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law of trichotomy
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(Definition)
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The law of trichotomy for a binary relation $R$ on a set $S$ is the property that
- for all $x,y\in S$ , exactly one of the following holds: $xRy$ or $yRx$ or $x=y$ .
A binary relation satisfying the law of trichotomy is sometimes said to be 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 axiom of choice.
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"law of trichotomy" is owned by yark.
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trichotomy, trichotomous |
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Cross-references: ZF, cardinal numbers, total order, transitive, finite, tournaments, binary relation
There are 14 references to this entry.
This is version 6 of law of trichotomy, born on 2004-03-06, modified 2006-11-25.
Object id is 5668, canonical name is LawOfTrichotomy.
Accessed 9159 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) | | | 06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order) |
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Pending Errata and Addenda
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