law of trichotomy

The law of trichotomy for a binary relationMathworldPlanetmath R on a set S is the property that

  • for all x,yS, 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to tournamentsMathworldPlanetmath, although the study of tournaments is usually restricted to the finite case.

A transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath trichotomous binary relation is called a total orderMathworldPlanetmath, and is typically written <.

The law of trichotomy for cardinal numbersMathworldPlanetmath is equivalent (in ZF) to the axiom of choiceMathworldPlanetmath (

Title law of trichotomy
Canonical name LawOfTrichotomy
Date of creation 2013-03-22 14:13:46
Last modified on 2013-03-22 14:13:46
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 06A05
Classification msc 03E20
Defines trichotomy
Defines trichotomous