Step 1 :Given the universal set U = \{1, 2, 3, 4, 5, 6, 7\}, and the sets A = \{1, 3, 6, 7\}, B = \{2, 6, 7\}, and C = \{2, 3, 4, 5, 7\}.
Step 2 :The question is asking for the complement of the union of sets A, B, and C. The union of sets A, B, and C would include all elements that are in any of these sets. The complement of this union would then be all elements in the universal set U that are not in the union of A, B, and C.
Step 3 :To solve this, we first find the union of sets A, B, and C. The union of A, B, and C is \{1, 2, 3, 4, 5, 6, 7\}.
Step 4 :Then, we find the complement of this union by finding all elements in U that are not in the union of A, B, and C. The complement of the union of sets A, B, and C is an empty set.
Step 5 :This means that all elements in the universal set U are included in the union of sets A, B, and C. Therefore, there are no elements in U that are not in the union of A, B, and C.
Step 6 :Final Answer: \(\boxed{(A \cup B \cup C)^{\prime}\text{ is the empty set}}\)