Given the following: \( { }_{19}^{41} \mathrm{X},{ }_{19}^{39} \mathrm{X},{ }_{20}^{41} \mathrm{X} \), and \( { }_{18}^{39} \mathrm{X} \). Which are isotopes of each other? \( \begin{array}{l}{ }_{19}^{41} \mathrm{X},{ }_{19}^{39} \mathrm{X},{ }_{20}^{41} \mathrm{X} \text {, and }{ }_{18}^{39} \mathrm{X} \text { are isotopes of each other. } \\ { }_{19}^{41} \mathrm{X} \text { and }{ }_{20}^{41} \mathrm{X} \text { are isotopes of each other, and }{ }_{19}^{39} \mathrm{X} \text { and }{ }_{18}^{39} \mathrm{X} \text { are isotopes of each other. } \\ { }_{19}^{41} \mathrm{X} \text { and }{ }_{19}^{39} \mathrm{X} \text { are isotopes of each other. } \\ \text { None are isotopes of each other. }\end{array} \).

Isotopes are nuclides of the same element, meaning they have the same number of protons (atomic number \( Z \)) but different mass numbers (\( A \)). Let's analyze each of the given nuclides: 1. **\( { }_{19}^{41}\mathrm{X} \) and \( { }_{19}^{39}\mathrm{X} \)**: - Both have \( Z = 19 \) (same element). - Different mass numbers: \( A = 41 \) and \( A = 39 \). - **These are isotopes of each other**. 2. **\( { }_{20}^{41}\mathrm{X} \) and \( { }_{18}^{39}\mathrm{X} \)**: - \( { }_{20}^{41}\mathrm{X} \) has \( Z = 20 \). - \( { }_{18}^{39}\mathrm{X} \) has \( Z = 18 \). - Different elements (different \( Z \)) and mass numbers. - **These are not isotopes**. 3. **Comparing across different \( Z \)**: - \( { }_{19}^{41}\mathrm{X} \) and \( { }_{20}^{41}\mathrm{X} \): Same mass number but different \( Z \) (different elements). - \( { }_{19}^{39}\mathrm{X} \) and \( { }_{18}^{39}\mathrm{X} \): Same mass number but different \( Z \) (different elements). - **None of these pairs are isotopes**. **Conclusion**: Only \( { }_{19}^{41}\mathrm{X} \) and \( { }_{19}^{39}\mathrm{X} \) are isotopes of each other because they share the same atomic number but have different mass numbers. **Answer**: Only \( { }_{19}^{41} \mathrm{X} \) and \( { }_{19}^{39} \mathrm{X} \) are isotopes of one another.