some theorems on strict betweenness relationsSomeTheoremsOnStrictBetweennessRelations2007-06-24 17:36:102008-05-01 11:25:55Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}Let $B$ be a strict betweenness relation. In the following the sets $B_{*pq}, B_{p*q}, B_{pq*}, B_{pq}, B(p,q)$
are defined in the entry about some theorems on the axioms of order.
\begin{thm}
Three elements are
in a strict betweenness relation only if they are pairwise distinct.
\end{thm}
\begin{thm}
If $B$ is strict, then $B_{*pq}$, $B_{p*q}$ and $B_{pq*}$ are pairwise disjoint.
Furthermore, if $p=q$ then all three sets are empty.
\end{thm}
\begin{thm}
If $B$ is strict, then $B_{pq}\cap B_{qp}=B_{p*q}$ and $B_{pq}\cup B_{qp}=B(p,q)$.
\end{thm}
\begin{thm}
If $B$ is strict, then for any $p,q\in A$, $p\ne q$, $B_{*pq}$, $B_{p*q}$ and $B_{pq*}$ are infinite.
\end{thm}