Talk:Set

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From peer review: CMummert As it stands, this article is extremely redundant with naive set theory, almost to the point of duplication. This article is an appropriate place to discuss the philosophical questions associated with sets. For example, Penelope Maddy is widely rumored to have claimed that sets exist in some "physical sense". There is no mention of extenstionality in the article, but this is one of the things that makes sets into sets. Like they say, there are no blue sets. The concept of the cumulative hierarchy is missing. Can a set be a member of itself? Of course the axiomatic treatment doesn't belong here, but some discussion does. The sections on basic operations could be grouped together (union, intersection, relative complement). There is no mention of Russell's paradox except as a "see also" link. This is, historically, an important step in the understanding of the technical limits of the natural language term "set". Trebor There is very little detail on the history of sets, or their application to Mathematics in general except in the lead. A bit more background information (with references) would balance the article and provide a more well-rounded look; at the moment, it's more of a textbook than an encyclopaedia article. Opabinia regalis FA and GA will whack you over lack of inline citations. Most of this stuff is trivially self-verifying, but when you add more background/history and more advanced material, make sure to add references where appropriate. Images are prime candidates for SVG-ifying. Automated peer review can be found here

Set has had a peer review by Wikipedia editors which is now archived. It may contain ideas you can use to improve this article.

This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B Class	High Importance	Field: Foundations, logic, and set theory
A vital article

Contents 1 Disambiguation 2 unheadered 3 list v. set 4 Set vs. Naive set theory 5 ∪ symbol displays as box? 6 symbols displaying as boxes 7 Symbols and the set theory 8 School curricula 9 if vs iff in mathematical definitions. 10 Symbols displayed as boxes 11 Some reverted edits 12 Disambig 13 Removed inappropriate (in my view) text 14 Living dragons 15 Lead section 16 Box-fixing 17 My reversion of edit to "definition" section 18 Thanks 19 "see link" 20 SECTION: Cardinality of a set 21 Unordered? 22 Improper Subsets 23 Contradiction / clarification?

[edit] Disambiguation

I started to do the disambiguation, but found that they are almost entirely for mathematical set. Would it not be better, and aid in accidental linking, if we just had "mathematical set" here and used a see also: for the other meanings?

Probably the mathematical meaning ought to be here, the other meanings being very much rarer in practice (as you have noticed). People will continue to link here by mistake anyway. Of couse, there's no harm in changing the links to set (mathematics) (or even the ugly mathematical set), since these are completely unambiguous and can be redirected wherever we want. --Zundark, Tuesday, April 9, 2002

If no one is going to fix the links, then I intend to move the page back again somewhen this week. --Zundark, Monday, April 15, 2002

I disambiguated all of the mythological links; I don't think that any of them are to the game. So somebody can either disambiguate the hundred or so remaining mathematical links, or Zundark can move the article back; I won't make that call. -- Toby Bartels 2002/04/17

[edit] unheadered

I'm all for putting real text about a clearly "primary" meaning into an article that also points to less common meanings. I'm not sure whose comment it was, but someone argued that the disambiguating pointers should be at the top of the article in that case, and I'm inclined to agree. Perhaps I'll write up these suggestions more clearly in the disambiguation page. --LDC

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{x : x is a primary color}

This is not a very precise definition. Depending on whether you consider additive or subtractive color models , either yellow or green are primary colors. -- JeLuF 09:49 19 Jun 2003 (UTC)

You can change "a primary color" to "an additive primary color", if you like. But other additive colour models are also possible, so even this isn't completely precise. Perhaps we should replace it with a better example. --Zundark 10:19 19 Jun 2003 (UTC)

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Can someone put in a description of nested sets and representation of tree structures with this. I can't find a decent reference anywhere for this. -- Chris.

[edit] list v. set

A suggested change:

By contrast, a collection of elements in which multiplicity but not order is relevant is called a multiset. A collection of elements in which multiplicity and order are relevant is called a list. Other related concepts are described below.

I believe that was the definition in my linear algebra textbook last semester. Goodralph 02:18, 21 Jul 2004 (UTC)

[edit] Set vs. Naive set theory

I think there is too much overlap between the articles Set and Naive set theory.

In reviewing the change history for Set, I find that the earliest versions of this article (can anyone tell me how to find the original version, the earliest I can find is as of 08:46, Sep 30, 2001) contained the following language prominently placed in the opening paragraph:

"For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set theory. Here we give only a brief overview of the concept." (The articles referred to have since been renamed as Naive set theory and Axiomatic set theory resp.)

