# 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: $xRy$ 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 (http://planetmath.org/AxiomOfChoice).

Title law of trichotomy LawOfTrichotomy 2013-03-22 14:13:46 2013-03-22 14:13:46 yark (2760) yark (2760) 9 yark (2760) Definition msc 06A05 msc 03E20 trichotomy trichotomous