Editing Transitive relation (section)

== Properties ==

=== Closure properties ===
* The [[converse relation|converse]] (inverse) of a transitive relation is always transitive. For instance, knowing that "is a [[subset]] of" is transitive and "is a [[superset]] of" is its converse, one can conclude that the latter is transitive as well.
* The intersection of two transitive relations is always transitive.<ref>{{Cite journal |last1=Bianchi |first1=Mariagrazia |last2=Mauri |first2=Anna Gillio Berta |last3=Herzog |first3=Marcel |last4=Verardi |first4=Libero |date=2000-01-12 |title=On finite solvable groups in which normality is a transitive relation |url=https://www.degruyter.com/document/doi/10.1515/jgth.2000.012/html |journal=Journal of Group Theory |volume=3 |issue=2 |doi=10.1515/jgth.2000.012 |issn=1433-5883 |access-date=2022-12-29 |archive-date=2023-02-04 |archive-url=https://web.archive.org/web/20230204151127/https://www.degruyter.com/document/doi/10.1515/jgth.2000.012/html |url-status=live }}</ref> For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and also has the same first name as" is also transitive.
* The union of two transitive relations need not be transitive. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. [[Herbert Hoover]] is related to [[Franklin D. Roosevelt]], who is in turn related to [[Franklin Pierce]], while Hoover is not related to Franklin Pierce.
* The complement of a transitive relation need not be transitive.<ref name="Derek.1964">{{Cite journal |last=Robinson |first=Derek J. S. |date=January 1964 |title=Groups in which normality is a transitive relation |url=https://www.cambridge.org/core/product/identifier/S0305004100037403/type/journal_article |journal=Mathematical Proceedings of the Cambridge Philosophical Society |language=en |volume=60 |issue=1 |pages=21–38 |doi=10.1017/S0305004100037403 |bibcode=1964PCPS...60...21R |s2cid=119707269 |issn=0305-0041 |access-date=2022-12-29 |archive-date=2023-02-04 |archive-url=https://web.archive.org/web/20230204151127/https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/groups-in-which-normality-is-a-transitive-relation/E1EECC9F60124437962FBF9FDD8E81BA |url-status=live }}</ref> For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

=== Other properties ===
A transitive relation is [[asymmetric relation|asymmetric]] if and only if it is [[irreflexive relation|irreflexive]].<ref>{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics - Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|url-status=dead|archive-url=https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|archive-date=2013-11-02}} Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".</ref>

A transitive relation need not be [[Reflexive relation|reflexive]]. When it is, it is called a [[preorder]]. For example, on set ''X'' = {1,2,3}:

* ''R'' = {{{Hair space}}(1,1), (2,2), (3,3), (1,3), (3,2){{Hair space}}} is reflexive, but not transitive, as the pair (1,2) is absent,
* ''R'' = {{{Hair space}}(1,1), (2,2), (3,3), (1,3){{Hair space}}} is reflexive as well as transitive, so it is a preorder,
* ''R'' = {{{Hair space}}(1,1), (2,2), (3,3){{Hair space}}} is reflexive as well as transitive, another preorder,
* ''R'' = {{{Hair space}}(1,2), (2,3), (1,3){{Hair space}}} is transitive, but not reflexive.

As a counter example, the relation <math> < </math> on the real numbers is transitive, but not reflexive.