Find the set $(A \cap C)^{\prime}$.
\[
\begin{array}{l}
U=\{1,2,3,4,5,6,7,8,9\} \\
A=\{3,4,7,9\} \\
C=\{1,3,4,7,9\}
\end{array}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $(A \cap C)^{\prime}=\{\}$ (Use a comma to separate answers as needed.)
B. $(A \cap C)^{\prime}$ is the empty set.

Step 1 ：Given the universal set U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, set A = \{3, 4, 7, 9\}, and set C = \{1, 3, 4, 7, 9\}.

Step 2 ：We are asked to find the set \((A \cap C)^{\prime}\). This represents the complement of the intersection of sets A and C.

Step 3 ：The intersection of A and C is the set of elements that are common to both A and C. So, we first find the intersection of A and C.

Step 4 ：The intersection of A and C is \{3, 4, 7, 9\}.

Step 5 ：The complement of a set is the set of all elements in the universal set that are not in the given set. So, we find the complement of the intersection in the universal set U.

Step 6 ：The complement of the intersection in the universal set U is \{1, 2, 5, 6, 8\}.

Step 7 ：So, \((A \cap C)^{\prime}\) = \{1, 2, 5, 6, 8\}.

Step 8 ：Final Answer: \(\boxed{\{1, 2, 5, 6, 8\}}\)