Recall De Morgan's laws in propositional logic: ¬(P ∧ Q) ≡ (¬P) ∨ (¬Q) and ¬(P ∨ Q) ≡ (¬P) ∧ (¬Q). Start by writing truth tables for both sides of each law to verify their equivalence.

Extend the truth table approach to predicate logic. Consider how quantifiers (∀, ∃) interact with negation. For example, analyze ¬(∀x P(x)) and ¬(∃x P(x)) to derive their equivalent forms.

Prove De Morgan's laws for quantifiers formally using the definitions of quantifiers and logical equivalence. Show that ¬(∀x P(x)) is equivalent to ∃x ¬P(x) and ¬(∃x P(x)) is equivalent to ∀x ¬P(x) by leveraging the properties of negation and quantifiers.