As subsequent editors, added new information to the beginning of the article, the placement of this "brief overview" language, gradually moved further into the article, until now it is "buried" as the last sentence of the "Definitions of sets" section. Consequently I suspect that some new editors are unaware that some of the material being added to this article is already in, or should be added to Naive set theory or even Axiomatic set theory (e.g. Well foundedness? Hypersets?).

If it is agreed that, Set is supposed to be a "brief overview" of the idea of a set, while Naive set theory and Axiomatic set theory give more detail, I propose two things:

1. Add something like: "This article gives only a brief overview of sets, for a more detailed discussion see Naive set theory and Axiomatic set theory." to the opening section of the article Set.
2. Move much of what is in the article Set to Naive set theory or Axiomatic set theory.

Comments?

Paul August 20:23, Aug 16, 2004 (UTC)

I have moved the sections on "Well-foundedness" and "Hypersets" to Axiomatic set theory, which I think is a more appropriate place for them - based on the idea expressed above that the Set article shold be a "brief overview". Paul August 07:34, Aug 18, 2004 (UTC)

I've made the the above proposed changes. Paul August 21:04, Aug 27, 2004 (UTC)

[edit] ∪ symbol displays as box?

Someone changed each set union symbol "∪" (i.e &cup) to an uppercase U, because they were displaying as boxes. Is there a problem with rendering ∪? It looks ok for me (Safari, IE, OmniWeb on MAC OSX). Does anybody else have problems with this? Paul August 19:34, Aug 31, 2004 (UTC)

[edit] symbols displaying as boxes

I've got symbols &cup,&sub, &empty and &sube displaying as boxes (IE 6.0), and it looks very annoying. The reason is maybe that I use Russian as a default languge (Regional and Language Options settings), and have also got a set of Russian fonts installed, but my Windows version is English (non-localized). Everything looks fine in Opera, though. What character set does IE use to process these symbols?

(I've recently changed my default language to English and it still does not work for IE). Igor

[edit] Symbols and the set theory

I see too many symbols in set theory as little squares. I think to be the one that changed the "union" symbol to an upper U, but nothing can be done for other symbols. Referring to the article on TeX markup, I think the reason is that the article on Sets is not written using the latter language. I tested it, without saving, starting with <,math> (please ignore the ,) and ending with <\,math> and all the formulas included in between went ok with the usual symbols. Somebody should patiently change the source language. demaag.

You need to get the proper fonts so they show up. I'm not sure how you can do this, perhaps someone else can clarify. Or try a different browser (like Mozilla Firefox). Dysprosia 09:08, 5 Sep 2004 (UTC)

[edit] School curricula

I think it is an interesting remark to make that set theory at one point was included in school curricula. As I understand it, this was (in the West) mostly a reaction to the Sputnik shock (I suppose the Soviet bloc school system included set theory in its curriculum?). I don't really know how things are today, other than that at least some countries seem to have largely eliminated set theory from their curricula and don't introduce set theory notation until university. Prumpf 00:07, 11 Sep 2004 (UTC)

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User: 84.65.179.65 took exception to the sentence:

Basic set theory, having only been invented at the end of the 19th century, is now part of the elementary school curriculum.

With the comment: Took out 'part of the elementary school curriculum'. Where? In America? Didn't do it at my school. What's the relevance of this to the article anyway?!?)

I've tried to address these concerns by replacing the above sentence with:

Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school.

I don't know whether set theory is usually taught in elementary school now, or if it is where (maybe someone will inform us?). I do know that is was introduced, in many (if not most) parts of the United States, into the elementary school curriculum in the 1960s, as part of what was called "new math". The relevance of this to the article is that, although it is a relatively recent mathematical development, it is now (or was?) thought to be so fundamental as to warrant teaching it in elementary school. Of course many parts of the "new math" curriculum fell out of favor, although I think that set theory was one of the least criticized parts. Paul August 01:42, Sep 11, 2004 (UTC)

There's a joke that pertains here:

Progress in mathematics education:

1950

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?

1960

A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?

1970

A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100 and each element is worth $1.

(a) Make 100 dots representing the elements of the set M

(b) The set C representing costs of production contains 20 fewer points than set M. Represent the set C as a subset of the set M.

(c) What is the cardinality of the set P of profits?

1980

A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.

1990

By cutting down a forest full of beautiful trees, a logger makes $20.

(a) What do you think of this way of making money?

(b) How did the forest birds and squirrels feel?

(c) Draw a picture of the forest as you'd like it to look.

