Let \( U=\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\}, A=\{0,2,4\}, B=\{0,2,4,6\}, C=\{0,1,2,3,4,5,6,7,8,9,10\} \). Use \( A, B \), and \( C \) to find the following. (A'UB) \( \cap C \)

To solve \( (A' \cup B) \cap C \), we first need to determine \( A' \), the complement of \( A \) with respect to \( U \). 1. The universal set \( U \) is \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} \). 2. The set \( A = \{0, 2, 4\} \). Therefore, the complement \( A' \) is: \[ A' = U - A = \{1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}. \] 3. Next, we find the union \( A' \cup B \), where \( B = \{0, 2, 4, 6\} \): \[ A' \cup B = \{1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} \cup \{0, 2, 4, 6\}. \] This results in: \[ A' \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}. \] 4. Now we need to find the intersection of this union with set \( C \): \[ C = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. \] 5. The intersection is: \[ (A' \cup B) \cap C = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} \cap \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. \] This gives: \[ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. \] Thus, the final result is: \[ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. \]