Editing Transitive relation (section)

== Related properties ==
[[File:Rock-paper-scissors.svg|alt=Cycle diagram|thumb|The [[Rock–paper–scissors]] game is based on an intransitive and antitransitive relation "''x'' beats ''y''".]]
A relation ''R'' is called ''[[intransitivity|intransitive]]'' if it is not transitive, that is, if ''xRy'' and ''yRz'', but not ''xRz'', for some ''x'', ''y'', ''z''.
In contrast, a relation ''R'' is called ''[[antitransitive]]'' if ''xRy'' and ''yRz'' always implies that ''xRz'' does not hold.
For example, the relation defined by ''xRy'' if ''xy'' is an [[even number]] is intransitive,<ref>since e.g. 3''R''4 and 4''R''5, but not 3''R''5</ref> but not antitransitive.<ref>since e.g. 2''R''3 and 3''R''4 and 2''R''4</ref> The relation defined by ''xRy'' if ''x'' is even and ''y'' is [[odd number|odd]] is both transitive and antitransitive.<ref>since ''xRy'' and ''yRz'' can never happen</ref> 
The relation defined by ''xRy'' if ''x'' is the [[successor function|successor]] number of ''y'' is both intransitive<ref>since e.g. 3''R''2 and 2''R''1, but not 3''R''1</ref> and antitransitive.<ref>since, more generally, ''xRy'' and ''yRz'' implies ''x''=''y''+1=''z''+2≠''z''+1, i.e. not ''xRz'', for all ''x'', ''y'', ''z''</ref> Unexpected examples of intransitivity arise in situations such as political questions or group preferences.<ref>{{Cite news|url=https://www.motherjones.com/kevin-drum/2018/11/preferences-are-not-transitive/|title=Preferences are not transitive|last=Drum|first=Kevin|date=November 2018|work=Mother Jones|access-date=2018-11-29|archive-date=2018-11-29|archive-url=https://web.archive.org/web/20181129113105/https://www.motherjones.com/kevin-drum/2018/11/preferences-are-not-transitive/|url-status=live}}</ref>

Generalized to stochastic versions (''[[stochastic transitivity]]''), the study of transitivity finds applications of in [[decision theory]], [[psychometrics]] and [[Utilitarianism|utility models]].<ref>{{Cite journal|last1=Oliveira|first1=I.F.D.|last2=Zehavi|first2=S.|last3=Davidov|first3=O.|date=August 2018|title=Stochastic transitivity: Axioms and models|journal=Journal of Mathematical Psychology|volume=85|pages=25–35|doi=10.1016/j.jmp.2018.06.002|issn=0022-2496}}</ref>

A ''[[quasitransitive relation]]'' is another generalization;<ref name="Derek.1964"/> it is required to be transitive only on its non-symmetric part. Such relations are used in [[social choice theory]] or [[microeconomics]].<ref>{{cite journal | last=Sen | first=A. | author-link=Amartya Sen | title=Quasi-transitivity, rational choice and collective decisions | zbl=0181.47302 | journal=Rev. Econ. Stud. | volume=36 | issue=3 | pages=381–393 | year=1969 | doi=10.2307/2296434 | jstor=2296434 }}</ref>

'''Proposition:'''  If ''R'' is a [[univalent relation|univalent]], then R;R<sup>T</sup> is transitive.
: proof: Suppose <math>x R;R^T y R;R^T z.</math> Then there are ''a'' and ''b'' such that <math>x R a R^T y R b R^T z .</math> Since ''R'' is univalent, ''yRb'' and ''aR''<sup>T</sup>''y'' imply ''a''=''b''. Therefore ''x''R''a''R<sup>T</sup>''z'', hence ''x''R;R<sup>T</sup>''z'' and R;R<sup>T</sup> is transitive.

'''Corollary''': If ''R'' is univalent, then R;R<sup>T</sup> is an [[equivalence relation]] on the domain of ''R''.
: proof: R;R<sup>T</sup> is symmetric and reflexive on its domain. With univalence of ''R'', the transitive requirement for equivalence is fulfilled.