Editing Transitive relation (section)

=== 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.