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Question

# Generalized De Morgan's laws for logic: $\neg (\forall xP(x))$ and $\exists x\neg P(x)$ have the same truth values. $\neg (\exists xP(x))$ and $\forall x\neg P(x)$ have the same truth values.

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$\textbf{De Morgan's Laws for Qualifiers}$:

$\neg \exists x P(x)\equiv \forall x \neg P(x)$

$\neg \forall x P(x)\equiv \exists x\neg P(x)$

$\textbf{Universal quantified statement}$ $\forall x P(x)$ is true if and only if $P(x)$ is true for all values of $x$ in the domain of discourse.

$\textbf{Existential quantified statement}$ $\exists x P(x)$ is true if and only if there exists at least one element $x$ in the domain for which $P(x)$ is true.

Thus the negation of a universal quantified statement of a proposition is the existential quantified statement of the negated proposition and the negation of a existential quantified statement of a proposition is the universal quantified statement of the negated proposition.

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