Talk:Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest

Probably a proof along the characteristics that the intersection of transitive relations is transitive, well, shortly, like Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets can be made. Nice and intrinsic. --Lord_Farin 18:07, 6 April 2012 (EDT)
 * I don't follow you ... --prime mover 18:09, 6 April 2012 (EDT)
 * Usually, there is some trivial structure (like the power set, or Cartesian product in the present case) both encompassing the structure that is to be expanded and having the desired properties.
 * Mostly, when one speaks about the smallest ladidada, this is WRT subset relation, and that usually implies that ladidada is preserved under arbitrary intersection. So the proof for ref'd results almost literally carries over. --Lord_Farin 18:14, 6 April 2012 (EDT)
 * I confess that this-all is actually somewhat outside my comfort zone - I posted it up when I was doing some specific research on graph theory for a java project for my day-job, and the pages on WP of R- S- and T-closure had direct relevance and looked interesting enough to post up. Beyond what I needed for work (which Neal Stephenson coincidentally covered directly in his Anathem novel) I know too little to be able to continue down this route. Feel free to take it from here. --prime mover 03:33, 7 April 2012 (EDT)

I put it up, does it make more sense now? --Lord_Farin 11:08, 19 April 2012 (EDT)
 * Works for me.--prime mover 12:48, 19 April 2012 (EDT)

Refactor
Since proof 1 was really about constructing the transitive closure by countable unions, with some repetition of proof 2 on top, I moved the good parts to Construction of Transitive Closure. I'm not sure that's really the best way to refactor, but I think this at least makes more sense than what came before. Proof 2 is clearly the simplest way by far to prove this theorem. --Dfeuer (talk) 21:17, 13 December 2012 (UTC)

The two proofs currently here are essentially the same. There are different proofs at Construction of Transitive Closure. Someone needs to decide how to arrange these things more sensibly. --Dfeuer (talk) 19:17, 19 January 2013 (UTC)


 * I think part/all of this was my fault, although I don't remember the exact chronology. BLARGH. --Dfeuer (talk) 19:19, 19 January 2013 (UTC)