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In mathematics, the Law of '''Trichotomy''' states that every real number is either positive, negative, or zero.<ref>http://mathworld.wolfram.com/TrichotomyLaw.html</ref> More generally, '''trichotomy''' is the property of an [[order relation]] < on a set ''X'' that for any ''x'' and ''y'', exactly one of the following holds: <math>x<y</math>, <math>x=y</math>, or <math>x>y</math>.
 
In [[mathematical notation]], this is
:<math>\forall x \in X \, \forall y \in X \, ( ( x < y \, \land \, \lnot (y < x) \, \land \, \lnot( x = y )\, ) \lor \, ( \lnot(x < y) \, \land \, y < x \, \land \, \lnot( x = y) \, ) \lor \, ( \lnot(x < y) \, \land \, \lnot( y < x) \, \land \, x = y \, \, ) ) \,.</math>
 
Assuming that the ordering is [[reflexive relation|irreflexive]] and [[transitive relation|transitive]], this can be simplified to
:<math>\forall x \in X \, \forall y \in X \, ( x < y \, \lor \, y < x \, \lor \, x = y ) \,.</math>
 
In classical logic, this '''axiom of trichotomy''' holds for ordinary comparison between [[real number]]s and therefore also for comparisons between [[integer]]s and between [[rational number]]s.  The law does not hold in general in [[intuitionistic logic]].
 
In [[Zermelo–Fraenkel set theory|ZF set theory]] and [[Von Neumann–Bernays–Gödel set theory|Bernays set theory]], the law of trichotomy holds between the [[cardinal number]]s of well-orderable sets even without the axiom of choice. If the [[axiom of choice]] holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).<ref>{{cite book | author=Bernays, Paul | title=Axiomatic Set Theory | publisher=Dover Publications | year=1991 | isbn=0-486-66637-9}}</ref>
 
More generally, a [[binary relation]] ''R'' on ''X'' is '''trichotomous''' if for all ''x'' and ''y'' in ''X'' exactly one of ''xRy'', ''yRx'' or ''x''=''y'' holds. If such a relation is also [[transitive relation|transitive]] it is a '''[[Total_order#Strict_total_order|strict total order]]'''; this is a special case of a [[strict weak  order]]. For example, in the case of three element set {''a'',''b'',''c''} the relation ''R'' given by ''aRb'', ''aRc'', ''bRc'' is a strict total order, while the relation ''R'' given by the cyclic ''aRb'', ''bRc'', ''cRa'' is a non-transitive trichotomous relation.
 
In the definition of an [[ordered integral domain]] or [[ordered field]], the law of trichotomy is usually taken as more foundational than the law of [[total order]].
 
Trichotomous relations cannot be [[reflexive relation|reflexive]], since ''xRx'' must be false. If transitive, they are trivially antisymmetric and also asymmetric, since ''xRy'' and ''yRx'' cannot both hold.
 
== See also ==
* [[Dichotomy]]
* [[Law of noncontradiction]]
* [[Law of excluded middle]]
 
== References ==
{{reflist}}
 
{{mathlogic-stub}}
 
[[Category:Order theory]]
[[Category:Mathematical relations]]
[[Category:Axiom of choice]]

Latest revision as of 03:33, 10 December 2014

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