# axiom of union

For any $X$ there exists a set $Y=\bigcup X$.

The Axiom of Union is an axiom of Zermelo-Fraenkel set theory^{}. In symbols, it reads

$$\forall X\exists Y\forall u(u\in Y\leftrightarrow \exists z(z\in X\wedge u\in z)).$$ |

Notice that this means that $Y$ is the set of elements of all elements of $X$. More succinctly, the union of any set of sets is a set. By Extensionality, the set $Y$ is unique. $Y$ is called the *union* of $X$.

In particular, the Axiom of Union, along with the Axiom of Pairing allows us to define

$$X\cup Y=\bigcup \{X,Y\},$$ |

as well as the triple

$$\{a,b,c\}=\{a,b\}\cup \{c\}$$ |

and therefore the $n$-tuple

$$\{{a}_{1},\mathrm{\dots},{a}_{n}\}=\{{a}_{1}\}\cup \mathrm{\cdots}\cup \{{a}_{n}\}$$ |

Title | axiom of union |
---|---|

Canonical name | AxiomOfUnion |

Date of creation | 2013-03-22 13:42:49 |

Last modified on | 2013-03-22 13:42:49 |

Owner | Sabean (2546) |

Last modified by | Sabean (2546) |

Numerical id | 8 |

Author | Sabean (2546) |

Entry type | Axiom |

Classification | msc 03E30 |

Synonym | union |