Determine if each of the following sets is closed under addition. \( \begin{array}{ll}\text { a) } L=\{0\} & \text { c) } N=\{6,9,11\} \\ \text { b) } S=\{0,6,12,18,24 \ldots\} & \text { d) }\{x \mid x \in \text { integers, } x>10\}\end{array} \) \( \begin{array}{ll}\text { a) } L=\{0\} \text { is closed under addition. } \\ \text { True } \\ \text { False }\end{array} \)

L is closed under addition because the only element in the set is 0, and \(0 + 0 = 0\). Therefore, the result remains within the set. So, the answer is **True**. Now, let's examine the other sets: For \(S=\{0,6,12,18,24, \ldots\}\): This set consists of all multiples of 6. If you add any two multiples of 6, say \(6m\) and \(6n\), their sum \(6m + 6n = 6(m+n)\) is also a multiple of 6. Therefore, the set \(S\) is closed under addition. Next, consider \(N=\{6, 9, 11\}\): If we take \(6 + 9 = 15\) (which is not in the set), we find that the set is not closed under addition. For the set of integers greater than 10: Take \(x = 11\) and \(y = 12\); \(x + y = 23\) which is still greater than 10. In fact, adding any two integers more than 10 will result in an integer greater than 10. Thus, this set is closed under addition. In summary: - \(L\) is closed under addition: **True**. - \(S\) is closed under addition: **True**. - \(N\) is NOT closed under addition: **False**. - The integers greater than 10: **True**.