TFAE 2002-06-05 11:48:30 2002-06-05 11:48:30 Definition % 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 \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\norm}[1]{\left\|#1\right\|} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} The abbreviation ``TFAE'' is shorthand for ``\textbf{t}he \textbf{f}ollowing \textbf{a}re \textbf{e}quivalent''. It is used before a set of equivalent conditions (each implies all the others). In a definition, when one of the conditions is somehow ``better'' (simpler, shorter, ...), it makes sense to phrase the definition with that condition, and mention that the others are equivalent. ``TFAE'' is typically used when none of the conditions can take priority over the others. Actually proving the claimed equivalence must, of course, be done separately.