EXERCISE 1.8.4: Using De Morgan's law for quantified statements to prove logical equivalence.Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences:(a) x (P(X)^-Q(x)) = 3x (-P(x) v Q(x))(b)x (P(x)→ Q(x)) = 3x (-P(x) ^ -Q(x))(c) -3x (-P(x) v (Q(x) A-R(x))) = x (P(x)^(-Q(x) v R(x)))

1.8.4: (a). To prove that, ¬∀x(P(x)∧¬Q(x))≡∃¬P(x)∨Q(x).

Answered: Use De Morgan's law for quantified…

Advanced Engineering Mathematics

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Chapter2: Second-order Linear Odes

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Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences: (a) x (P(x)^-Q(x)) = 3x (-P(x) v Q(x)) (b) x (P(x)→→ Q(x)) = 3x (-P(x) ^ ¬Q(x)) (c) -3x (-P(x) v (Q(x) ^-R(x))) = x (P(x) ^ (-Q(x) v R(x)))