World Library  
Flag as Inappropriate
Email this Article

Trichotomy (mathematics)

Article Id: WHEBN0000373956
Reproduction Date:

Title: Trichotomy (mathematics)  
Author: World Heritage Encyclopedia
Language: English
Subject: Axiom of choice, Total order, Cardinality, Interval (mathematics), Partially ordered set
Publisher: World Heritage Encyclopedia

Trichotomy (mathematics)

In mathematics, the Law of Trichotomy states that every real number is either positive, negative, or zero.[1] 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: x, x=y, or x>y.

In mathematical notation, this is

\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 \, \, ) ) \,.

Assuming that the ordering is irreflexive and transitive, this can be simplified to

\forall x \in X \, \forall y \in X \, ( x < y \, \lor \, y < x \, \lor \, x = y ) \,.

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers. The law does not hold in general in intuitionistic logic.

In ZF set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers 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).[2]

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 it is a 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.

A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it is trivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

See also


  1. ^
  2. ^ Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.