Paul August 01:56, Sep 11, 2004 (UTC)

Some version of this problem should also be politically correct, like 'His/her cost of production is $80 and his/her profit is $20'

[edit] if vs iff in mathematical definitions.

In mathematics, the use of "if" is in definitions is the common practice, and it is perfectly "precise". Quoting from Talk:iff:

Regarding "if/iff" convention for defs: I've reinserted the comment about "if" being used conventionally in math defs. I'm sorry, I've read a lot of math books, and this is a common convention. Many definitions use the terminology "if", in the sense of "If P(X), then X is called blah" or "X is said to be blah if P(X)", yet not every definition uses "iff", and all definitions are intended to be "iff", because that's what definitions are. (To counter your remark, definitions are not intended to assert equivalencies; an equivalence is usually meant to indicate a statement saying two things imply each other that has to be PROVED...definitions aren't proved, they're declared, so it doesn't make sense to say e.g. "'R is an integral domain' is equivalent to 'R is a commutative ring with identity'" because these statements aren't "equivalent" in the ordinary sense of the term, one does not PROVE they're equivalent, that simply IS the definition of an integral domain. Here are several cases where the "if" convention is used in the wikipedia itself...

• "A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n" (from prime number)

• "If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b." (from integral domain...notice, the first use of the word is in the sense of a definition, hence only "if" is used (although "iff" would be correct as well), but the second IS an actual theorem (result) because the equivalent condition requires proof. So, for the second statement, the meaning would change if "iff" were replaced by "if", although for the first statement it doesn't matter.

• "In complex analysis, a function is called entire if it is defined on the whole complex plane and is holomorphic everywhere" (from entire function).

The list could go on. Revolver

Revolver is correct. "If" is used everywhere in mathematical definitions, both in Wikipedia and elsewhere. "iff" or "if and only if" is commonly reserved for use in biconditionals, which definitions are not. The use of "iff" or ""if and only if" here is particularly inappropriate as this is intended as a basic introductory and elementary level article, which could be read even by a grade school student. Paul August 14:43, Nov 9, 2004 (UTC)

I would say, in biconditionals and in theorems (although formally a theorem is a biconditional; but you are talking about grade school students here...). Mikkalai 18:45, 7 Jan 2005 (UTC)

[edit] Symbols displayed as boxes

I confirm that under a French localized Windows XP/Internet Explorer 6.0, many symbols (like the U for union) are displayed as boxes. --Didier

[edit] Some reverted edits

Recently User:Kendrick Hang made some changes, some of which I have just reverted. The reversions were made, primarily to keep the article "a brief and basic introduction" and to reserve more detailed and complete treatments of these ideas for other articles (Naive set theory, Cardinality, Complement (set theory) etc.) as stated in the articles lead section and discussed above on this talk page. If anyone wants to discuss these changes further I'd be happy to do so. Paul August ☎ 17:57, Jan 7, 2005 (UTC)

I understand the need to keep the article to the basics, but wouldn't one assume that the basic set operations that most people read in an introductory discrete mathematics text are union, intersection, and difference? Maybe we could at least mention that relative complement is also known as a difference operation? If difference doesn't belong here, maybe we could put a link to where someone would be able to find more about it? -- Kendrick

Yes, union, intersection and difference are the most basic set operations. Although in my experience, the term "complement" is, by far, more commonly used than the term "difference". Following your suggestion, however, I've added that "relative complement" is also called "set theoretic difference". As for links, there is a link to Complement (set theory) which has a more detailed treatment of complements including a link to symmetric difference. Paul August ☎ 02:15, Jan 9, 2005 (UTC)

[edit] Disambig

i propose moving this to "set (mathematics)" and turning "set" into the disambig page, as there are currently 9 different entries linked to the disambig page. any serious objections? --Heah 17:19, 25 Apr 2005 (UTC)

I am not in principle opposed to doing the move, although I don't quite see the gain. The big question is, who is going to fix all the links, there are hundreds of them pointing to set. Oleg Alexandrov 17:29, 25 Apr 2005 (UTC)

There are just a bunch of articles with the name "set", and it would seem prudent and generally time saving to have that as the disambig page. Not a huge gain, but it's there. There certainly are a whole lot of pages that link here! Although looking at that list makes this whole thing less appealing, i'd be willing to fix the links, i guess, as i'm the one proposing the move. It'll take a lot of time but imo will be beneficial in the long run. --Heah 17:47, 25 Apr 2005 (UTC)

I think this is probably not a good idea. There are currently over 500 pages which link to Set, this is far more than the number of pages that link to all other entries for "set" on the disambiguation page, combined. The current disambiguation of set is an example of what is called "primary topic" disambiguation, which I think is the appropriate type of disambiguation in this case. Quoting from: Wikipedia:Disambiguation#Types of disambiguation:

"Primary topic" disambiguation: if one meaning is clearly predominant, it remains at "Mercury", the general title. The top of the article provides a link to the other meanings, or if there are a large number, to a page named "Mercury (disambiguation)". For example: the page Rome has a link at the top to a page named "Rome (disambiguation)" which lists other cities named Rome. The page Cream has a link to the page Cream (band) at the top.

