Given \( U=\{1,2,3,4,5,6,7,8,9\}, A=\{1,2,3,4\}, B=\{2,4,6\}, C=\{1,3,6,9\} \) Find \( A^{c} \cap B \)

To find \( A^{c} \cap B \), we first need to determine the complement of set \( A \) with respect to the universal set \( U \). The universal set \( U \) is \( \{1,2,3,4,5,6,7,8,9\} \) and the set \( A \) is \( \{1,2,3,4\} \). Therefore, the complement of \( A \) (denoted as \( A^{c} \)) consists of all elements in \( U \) that are not in \( A \): \[ A^{c} = U - A = \{5, 6, 7, 8, 9\} \] Next, we have set \( B = \{2, 4, 6\} \). Now we need to find the intersection of \( A^{c} \) and \( B \): \[ A^{c} \cap B = \{5, 6, 7, 8, 9\} \cap \{2, 4, 6\} \] The only element that is common in both sets \( A^{c} \) and \( B \) is \( 6 \). Therefore, \[ A^{c} \cap B = \{6\} \]