Exercise · ZW-NEGATION-OF-QUANTIFIERS-009 · ★★★ Advanced
Select the correct equivalence.
Choose one option.
Show answer
- A. \(\neg\forall x\,P(x)\equiv\exists x\,\neg P(x)\)
Explanation
Negating a universal gives an existential with the property negated, so \(\neg\forall x\,P(x)\equiv\exists x\,\neg P(x)\). The others swap the quantifier wrongly or fail to negate the property.
Review the concept: Negation of quantifiers
For a predicate \(P(x)\) over \(D\), negating a quantifier swaps the quantifier and negates the predicate.
\[\begin{gathered}\neg\forall x P(x)\equiv\exists x\neg P(x)\\\neg\exists x P(x)\equiv\forall x\neg P(x)\end{gathered}\]
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