law of trichotomy

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: $x R y$ or $y R x$ 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 (http://planetmath.org/AxiomOfChoice).

单词	LawOfTrichotomy
释义	law of trichotomy 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: $x R y$ or $y R x$ 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 (http://planetmath.org/AxiomOfChoice).
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