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# 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

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

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

$\{a_{1},\ldots,a_{n}\}=\{a_{1}\}\cup\cdots\cup\{a_{n}\}$ |

Synonym:

union

Type of Math Object:

Axiom

Major Section:

Reference

## Mathematics Subject Classification

03E30*no label found*

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