# some theorems on strict betweenness relations

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.

###### Theorem 1.

Three elements are in a strict betweenness relation only if they are pairwise distinct.

###### Theorem 2.

If $B$ is strict, then ${B}_{\mathrm{*}p\mathit{}q}$, ${B}_{p\mathrm{*}q}$ and ${B}_{p\mathit{}q\mathit{}\mathrm{*}}$ are pairwise disjoint. Furthermore, if $p\mathrm{=}q$ then all three sets are empty.

###### Theorem 3.

If $B$ is strict, then ${B}_{p\mathit{}q}\mathrm{\cap}{B}_{q\mathit{}p}\mathrm{=}{B}_{p\mathrm{*}q}$ and ${B}_{p\mathit{}q}\mathrm{\cup}{B}_{q\mathit{}p}\mathrm{=}B\mathit{}\mathrm{(}p\mathrm{,}q\mathrm{)}$.

###### Theorem 4.

If $B$ is strict, then for any $p\mathrm{,}q\mathrm{\in}A$, $p\mathrm{\ne}q$, ${B}_{\mathrm{*}p\mathit{}q}$, ${B}_{p\mathrm{*}q}$ and ${B}_{p\mathit{}q\mathit{}\mathrm{*}}$ are infinite^{}.

Title | some theorems on strict betweenness relations |
---|---|

Canonical name | SomeTheoremsOnStrictBetweennessRelations |

Date of creation | 2013-03-22 17:18:59 |

Last modified on | 2013-03-22 17:18:59 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 6 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 51G05 |

Related topic | StrictBetweennessRelation |