Solved: Rewrite each of these statements so that negations

QUESTION:

Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

a) \(\neg \forall x \forall y P(x, y)\)

b) \(\neg \forall y \exists x P(x, y)\)

c) \(\neg \forall y \forall x(P(x, y) \vee Q(x, y))\)

d) \(\neg(\exists x \exists y \neg P(x, y) \wedge \forall x \forall y Q(x, y))\)

e) \(\neg \forall x(\exists y \forall z P(x, y, z) \wedge \exists z \forall y P(x, y, z))\)

Equation Transcription:

¬∀x∀yP (x, y)

¬∀y∃xP (x, y)

¬∀y∀x(P (x, y) ∨ Q(x, y))

¬(∃x∃y¬P (x, y) ∧ ∀x∀yQ(x, y))

¬∀x(∃y∀zP (x, y, z) ∧ ∃z∀yP (x, y, z))

Text Transcription:

\neg \forall x \forall y P(x, y)

\neg \forall y \exists x P(x, y)

\neg \forall y \forall x(P(x, y) \vee Q(x, y))

\neg(\exists x \exists y \neg P(x, y) \wedge \forall x \forall y Q(x, y))

\neg \forall x(\exists y \forall z P(x, y, z) \wedge \exists z \forall y P(x, y, z))