example of law of trichotomy on <
ExampleOfLawOfTrichotomyOn
2007-06-13 15:45:06
2007-06-13 17:49:39
Example
law of trichotomy
\usepackage{amssymb}
%\usepackage{amsmath}
%\usepackage{amsfonts}
%\usepackage{amsthm}
%Named sets
\newcommand{\R}{\mathbb{R}} %Real numbers
%\newcommand{\C}{\mathbb{C}} %Complex numbers
%Functions
%\newcommand{\modulus}[1]{\left|{#1}\right|} %|z|
%Numbers
%\newcommand{\I}{\mathrm{i}} %sqrt{-1}
%\newcommand{\e}{\mathrm{e}} $exponential
%Greek
%\newcommand{\ve}{\varepsilon} %nice epsilon
From the axiomatic definition of the real numbers, ``$<$'' is a relation on
$\R$ which satisfies the law of trichotomy. That is, for all $a,b\in\R$,
exactly one of the following is true:
\begin{itemize}
\item $a<b$
\item $b<a$
\item $a=b$
\end{itemize}
As an axiom, this is sometimes expressed with $b=0$. That is, for all $a\in\R$, $a$ is either positive, negative, or zero.