Math 2.0

Mathematical logic and foundations Concept

Order of quantifiers

Builds on Universal quantifierExistential quantifier

Summary

In a statement with several quantifiers, the order of different quantifiers changes the meaning. For a predicate P(x,y)P(x, y), the statement ∃y ∀x P(x,y)\exists y \, \forall x \, P(x, y) implies ∀x ∃y P(x,y)\forall x \, \exists y \, P(x, y), but the converse is false in general. Two quantifiers of the same type commute: ∀x ∀y P(x,y)\forall x \, \forall y \, P(x, y) is equivalent to ∀y ∀x P(x,y)\forall y \, \forall x \, P(x, y), and the same holds for ∃\exists.

Remarks

On the integers, ∀x ∃y (y>x)\forall x \, \exists y \, (y > x) is true, but ∃y ∀x (y>x)\exists y \, \forall x \, (y > x) is false. In ∀x ∃y\forall x \, \exists y, the choice of yy can depend on xx. The definitions of continuity and of uniform continuity differ only in the order of quantifiers.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionOrder of quantifiers, concept, no evidence levelOrder of quantifiersconcept
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. Predicate definition
  3. Existential quantifier definition
  4. Universal quantifier definition
  5. Order of quantifiers concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantifier_(logic), lead section
About this page
Tag0FYQMKFE77
URIm2://0FYQMKFE77
Aliasesquantifier-order
Kindconcept
WritingW1 stated
MSC03B10
UsesUniversal quantifier, Existential quantifier
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