Prove the generalization of DeMorgan’s laws to any finite collection of sets:

\[\overline{A_1 \cap A_2 \cap \cdots \cap A_n} = \overline{A_1} \cup \overline{A_2} \cup \cdots \cup \overline{A_n}\] \[\overline{A_1 \cup A_2 \cup \cdots \cup A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots \cap \overline{A_n}\]

B.2 tells us that \(\overline{B \cap C} = \overline{B} \cup \overline{C}\). Suppose that set \(\overline{B}\) is composed of two smaller sets, call them \(\overline{A_1}\) and \(\overline{A_2}\). Then the following is true: \(\overline{A_1} \cup \overline{A_2} \cup \overline{C} = \overline{A_1 \cap A_2 \cap C} = \overline{B \cap C}\). Now suppose \(C\) is composed of an arbitrary number of sets, call these \(\overline{A_3}, \dots, \overline{A_n}\). Then, by similar logic we see that \(\overline{A_1} \cap \overline{A_2} \cup \overline{A_3} \cup \cdots \cup \overline{A_n} = \overline{A_1 \cap A_2 \cap A_3 \cap \cdots \cap A_n}\).

B.2 also tells us that \(\overline{B \cup C} = \overline{B} \cap \overline{C}\) which we can use to craft an equivalent argument showing \(\overline{A_1 \cup A_2 \cup \cdots \cup A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots \cap \overline{A_n}\)