Transitivity: Difference between revisions

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In mathematics, the word transitive admits at least two distinct meanings:

A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. See group action. A somewhat related meaning is explained at ergodic theory.

A binary relation is transitive if whenever A is related to B and B is related to C, then A is related to C, for all A, B, and C in the domain of the relation. See transitive relation.

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+In [[mathematics]], the word '''''transitive''''' admits at least two distinct meanings:
-{{Wiktionarypar|transitivity}}
-The word '''transitivity''' is used in severals contexts:
-* In [[grammar]], a verb is '''transitive''' if it takes an object.  See [[Transitivity (grammatical category)]] and [[transitive verb]].
-* In [[logic]] and [[mathematics]], a [[binary relation]] ''R'' is '''transitive''' if ''xRy'' and ''yRz'' together imply ''xRz''.  For example, the ''less-than'' relation is transitive.  See [[transitive relation]], [[intransitivity]].
+* A [[group (mathematics)|group]] ''G'' acts '''transitively''' on a [[set]] ''S'' if for any ''x'', ''y'' ∈ ''S'', there is some ''g'' ∈ ''G'' such that ''gx'' = ''y''.  See [[group action]].  A somewhat related meaning is explained at [[ergodic theory]].
-* In [[mathematics]], a [[group action]] is '''transitive''' if it has just one [[orbit (group theory)|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''.
+* A [[binary relation]] is '''transitive''' if whenever A is related to B and B is related to C, then A is related to C, for all A, B, and C in the domain of the relation.  See [[transitive relation]].
+==See also==
-* In [[set theory]], a set is '''transitive''' if it contains all [[Element (mathematics)|element]]s of its elements.  See [[transitive set]].
+* [[transitive closure]]
+* [[intransitivity]]
+{{mathdab}}
+[[de:Transitivität (Mathematik)]]
-'''Transitive''' may refer to:
+[[uk:Транзитивність]]
-* [[Transitive Corporation]], a [[computer software]] firm based in [[Manchester]], [[England]] and [[Los Gatos]], [[California]].
-{{disambig-cleanup}}
-[[de:Transitivität]]
-[[es:Transitividad]]
-[[fr:Transitivité]]