Use DeMorgan's law for quantified statements and the laws of propositional logic to show the following equivalences:(a) ¬∀x (P(x ∧ ¬Q(x)) ≡ ∃x (¬P(x) ∨ Q(x)) (b) ¬∀x (¬P(x → Q(x)) ≡ ∃x (¬P(x) ∧ ¬Q(x)) (c) ¬∃x (¬P(x) ∨ (Q(x) ∧ ¬R(x))) ≡ ∀x (P(x) ∧ (¬Q(x) ∨ R(x)))

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Use DeMorgan's law for quantified statements and the laws of propositional logic to show the following equivalences:
(a) ¬∀x (P(x ∧ ¬Q(x)) ≡ ∃x (¬P(x) ∨ Q(x))

(b) ¬∀x (¬P(x → Q(x)) ≡ ∃x (¬P(x) ∧ ¬Q(x))

(c) ¬∃x (¬P(x) ∨ (Q(x) ∧ ¬R(x))) ≡ ∀x (P(x) ∧ (¬Q(x) ∨ R(x)))

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