Transitive relation

From Biocrawler, the free encyclopedia.

In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c.

In mathematical notation, this is:

For example, "is greater than" and "is equal to" are transitive relations: if a = b and b = c, then a = c.

On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire.

Examples of transitive relations include:

• "is equal to" (equality)
• "is a subset of" (set inclusion)
• "is less than" and "is less than or equal to" (inequality)
• "divides" (divisibility)

A transitive relation that is also reflexive is a preorder. A preorder that is antisymmetric is a partial order. A preorder that is symmetric, is an equivalence relation.

See also

• transitive closure
• intransitivity
• reflexive relation
• symmetric relation

External link

• Transitivity in Action (http://www.cut-the-knot.org/triangle/remarkable.shtml)es:Relación transitiva

fr:Transitivité_(mathématiques) zh:传递关系

Wikipedia (http://en.wikipedia.org/wiki/Main_Page) Transitive_relation (http://en.wikipedia.org/wiki/Transitive_relation) version history (http://en.wikipedia.org/w/index.php?title=Transitive_relation&action=history) GNU Free Documentation Lizenz (http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License) CC-by-sa (http://creativecommons.org/licenses/by-sa/2.5/)

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