AxiomOfUnion 2003-06-25 07:12:04 2003-06-26 05:08:42 Axiom % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \emph{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 \[ \{ a_1, \ldots, a_n \} = \{ a_1 \} \cup \cdots \cup \{ a_n \} \]