1. We need to prove the logical equivalence using truth tables. 2. For the first formula, De Morgan's law states: $$\neg (P \wedge Q) = \neg P \vee \neg Q$$. We construct the truth table columns for $P$, $Q$, $P \wedge Q$, $\neg (P \wedge Q)$, $\neg P$, $\neg Q$, and finally $\neg P \vee \neg Q$ to show both sides have identical truth values. 3. For the second formula: $$\neg (A \vee B \vee C) = \neg A \wedge \neg B \wedge \neg C$$. Similarly, build the truth table with columns for $A$, $B$, $C$, $A \vee B \vee C$, $\neg (A \vee B \vee C)$, $\neg A$, $\neg B$, $\neg C$, and $\neg A \wedge \neg B \wedge \neg C$ to verify equivalence. 4. For the third formula: $$\neg (A \wedge (B \vee C)) \equiv \neg A \vee (\neg B \wedge \neg C)$$, compose the truth table with columns for $A$, $B$, $C$, $B \vee C$, $A \wedge (B \vee C)$, $\neg (A \wedge (B \vee C))$, $\neg A$, $\neg B$, $\neg C$, and $\neg A \vee (\neg B \wedge \neg C)$ to confirm the equivalence. 5. Each truth table demonstrates that the expressions on both sides of the equivalence yield the same truth value for all possible inputs. 6. Therefore, all three logical equivalences are proven using truth tables.