AxiomOfPowerSet 2003-06-26 05:20:19 2006-12-12 11:12:04 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 The \emph{axiom of power set} is an axiom of Zermelo-Fraenkel set theory which postulates that for any set $X$ there exists a set $\mathcal{P}(X)$, called the \emph{power set} of $X$, consisting of all subsets of $X$. In symbols, it reads: \[ \forall X \exists \mathcal{P}(X) \forall u (u \in \mathcal{P}(X) \leftrightarrow u \subseteq X). \] In the above, $u \subseteq X$ is defined as $\forall z(z \in u \rightarrow z \in X)$. By the extensionality axiom, the set $\mathcal{P}(X)$ is unique. The Power Set Axiom allows us to define the Cartesian product of two sets $X$ and $Y$: \[ X \times Y = \{ (x, y) : x \in X \land y \in Y \}. \] The Cartesian product is a set since \[ X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). \] We may define the Cartesian product of any finite collection of sets recursively: \[ X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. \]