Mark the correct options:

Options

A. A system X is in the thermal equilibrium with Y but not with Z. Systems Y and Z may be in thermal equilibrium with each other.
B. A system X is in thermal equilibrium with Y but not with Z. Systems 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 systems 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 systems Y and Z may be in thermal equilibrium with each other.

Answer

To solve the question regarding thermal equilibrium, we need to analyze the given options step by step.

### Step-by-Step Solution:

1. **Understanding Thermal Equilibrium**:
- Two systems are in thermal equilibrium if they are at the same temperature. If system X is in thermal equilibrium with system Y, it means they have the same temperature (let's denote it as T).

2. **Analyzing the First Option**:
- The first option states that "A system X is in thermal equilibrium with Y but not with Z."
- Since X and Y are in thermal equilibrium, they share the same temperature (T). However, since X is not in thermal equilibrium with Z, it implies that Z must have a different temperature (let's denote it as T').
- Therefore, Y and Z cannot be in thermal equilibrium because Y has the temperature T and Z has the temperature T'.
- **Conclusion**: This option is correct.

3. **Analyzing the Second Option**:
- The second option states that "A system X is neither in thermal equilibrium with Y nor Z."
- If X is not in thermal equilibrium with Y, it means they have different temperatures. However, if X is not in thermal equilibrium with Y, it does not necessarily imply that Y and Z cannot be in thermal equilibrium.
- For example, if X has a temperature of 5°C, Y could have 10°C, and Z could have 10°C as well, which means Y and Z are in thermal equilibrium.
- **Conclusion**: This option is incorrect because Y and Z can still be in thermal equilibrium.

4. **Analyzing the Third Option**:
- The third option states that "The systems Y and Z must be in thermal equilibrium."
- This statement is not necessarily true. While Y and Z can be in thermal equilibrium, there is no information provided that confirms they must be. They could have different temperatures as well.
- **Conclusion**: This option is incorrect.

5. **Analyzing the Fourth Option**:
- The fourth option states that "System X is neither in thermal equilibrium with Y nor with Z."
- This option is similar to the second option. If X is not in thermal equilibrium with Y, it does not imply that Y and Z cannot be in thermal equilibrium.
- **Conclusion**: This option is also incorrect.

### Final Summary of Correct Options:
- The only correct option is the first one: "A system X is in thermal equilibrium with Y but not with Z."