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 \( A \) within the universal set \( U \). The complement \( A^{c} \) consists of all the elements in \( U \) that are not in \( A \): - \( U = \{1,2,3,4,5,6,7,8,9\} \) - \( A = \{1,2,3,4\} \) So, \[ A^{c} = U - A = \{5, 6, 7, 8, 9\} \] Next, we find the intersection of \( A^{c} \) and \( B \): - \( B = \{2,4,6\} \) - \( A^{c} = \{5, 6, 7, 8, 9\} \) Now, we find \( A^{c} \cap B \): \[ A^{c} \cap B = \{5, 6, 7, 8, 9\} \cap \{2, 4, 6\} = \{6\} \] Thus, the result is: \[ A^{c} \cap B = \{6\} \]