axiom of extensionality
If X and Y have the same elements, then X=Y.
The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:
∀u(u∈X↔u∈Y)→X=Y. |
Note that the converse,
X=Y→∀u(u∈X↔u∈Y) |
is an axiom of the predicate calculus. Hence we have,
X=Y↔∀u(u∈X↔u∈Y). |
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 |