Transitivity: Difference between revisions
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In [[mathematics]], '''transitivity''' is a mathematical property of [[binary relation]]s such that if A and B are related, and B and C are related, then it follows that A and C are also related, for all A, B, and C for which the relation may apply. The relation is then said to be '''transitive'''. |
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* In [[grammar]], a verb is '''transitive''' if it takes an object. See [[transitive verb]]. |
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In notation, this is: |
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* In [[mathematics]], a [[binary relation]] ''R'' is '''transitive''' if ''xRy'' and ''yRz'' together imply ''xRz''. For example, the ''less-than'' relation is transitive. See [[transitivity]]. |
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:<math>\forall a,b,c:\,a\,\mathbf{R}\,b\,\wedge\,b\,\mathbf{R}\,c\,\Rightarrow\,a\,\mathbf{R}\,c</math> |
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* In [[mathematics]], a [[group action]] is '''transitive''' if it has just one [[orbit (mathematics)|orbit]]. It is called '''doubly transitive''' if it is transitive on ordered pairs of distinct elements; and so on for '''triply transitive''', etc.. An [[ergodic theory|ergodic]] group action is also called ''metrically transitive''. |
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For example, "is greater than" and "is equal to" are transitive relations: if a=b and b=c, then a=c. |
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{{msg:disambig}} |
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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. |
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Example of transitive relations include: |
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* "is equal to" -- [[equality]] |
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* "is a [[subset]] of" |
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If a transitive relation is also [[reflexive]] and symmetric, then it is said to be an [[equivalence relation]]. |
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See also [[transitive closure]]. |
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Revision as of 22:14, 15 March 2004
In mathematics, transitivity is a mathematical property of binary relations such that if A and B are related, and B and C are related, then it follows that A and C are also related, for all A, B, and C for which the relation may apply. The relation is then said to be transitive.
In 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.
Example of transitive relations include:
If a transitive relation is also reflexive and symmetric, then it is said to be an equivalence relation.
See also transitive closure.