Math 2.0

Mathematical logic and foundations Theorem

Negation of an existential statement

Builds on Universal quantifierExistential quantifierNegation

Statement

Let XX be a set and let P(x)P(x) be a predicate on XX. The negation of ∃x∈X, P(x)\exists x \in X, \, P(x) is logically equivalent to ∀x∈X, ¬P(x)\forall x \in X, \, \neg P(x). ¬(∃x∈X, P(x))≡∀x∈X, ¬P(x)\neg \bigl( \exists x \in X, \, P(x) \bigr) \equiv \forall x \in X, \, \neg P(x)

Remarks

This equivalence is valid in intuitionistic logic as well as in classical logic. Together with the negation of a universal statement, it gives a rule. To negate a quantified statement, swap each ∀\forall and ∃\exists and negate the predicate.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Proposition, concept, no evidence levelPropositionconceptNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionNegation of an existential statement, statement, no evidence levelNegation of an existe…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Negation definition
  3. Predicate definition
  4. Existential quantifier definition
  5. Universal quantifier definition
  6. Negation of an existential statement statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantifier_(logic), lead section
About this page
Tag05SZQKH75K
URIm2://05SZQKH75K
Kindstatement, theorem
WritingW0 stub
MSC03B10
StatusThis copy of the site has no build data. Run m2 status 05SZQKH75K.
UsesUniversal quantifier, Existential quantifier, Negation

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