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

Let's evaluate each set to determine whether it is **closed under addition**. A set is **closed under addition** if the sum of any two elements in the set also belongs to the set. ### a) \( P = \{0\} \) - **Evaluation:** The only element in \( P \) is \( 0 \). Adding \( 0 \) to itself: \[ 0 + 0 = 0 \in P \] - **Conclusion:** **True.** The set \( P \) is closed under addition. --- ### b) \( S = \{0, 4, 8, 12, 16, \ldots\} \) - **Evaluation:** This set consists of all non-negative multiples of \( 4 \). Adding any two elements: \[ 4k + 4m = 4(k + m) \] where \( k, m \) are non-negative integers. The sum \( 4(k + m) \) is also a multiple of \( 4 \) and hence belongs to \( S \). - **Conclusion:** **True.** The set \( S \) is closed under addition. --- ### c) \( N = \{4, 7, 9\} \) - **Evaluation:** Let's check all possible sums: \[ 4 + 4 = 8 \quad (\text{Not in } N) \\ 4 + 7 = 11 \quad (\text{Not in } N) \\ 4 + 9 = 13 \quad (\text{Not in } N) \\ 7 + 7 = 14 \quad (\text{Not in } N) \\ 7 + 9 = 16 \quad (\text{Not in } N) \\ 9 + 9 = 18 \quad (\text{Not in } N) \] - **Conclusion:** **False.** The set \( N \) is **not** closed under addition, as the sums do not all belong to \( N \). --- ### d) \( \{x \mid x \text{ is an integer},\ x > 18\} \) - **Evaluation:** Let \( x \) and \( y \) be any two integers greater than \( 18 \). Then: \[ x + y > 18 + 18 = 36 \] Since \( 36 > 18 \), \( x + y \) is also an integer greater than \( 18 \). - **Conclusion:** **True.** The set is closed under addition. --- ### Summary - **a) True** - **b) True** - **c) False** - **d) True**