# Definition talk:Pairwise Disjoint

Jump to navigation
Jump to search

Does it give us an advantage to change the definition from a set of sets to a family of sets? Before you can define a family you need to have defined the concept of a mapping, which is further down the road of complexity. What would be good to strive for would be a definition from as basic a level as possible. --prime mover 02:16, 27 April 2012 (EDT)

- I can see your point, and I agree. So it would become 'For $S, T \in \Bbb S$ such that $S \ne T$, $S \cap T = \varnothing$.' is it not? --Lord_Farin 04:02, 27 April 2012 (EDT)
- Effectively. It means reverting back to the last version of Dec 2011. --prime mover 08:35, 27 April 2012 (EDT)
- Isn't the current version more general? Maybe we could include both cases? --abcxyz 10:33, 27 April 2012 (EDT)

- Effectively. It means reverting back to the last version of Dec 2011. --prime mover 08:35, 27 April 2012 (EDT)

I wouldn't say that a set can be generalised by indexing it; I recall a theorem saying that any set may be indexed, but I can't find it atm. Both cases would offer a solution. Something along the lines of 'when $\Bbb S$ is an indexed set, ... $S_i \cap S_j = \varnothing$ if $i \ne j$' --Lord_Farin 10:57, 27 April 2012 (EDT)

- My concern was that, for example, using the "indexed family" version, the ordered pair $\left({S, S}\right)$ where $S$ is non-empty would not be considered as pairwise disjoint, whereas if one uses the "set of sets" version, the set under consideration would be just the singleton $\left\{{S}\right\}$, which would be considered as pairwise disjoint.
- I've changed the definition page. Is it going in the right direction? --abcxyz 11:17, 27 April 2012 (EDT)

- Ahaha. I hadn't thought of that. In that case it's worth splitting the definitions out into separate transcluded pages for both the "set of sets" and "family of sets" definitions, and specifically adding a page (visibly linked to) which makes that distinction plain. It's subtle enough to be easily missed. Good call.

- Also worth mentioning that a "pairwise disjoint family" includes the concept of a "pairwise distinct sequence" (also possibly worth including as a page of its own), which is used in (for example) Definition:Dynkin System. --prime mover 07:20, 28 April 2012 (EDT)

- I also missed that intricate point, good call. --Lord_Farin 07:26, 28 April 2012 (EDT)