Definition talk:Transitive Closure (Relation Theory)

Even this term is somewhat ambiguous. Transitive closure may also refer to the smallest transitive set containing a given set. --Andrew Salmon 06:04, 26 July 2012 (UTC)


 * We could then make Definition:Transitive Closure (Set Theory) and Definition:Transitive Closure (Relation Theory). As it stands I imagine the former instantiates the latter, but I'm not entirely sure. --Lord_Farin 10:31, 2 August 2012 (UTC)


 * I just realized that a class $A$ is a transitive class iff $\Epsilon \restriction A$ is a transitive relation. This provides a clear connection between transitive classes and transitive relations.  Perhaps this could make the proof that every set has a transitive closure slightly easier.  Just take the domain of the transitive closure of the relation $\Epsilon \restriction A$ and you have the smallest transitive set containing $A$! --Andrew Salmon 02:27, 15 August 2012 (UTC)


 * That appears to work indeed; of course, some work is to be done to extend the 'Transitive Closure Exists' theorem to class relations. --Lord_Farin 07:16, 15 August 2012 (UTC)