AxiomOfExtensionality 2003-06-24 18:28:55 2003-06-24 18:51:52 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 If $X$ and $Y$ have the same elements, then $X = Y$. The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads: \[ \forall u(u \in X \leftrightarrow u \in Y) \rightarrow X = Y. \] Note that the converse, \[ X = Y \rightarrow \forall u(u \in X \leftrightarrow u \in Y) \] is an axiom of the predicate calculus. Hence we have, \[ X = Y \leftrightarrow \forall u(u \in X \leftrightarrow u \in Y). \] Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.