Talk:Algebra of sets

Untitled

The following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the debate was no move. -- tariqabjotu 02:30, 24 August 2006 (UTC)

Requested move

Algebra of sets → Set algebra … Rationale: Confer math texts and ghits.

Survey

Add "* Support" or "* Oppose" followed by an optional one-sentence explanation, then sign your opinion with ~~~~

• Oppose. "Set algebra" might mean something completly different, and probably should redirect to Field of sets. — Arthur Rubin | (talk) 22:00, 14 August 2006 (UTC)
• Weak oppose; I don't see that much difference, but "algebra of sets" feels more idiomatic. Septentrionalis 12:46, 18 August 2006 (UTC)

Discussion

• Propose name change to Set algebra (or maybe even Set arithmetic?), which previously redirected to Field of sets for reasons I still don't understand 74.132.209.231 00:26, 13 August 2006 (UTC)

The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Prop #8

I think in "preposition 8" since $A\subset B$ so $A-B=\emptyset$ not $B-A=\emptyset$

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

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 ☎

Subtraction?

There is no where in the article a definition of the "substruction" operation appearing, namely \ Diam - 18 Feb 2011

See relative complement for the definition. Paul August ☎ 12:01, 18 February 2011 (UTC)

Notation

I know the choice of symbols doesn't really matter much, but isn't it fairly standard to call the universal set Capital Omega?--66.153.117.118 (talk) 01:18, 24 February 2011 (UTC)

No, Capital Omega is not a standard symbol for a universal set. In fact, U seems to be universally used. For example, do a search for "universal set venn diagram" in the images part of THE search engine and you will see a sea of U's, but no Capital Omegas.

Netrapt (talk) 13:45, 16 March 2012 (UTC)

Didactic tone

The article had a very didactic tone. I have tried to make it more descriptive and less didactic. I am not too happy with the numbered "propositions", though I can't put my finger on why. I suppose it's because it's not clear whether the article is taking a set-theoretic point of view, where these are consequences of logical definitions like A U B = {x | x in A or x in B}, or an algebraic point of view, in which case some of these are axioms and others are theorems. Under "proposition 3", it says that these results "can be derived from the five fundamental pairs...", but the fundamental pairs weren't defined as fundamental (except by the title of the section). The article in general is trying to stay very elementary, but it wikilinks to Duality (order theory), which isn't very elementary. Set difference is mentioned but not defined. --Macrakis (talk) 20:34, 14 September 2013 (UTC)

Missing definition of f inverse

https://en.wikipedia.org/wiki/Algebra_of_sets#Sets_and_maps seems to be missing a definition of $f^{-1}(X)$ --Intellec7 (talk) 05:20, 1 September 2020 (UTC)

 Done Added a link to preimage. - Jochen Burghardt (talk) 06:08, 1 September 2020 (UTC)

Questionable Analogy

"The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over union. However, unlike addition and multiplication, union also distributes over intersection." Is the only sense of the analogy that multiplication distributes over addition? If so, then I think this should be removed. If the goal of making an analogy here is to explain to a reader what it means for an operation to distribute over another operation, I think that should probably be left for the article for the distributive property.

Perhaps it could be replaced in favor of another analogy:

$\{1,2,\ldots ,n\}=S(n)$

$S(a)\cap S(b)=S(\min(a,b))$

$S(a)\cup S(b)=S(\max(a,b))$

As far as I know, this is an isomorphism, so quite a bit more than an analogy, and might not necessarily help a reader understand at this point, but could be included in the article elsewhere. Note that $\min$ and $\max$ distribute over each other. --Intellec7 (talk) 16:33, 1 September 2020 (UTC)

The analogy extends farther than just the distributive property, e.g $A\cup \varnothing =A$ and $A\cap \varnothing =\varnothing$ (where $\varnothing$ is the analogue of zero), among others. Note also that the union of and intersection sets A and B is also called their "sum" and "product", and written A+B and AB. Paul August ☎ 17:23, 1 September 2020 (UTC)

Recent changes

I think that recent changes have taken this article in the wrong direction. This article is intended to be about the general properties of the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion, which are sometimes described as the "algebra of sets". It is specifically *not* intended to be about the abstract object sometimes called an "algebra of sets", which is covered by the article field of sets. Paul August ☎ 13:44, 21 October 2020 (UTC)

Concerning the relations between both articles, isn't the algebra of sets a particular case of an algebra of sets (as handled at field of sets)? I guess the former is obtained by choosing ${\mathcal {F}}$ to be the powerset of $X$ in the definition of the latter. If that is right, it should be mentioned in both articles. - Jochen Burghardt (talk) 14:54, 7 November 2020 (UTC)

@Jochen Burghardt: The phase "the algebra of sets" (as used here in the article) isn't referring to any formal mathematical object, rather it's just a descriptive name for the algebraic properties of the various set operations. Granted these properties could all be considered as following from the collection of specific objects you mention above, but they don't need to be, and can be considered without regard to any such formal objects. As for mentioning of the objects: a field of sets (or an algebra of sets), note that the lede does says:

In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines ...

Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being $\varnothing$ and the top being the universe set under consideration.

Do you think it should say more? Paul August ☎ 16:32, 7 November 2020 (UTC)