Let \( U \) be the universal set, where: \( U=\{1,2,3,4,5,6,7,8,9\} \) Let sets \( A \) and \( B \) be subsets of \( U \), where: \( A=\{1,3,4,5,8\} \) and \( B=\{2,3,4,7\} \) Find each of the given sets. \( (A \cup B)^{\prime}=\{ \) Enter the elements as a list, separated by commas with no spaces, in order from smallest to largest. If the \( A^{\prime} \cap B^{\prime}=\{ \) result is the empty set, enter DNE You may want to draw a Venn Diagram to help answer this question.

Let's find each of the given sets step by step. ### 1. \( (A \cup B)^{\prime} \) **Step 1:** Find \( A \cup B \). \[ A = \{1, 3, 4, 5, 8\} \] \[ B = \{2, 3, 4, 7\} \] \[ A \cup B = \{1, 2, 3, 4, 5, 7, 8\} \] **Step 2:** Find the complement of \( A \cup B \) with respect to the universal set \( U \). \[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \] \[ (A \cup B)^{\prime} = U \setminus (A \cup B) = \{6, 9\} \] ### 2. \( A^{\prime} \cap B^{\prime} \) **Step 1:** Find \( A^{\prime} \) and \( B^{\prime} \). \[ A^{\prime} = U \setminus A = \{2, 6, 7, 9\} \] \[ B^{\prime} = U \setminus B = \{1, 4, 5, 6, 8, 9\} \] **Step 2:** Find the intersection of \( A^{\prime} \) and \( B^{\prime} \). \[ A^{\prime} \cap B^{\prime} = \{6, 9\} \] ### Final Answers \[ (A \cup B)^{\prime} = \{6,9\} \] \[ A^{\prime} \cap B^{\prime} = \{6,9\} \] **Entered as lists:** - \( (A \cup B)^{\prime} = \{6,9\} \) - \( A^{\prime} \cap B^{\prime} = \{6,9\} \)