Given the following sets, find the set $(A \cup B)^{\prime} \cap C$.
Points: 0 of 1
\[
\begin{array}{l}
U=\{1,2,3, \ldots, 6\} \\
A=\{2,3,5,6\} \\
B=\{1,3,5\} \\
C=\{1,2,3,4,5\}
\end{array}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $(A \cup B)^{\prime} \cap \mathrm{C}=$
(Use a comma to separate answers as needed. Use ascending order.)
B. $(A \cup B)^{\prime} \cap C$ is the empty set.

Step 1 ：Given the universal set \(U = \{1, 2, 3, 4, 5, 6\}\), and the sets \(A = \{2, 3, 5, 6\}\), \(B = \{1, 3, 5\}\), and \(C = \{1, 2, 3, 4, 5\}\).

Step 2 ：First, we find the union of sets A and B, denoted as \(A \cup B\). This is the set of elements which are in A, or B, or in both. So, \(A \cup B = \{1, 2, 3, 5, 6\}\).

Step 3 ：Next, we find the complement of the union of A and B, denoted as \((A \cup B)^{\prime}\). This is the set of elements in the universal set U that are not in \(A \cup B\). So, \((A \cup B)^{\prime} = \{4\}\).

Step 4 ：Finally, we find the intersection of \((A \cup B)^{\prime}\) and C, denoted as \((A \cup B)^{\prime} \cap C\). This is the set of elements which are in both \((A \cup B)^{\prime}\) and C. So, \((A \cup B)^{\prime} \cap C = \{4\}\).

Step 5 ：Final Answer: \((A \cup B)^{\prime} \cap C = \boxed{4}\)