Math 2.0

Mathematical logic and foundations Axiom

Axiom of extensionality

Builds on SetElementBiconditionalUniversal quantifier

Statement

Let AA and BB be sets. If AA and BB have the same elements, then A=BA = B: ∀A ∀B (∀x (x∈A↔x∈B)→A=B).\forall A \, \forall B \, \bigl( \forall x \, (x \in A \leftrightarrow x \in B) \to A = B \bigr).

Remarks

It is one of the axioms of Zermelo–Fraenkel set theory. The converse is a rule of logic: equal sets have the same elements. In practice, A=BA = B is often proved with A⊆BA \subseteq B and B⊆AB \subseteq A. See subset.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…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. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Implication definition
  6. Predicate definition
  7. Biconditional definition
  8. Universal quantifier definition
  9. Axiom of extensionality statement

Where this leads

31 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_of_extensionality, lead section
About this page
Tag0TB16T2C97
URIm2://0TB16T2C97
Aliasesextensionality
Kindstatement, axiom
WritingW1 stated
MSC03E30
StatusThis copy of the site has no build data. Run m2 status 0TB16T2C97.
UsesSet, Element, Biconditional, Universal quantifier
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