Mark the correct options

Options

A. A system X is in thermal equilibrium with Y but not with Z. The systems Y and Z may be in thermal equilibrium with each other.
B. A system X is in thermal equilibrium with but not with Z. The system Y and Z are not in thermal equilibrium with each other.
C. A system X is neither in thermal equilibrium with Y nor with Z. The system Y and Z must be in thermal equilibrium with each other.
D. A system X is neither in thermal equilibrium with Y nor with Z. The system Y and Z may be in thermal equilibrium with each other.

Explanation

To solve the question regarding the Zeroth Law of Thermodynamics and determine the correct options, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding the Zeroth Law of Thermodynamics**:
- The Zeroth Law states that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is also in thermal equilibrium with system C. This principle helps us understand the relationships between different systems in terms of temperature.

2. **Analyzing Option A**:
- Statement: "A system X is in thermal equilibrium with Y but not with Z."
- Since X is in thermal equilibrium with Y, it follows that Y could potentially be in thermal equilibrium with Z. However, the statement does not confirm that Y and Z are in thermal equilibrium. Therefore, this option is incorrect.

3. **Analyzing Option B**:
- Statement: "A system X is in thermal equilibrium with Y but not with Z."
- Similar to Option A, if X is in thermal equilibrium with Y, it does not imply anything about the relationship between Y and Z. Therefore, this option is also incorrect.

4. **Analyzing Option C**:
- Statement: "A system X is neither in thermal equilibrium with Y nor with Z."
- This statement implies that since X is not in equilibrium with either Y or Z, we cannot conclude anything about the relationship between Y and Z. Therefore, this option is incorrect.

5. **Analyzing Option D**:
- Statement: "A system X is neither in thermal equilibrium with Y nor with Z."
- Similar to Option C, this statement also does not provide any information about the relationship between Y and Z. Therefore, this option is also incorrect.

### Final Conclusion:
After analyzing all the options, we find that:
- **Correct Options**: None of the options A, B, C, or D are correct based on the Zeroth Law of Thermodynamics.