Difference between revisions of "Transitive relation"
From Encyclopedia of Mathematics
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| − | One of the most important properties of a [[binary relation]]. A relation $R$ on a set $A$ is called transitive if, for any $a,b,c\in A$, the conditions $aRb$ and $bRc$ imply $aRc$. [[Equivalence relation]]s and [[Order (on a set)|orderings]] are examples of transitive relations. The universal relation, $a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty relation are transitive. | + | One of the most important properties of a [[binary relation]]. A relation $R$ on a set $A$ is called transitive if, for any $a,b,c\in A$, the conditions $aRb$ and $bRc$ imply $aRc$. [[Equivalence relation]]s and [[Order (on a set)|orderings]] are examples of transitive relations. The universal relation, $a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty (nil) relation are transitive. |
The intersection of transitive relations on a set is again transitive. The ''transitive closure'' $R^*$ of a relation $R$ is the smallest transitive relation containing $R$. It can be described as $a R^* b$ if there exists a finite chain $a = a_0, a_1, \ldots, a_n = b$ such that for each $i=1,\ldots,n$ we have $a_{i-1} R a_i$. | The intersection of transitive relations on a set is again transitive. The ''transitive closure'' $R^*$ of a relation $R$ is the smallest transitive relation containing $R$. It can be described as $a R^* b$ if there exists a finite chain $a = a_0, a_1, \ldots, a_n = b$ such that for each $i=1,\ldots,n$ we have $a_{i-1} R a_i$. | ||
Revision as of 18:36, 19 October 2014
One of the most important properties of a binary relation. A relation $R$ on a set $A$ is called transitive if, for any $a,b,c\in A$, the conditions $aRb$ and $bRc$ imply $aRc$. Equivalence relations and orderings are examples of transitive relations. The universal relation, $a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty (nil) relation are transitive.
The intersection of transitive relations on a set is again transitive. The transitive closure $R^*$ of a relation $R$ is the smallest transitive relation containing $R$. It can be described as $a R^* b$ if there exists a finite chain $a = a_0, a_1, \ldots, a_n = b$ such that for each $i=1,\ldots,n$ we have $a_{i-1} R a_i$.
References
| [a1] | R. Fraïssé, Theory of Relations, Studies in Logic and the Foundations of Mathematics, Elsevier (2011) ISBN 0080960413 |
How to Cite This Entry:
Transitive relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transitive_relation&oldid=33954
Transitive relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transitive_relation&oldid=33954
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article