Given the following sets, find the set \( (A \cup B) \cap(A \cup C) \). \( \begin{array}{l}U=\{1,2,3, \ldots, 10\} \\ A=\{1,2,3,5\} \\ B=\{4,7,10\} \\ C=\{2,5,6,8,9\}\end{array} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (A B B) \( \cap(A \cup C)=\{ \) (Use a comma to separate answers as needed. Use ascending order.) B. (A B \( \cap(A \cup C) \) is the empty set.

To find the set \( (A \cup B) \cap (A \cup C) \), we first need to determine \( A \cup B \) and \( A \cup C \). 1. **Calculate \( A \cup B \)**: \[ A = \{1, 2, 3, 5\} \] \[ B = \{4, 7, 10\} \] \[ A \cup B = \{1, 2, 3, 5\} \cup \{4, 7, 10\} = \{1, 2, 3, 4, 5, 7, 10\} \] 2. **Calculate \( A \cup C \)**: \[ C = \{2, 5, 6, 8, 9\} \] \[ A \cup C = \{1, 2, 3, 5\} \cup \{2, 5, 6, 8, 9\} = \{1, 2, 3, 5, 6, 8, 9\} \] 3. **Now calculate \( (A \cup B) \cap (A \cup C) \)**: \[ A \cup B = \{1, 2, 3, 4, 5, 7, 10\} \] \[ A \cup C = \{1, 2, 3, 5, 6, 8, 9\} \] The intersection \( (A \cup B) \cap (A \cup C) \) includes elements that are in both sets: \[ (A \cup B) \cap (A \cup C) = \{1, 2, 3, 5\} \] Thus, the final answer is: \[ \{1, 2, 3, 5\} \] So the correct choice is: A. \( (A \cup B) \cap (A \cup C) = \{1, 2, 3, 5\} \)