. **Analyze the Third Set:**
- The third set is defined as all x such that x² = 4 or x = 3.
- The solutions to x² = 4 are x = 2 and x = -2.
- Therefore, the set includes {2, -2, 3}.
- **Conclusion:** This set contains three elements, so it is not a null set.

4. **Analyze the Fourth Set:**
- The fourth set is defined as all x such that x² + 1 = 0.
- Rearranging gives x² = -1, which has no real solutions.
- The solutions are x = ±i (where i is the imaginary unit), which are not real numbers.
- Since we are considering x belonging to the set of real numbers (R), there are no real solutions.
- **Conclusion:** This set contains no elements, making it a null set.

### Final Answer:
The fourth option, where x² + 1 = 0, is a null set.