Transitivity: Difference between revisions

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In mathematics, the word transitive admits at least three 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.
A transitive set is a set A such that whenever x ∈ A, and y ∈ x, then y ∈ A. The smallest transitive set containing a set A is called the transitive closure of A.

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 * 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]].
-* A [[transitive set]] is a [[set]] A such that whenever ''x'' ∈ ''A'', and ''y'' ∈ ''x'', then ''y'' ∈ ''A''. The smallest transitive set containing a set A is called the [[transitive closure]] of A.
+* A '''[[transitive set]]''' is a [[set]] A such that whenever ''x'' ∈ ''A'', and ''y'' ∈ ''x'', then ''y'' ∈ ''A''. The smallest transitive set containing a set A is called the [[transitive closure]] of A.
 ==See also==