# 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

 $\{a_{1},\ldots,a_{n}\}=\{a_{1}\}\cup\cdots\cup\{a_{n}\}$
Title axiom of union AxiomOfUnion 2013-03-22 13:42:49 2013-03-22 13:42:49 Sabean (2546) Sabean (2546) 8 Sabean (2546) Axiom msc 03E30 union