Talk:Algebra of sets

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Contents 1 Requested move 1.1 Survey 1.2 Discussion 2 Prop #8 3 Three questions

[edit] Prop #8

I think in "preposition 8" since so not

Yes! Paul August 19:50, Sep 8, 2004 (UTC)

[edit] Three questions

1. Where does the terminology "algebra of sets" come from?
2. Is there any sort of interesting relationship between this theory and topos theory, another algebraic (or at least categorical) theory of simple collections?
3. Same qn, v.a.v. Tarski's Calculus of Relations (which doesn't seem to be documented here at Wikipedia, but it may be the same theory as relational algebra)? --- Charles Stewart 20:03, 27 September 2005 (UTC)

Charles, I don't know the answers to questions 2 and 3. By way of trying to shed some light on question 1., I will copy here a discussion that took place at Wikipedia talk:WikiProject Mathematics:

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I've just created a new "set theory" article: The algebra of sets I'd be interested if anyone has any comments. In a sense it's an expanded version of Simple theorems in the algebra of sets the latter being primarily just a list. One could argue that consequently the latter article is no longer necessary. But I can see the possible use of an article which simply lists results. Comments? Paul August 03:53, Sep 6, 2004 (UTC)

Hmmm - a few questions relative to the integration with the rest of WP. What you mean mostly is 'here is some explicit information about the Boolean algebra of sets'. Which might be useful to some people, indeed. Since the 'set of all sets' is chimerical, your 'algebra' is not precisely a Boolean algebra; the subsets of a given set X would give a Boolean algebra. I think this kind of placing would be helpful; and probably renaming the page.

Charles Matthews 08:07, 6 Sep 2004 (UTC)

Charles, thank you for your comments. As to the title, I took my lead from Simple theorems in the algebra of sets. The word "algebra" here is not being used as a technical term, as say in "Boolean algebra" or "linear algebra" but rather as a descriptive term, for this collection of facts concerning "the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion." The motivation for using the word "algebra" beyond it's descriptiveness, is to help the reader make the connection to the perhaps more familiar notion of algebra of numbers. It is a relatively common way of describing this material. For example Robert R. Stoll in Set Theory and Logic has a section titled "The Algebra of Sets", as does Seymore Lipschutz in his Set Theory and Related Topics (Schaum's Outline Series). Having said that I'm not opposed to finding a better name for the article. I had also considered simply "Set algebra" as an alternative name. What name are you proposing? As you say, and as is pointed out in the article, the power set of a given set is a Boolean algebra. As to your other suggestion of "this kind of placing would be helpful" I'm not sure what this means, could you please be more specific? Thank you again. Paul August 16:41, Sep 6, 2004 (UTC)

I would like to see even more analogies with usual algebra. A) You never say explicitly which operation is the analog of addition and which of multiplication (does this make sense? If not, the article should explain that too). B) Analogs of (a <= b) => (a+c <= b+c) should be highlighted. C) perhaps to put to the right of every inequality the anaolg (if it exists) in usual algebra? Arrange everything in comparison tables? I feel I'm starting to float. Think about these. Gadykozma 12:13, 6 Sep 2004 (UTC)

Gadykozma, thanks for your comments. As far as the analogy holds, union is the analog of addition (in fact the union of two sets has been sometimes called their "sum") and Intersection is the analog of multiplication. The article used the order of their mention to try to make this clear (perhaps a well placed "respectively" is needed.) As I partially said above, the use of this analogy is to help motivate these ideas for the reader, and to help place these facts concerning set theory in an appropriate setting. Including the fact that A ⊆ B ⇒ (A ∪ C) ⊆ (B ∪ C) is probably good in it's own right, that it continues the analogy is also nice. But I think we should be careful about relying too heavily on the analogy. It is not meant (by me at least ;-)) to be an article about the analogy. Paul August 16:41, Sep 6, 2004 (UTC)

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Paul August ☎ 20:32, 27 September 2005 (UTC)

Paul, thanks very much. If I am not mistakened, this structure is the same thing as a power set algebra. --- Charles Stewart 21:48, 28 September 2005 (UTC)

Your welcome. Well, yes the usual Boolean algebra on the power set of a set is sometimes called the "power set algebra". Are you suggesting that that might be a better name for this article? Perhaps, but I didn't (and don't, but maybe I should) conceive of this article being about a particular Boolean algebra, rather, as I wrote above, I think of it as about "the basic properties and laws of sets …". Perhaps it would be better to just call it the "laws of sets" ;-) I think I'm beginning to develop a dislike for the word "algebra" altogether ;-) Paul August ☎ 22:38, 28 September 2005 (UTC)

It is clearly time to take a stand. We live in the enlightened age where the true definition of algebra has been revealed, in the Book of Lawvere, and we must stop indulging the sinful notion that the definitions in such benighted articles such as monoid and ring theory are of algebraic structures. That's what you had in mind, right? --- Charles Stewart 14:00, 29 September 2005 (UTC)

Something like that yes ;-) Paul August ☎