If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$,$A = \{2, 4, 6, 8\}$,and $B = \{2, 3, 5, 7\}$,verify that $(A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}$.

Given $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$,$A = \{2, 4, 6, 8\}$,and $B = \{2, 3, 5, 7\}$.
First,find $A \cap B = \{2\}$.
Then,$(A \cap B)^{\prime} = U \setminus \{2\} = \{1, 3, 4, 5, 6, 7, 8, 9\}$.
Next,find $A^{\prime} = U \setminus A = \{1, 3, 5, 7, 9\}$.
Find $B^{\prime} = U \setminus B = \{1, 4, 6, 8, 9\}$.
Then,$A^{\prime} \cup B^{\prime} = \{1, 3, 5, 7, 9\} \cup \{1, 4, 6, 8, 9\} = \{1, 3, 4, 5, 6, 7, 8, 9\}$.
Since $(A \cap B)^{\prime} = \{1, 3, 4, 5, 6, 7, 8, 9\}$ and $A^{\prime} \cup B^{\prime} = \{1, 3, 4, 5, 6, 7, 8, 9\}$,it is verified that $(A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}$.