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'law of trichotomy'
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| Title of object: |
law of trichotomy |
| Canonical Name: |
LawOfTrichotomy |
| Type: |
Definition |
| Created on: |
2004-03-06 07:05:05 |
| Modified on: |
2004-03-06 13:14:02 |
| Classification: |
msc:03E20, msc:06A05 |
| Defines: |
trichotomy, trichotomous |
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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}.
Note that 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 to the axiom of choice. |
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