Determine if each of the following sets is closed under addition. \( \begin{array}{ll}\text { a) } P=\{0\} & \text { c) } \mathrm{N}=\{4,7,9\} \\ \text { b) } \mathrm{S}=\{0,4,8,12,16 \ldots\} & \text { d) }\{x \mid x \in \text { integers, } x>18\}\end{array} \) \( \begin{array}{ll}\text { a) } \mathrm{P}=\{0\} \text { is closed under addition. } \\ \text { False } \\ \text { True }\end{array} \)

The set \( P = \{0\} \) is indeed closed under addition because adding 0 to itself results in 0, which is still in the set. As for \( S = \{0, 4, 8, 12, 16, \ldots\} \), it is also closed under addition since the sum of any two multiples of 4 is another multiple of 4, which remains in the set. For \( N = \{4, 7, 9\} \), it is not closed under addition. For instance, adding \( 4 + 7 = 11 \), which is not in the set. Finally, the set \( \{x \mid x \in \text{integers}, x > 18\} \) is closed under addition since adding any two integers greater than 18 will always result in another integer greater than 18.