Exercise · ZW-NEGATION-OF-QUANTIFIERS-005 · ★☆☆ Basic
Which equivalence is correct?
Choose one option.
Show answer
- A. \(\neg\exists x\,P(x)\equiv\forall x\,\neg P(x)\)
Explanation
Negating 'there exists \(x\) with \(P(x)\)' gives 'for all \(x\), not \(P(x)\)': \(\neg\exists x\,P(x)\equiv\forall x\,\neg P(x)\).
Review the concept: Negation of quantifiers
For a predicate \(P(x)\) over \(D\), negating a quantifier swaps the quantifier and negates the predicate.
\[\begin{gathered}\neg\forall x P(x)\equiv\exists x\neg P(x)\\\neg\exists x P(x)\equiv\forall x\neg P(x)\end{gathered}\]
More practice problems: negation of quantifiers
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- ZW-NEGATION-OF-QUANTIFIERS-004 What operation appears when negating a quantifier?
- ZW-NEGATION-OF-QUANTIFIERS-006 The negation of ∀ x P(x) is equivalent to...
- ZW-NEGATION-OF-QUANTIFIERS-007 The negation of ∃ x P(x) is equivalent to...
- ZW-NEGATION-OF-QUANTIFIERS-008 To negate 'every number is positive' it is enough to state...
- ZW-NEGATION-OF-QUANTIFIERS-009 Select the correct equivalence.
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