Talk:Partition of a set

Displayed TeX whithin a link

I just included some displayed TeX within in a link, thus:

The [[integer partition | partition

:<math>etc. etc.</math>

of the integer]] n ...

It seems to work: you can click on the word partition, or on the displayed TeX, or on the following phrase, and get the integer partition page. Michael Hardy 20:07, 5 Feb 2004 (UTC)

Definition correct?

Is the definition of partition entirely correct / undisputed?

specifically I wonder about the requirement that the elements of the partition needing to be non-empty.

The definition given in Potter, M. "Set theory and its philosophy", ( 2004, Oxford University Press ), on p. 130 is:

"Definition. A collection B of subsets of A is a partition of A if each element of A belongs to exactly one element of B.

According to this definition, A is a partition of itself, since it certainly is a subset of itself, and furthermore { A, {null} } is also a partition of A - since there is no restriction on the elements being null.

This does not seem to quite capture the essence of partition to me, but it does seem a commonly used definition - ie see the MathWorld definition: http://mathworld.wolfram.com/SetPartition.html

so I'm wondering whether the article should be updated to clarify this ambiguity - that there are at least two alternate definitions being used.

--Corvi42 04:15, 10 Jun 2005 (UTC)

Ok, I went ahead and added a small addendum to axiom 1 saying that it is not included in all definitions.

Also, how is it that Axiom 1 can be vacuously true for the empty set - it contradicts Axiom 1!

--Corvi42 04:27, 10 Jun 2005 (UTC)

Again, I went ahead and corrected the erroneous example of the empty set, and added a few more examples. --Corvi42 15:02, 10 Jun 2005 (UTC)

You are wrong about the empty set. The partition with 0 blocks satisfies all the axioms including Axiom 1. --Zero 16:55, 10 Jun 2005 (UTC)

Where does axiom 1 come from?

Just out of curiosity - where does axiom 1 come from? So far I haven't found any definition which has this axiom other than this page and other online definitions which are obviously copies of this page. The definition in Britannica doesn't include this restriction either: set partition (http://www.britannica.com/eb/article?tocId=9058597&query=set%20partition&ct=)

The first four books I pulled from my library shelves, all define a partition as a collection of non-empty sets:

• Naive Set Theory, Paul Halmos
• Set Theory and Logic, Robert R. Stoll
• Axiomatic Set Theory, Patrick Suppes
• Set Theory, Seymour Lipschutz

See also: PlanetMath (http://planetmath.org/encyclopedia/Partition.html)

The definition you mention above from Mathworld (http://mathworld.wolfram.com/SetPartition.html) is inconsistent with Mathworld's own definition of Bell number (http://mathworld.wolfram.com/SetPartition.html). It is a common oversight by authors to forget to take into account the empty set. Where else have you found the definition not to include axiom 1?

Paul August ☎ 22:57, Jun 10, 2005 (UTC)

---

Thanks for the references Paul. I suspected that there ought to be many, but couldn't find any myself, which got me curious. Given the "intuitive" interpretation of what a partition is, axiom 1 certainly makes sense - but it does add some theoretical complications for general theorems like "any subset and its complement is a partition" - so I can also see why some might choose not to include it.

As I mentioned above, my sources which did not use axiom 1 were the Mathworld reference, which seems to be problematic from what you say, also the Britannica reference and the Potter, M book "Set theory and its philosophy".

Anyway - thanks for the clarification. --Corvi42 22:38, 12 Jun 2005 (UTC)

You're very welcome, glad to help. Paul August ☎ 00:36, Jun 13, 2005 (UTC)