# 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:

$$\forall u(u\in X\leftrightarrow u\in Y)\to X=Y.$$ |

Note that the converse^{},

$$X=Y\to \forall u(u\in X\leftrightarrow u\in Y)$$ |

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

$$X=Y\leftrightarrow \forall u(u\in X\leftrightarrow u\in Y).$$ |

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 |