Exercise · ZW-NEGATION-OF-QUANTIFIERS-010 · ★★☆ Intermediate
Confusing \(\neg\forall x\,P\) with \(\forall x\,\neg P\) turns 'there is a counterexample' into...
Choose one option.
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- D. 'all are counterexamples'
Explanation
\(\neg\forall x\,P\) means there is a counterexample, while \(\forall x\,\neg P\) says all are counterexamples; confusing them changes the meaning.
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
- ZW-NEGATION-OF-QUANTIFIERS-001 The negation of ∀ x P(x) is...
- ZW-NEGATION-OF-QUANTIFIERS-002 The negation of ∃ x P(x) is...
- ZW-NEGATION-OF-QUANTIFIERS-003 "Not all students passed" is equivalent to...
- ZW-NEGATION-OF-QUANTIFIERS-004 What operation appears when negating a quantifier?
- ZW-NEGATION-OF-QUANTIFIERS-005 Which equivalence is correct?
- 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...