axiom of extensionality

If X and Y have the same elements, then X=Y.

The Axiom of ExtensionalityMathworldPlanetmath is one of the axioms of Zermelo-Fraenkel set theoryMathworldPlanetmath. In symbols, it reads:


Note that the converseMathworldPlanetmath,


is an axiom of the predicate calculus. Hence we have,


Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.

Title axiom of extensionality
Canonical name AxiomOfExtensionality
Date of creation 2013-03-22 13:42:40
Last modified on 2013-03-22 13:42:40
Owner Sabean (2546)
Last modified by Sabean (2546)
Numerical id 5
Author Sabean (2546)
Entry type Axiom
Classification msc 03E30
Synonym extensionality