LawOfTrichotomy 2004-03-06 07:05:05 2006-11-25 13:36:34 Definition trichotomy trichotomous \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{case} \PMlinkescapeword{equivalent} \PMlinkescapeword{property} \PMlinkescapeword{restricted} The \emph{law of trichotomy} for a binary relation $R$ on a set $S$ is the property that \begin{itemize} \item for all $x,y\in S$, exactly one of the following holds: $xRy$ or $yRx$ or $x=y$. \end{itemize} A binary relation satisfying the law of trichotomy is sometimes said to be \emph{trichotomous}. Trichotomous binary relations are equivalent to tournaments, although the study of tournaments is usually restricted to the finite case. A transitive trichotomous binary relation is called a total order, and is typically written $<$. The law of trichotomy for cardinal numbers is equivalent (in ZF) to the \PMlinkname{axiom of choice}{AxiomOfChoice}.