Paul August ☎ 18:01, Apr 25, 2005 (UTC)

[edit] Removed inappropriate (in my view) text

I've removed the following text from the "introduction" section:

"The informality of this 'definition' of a set leaves clear that different sets are different; so the definition of a set goes hand in hand with a classification of its objects. Of course, sets share properties; but these properties are tightly connected with provisions in the definition of any given set. For example, we can't speak of combinatorics (see "cardinality" below) of uncountable sets, like the set of real numbers."

This text seems more like philosophy than mathematics, and frankly I don't really understand what exactly it is trying to say. In any case I think it is out of place here. The purpose of this article is to give "a brief and basic introduction" to sets.

Paul August ☎ 16:13, Apr 29, 2005 (UTC)

[edit] Living dragons

How do we know that set A is equal to the null set (where A is the set of living dragons)?

[edit] Lead section

The lead section currently doesn't say what a set is, only that it is a concept in mathematics. Why not have the definition there? - Fredrik | talk 8 July 2005 07:25 (UTC)

I myself like it that way. The concept of set is a rather abstract one. I think it is good to have some rambling about its importance before getting down to business. But it was not me who wrote that, so let's see what others have to say. Oleg Alexandrov 8 July 2005 15:30 (UTC)

Oleg, you liike it which way? As it is, or as Fredrik suggests, with the content in the "definition" section moved to the lead? Paul August ☎ July 8, 2005 16:44 (UTC)

OK, I like it the way it is. :) Oleg Alexandrov 8 July 2005 17:28 (UTC)

I agree that the lead section could say something more about what a set is — but I think the content in the "definition" section (or something very much like it) should stay where it is. I will try to rework the lead and "definition" sections over the weekend. Paul August ☎ July 8, 2005 16:44 (UTC)

Well, the "definition" goes "Informally, ...". Fredrik | talk 8 July 2005 21:37 (UTC)

By the way I've often thought it might be nice if this article could be an FA. It would be nice to have a mathematics FA that was accessible to the general reader. What do you guys think? Paul August ☎ July 8, 2005 16:44 (UTC)

The thought occurred to me as well; this article seems quite accessible. Some more detailed history would be required, to elaborate on the importance assigned to sets in the intro paragraph. Fredrik | talk 8 July 2005 21:37 (UTC)

I'm not sure that this article, being a "brief and basic introduction" is the most appropriate article for much on the history of set theory. Paul August ☎ 18:57, July 11, 2005 (UTC)

OK I've had a go at expanding the lead and "definition" sections. Comments? Paul August ☎ 18:33, July 11, 2005 (UTC)

[edit] Box-fixing

I didn't think this was controversial, but when 95% of our readers use standard versions of IE with no extra fonts, I think it is us who should be conforming to their needs. IE does support empty set and intersection characters, but not union or subset. I've replaced any union characters with a capital U (which looks pretty good in a sans serif font) and I replaced some subset symbols with "is a subset of", and others with images using math tags. I'd advise similar practices be followed in other articles. Note that if you set your preferences to "HTML if possible", the formulas in math tags should be rendered correctly as HTML for people with browsers supporting all these characters. Deco 20:56, 11 July 2005 (UTC)

[edit] My reversion of edit to "definition" section

I've reverted the recent edit of the "definition" section by User:Peak. My changes, and reasons for each, are the following:

• I reinserted the first sentence of the section: Like the concepts of point and line in Euclidian geometry, in mathematics, the terms "set" and "set membership" are fundamental objects used to define other mathematical objects, and so are not themselves formally defined. The purpose of this sentence is to make clear that while the section is titled "Definition", what follows is not strictly speaking a definition. It also explains the fundamental nature of sets.

• I changed the second sentence from: A set can be thought of as a well-defined collection of entities or objects. back to However, Informally, a set can be thought of as a well-defined collection of objects considered as a whole. I think this is better because:

• "informally" again helps make it clear, that what follows is not a formal definition.
• The qualification "considered as a whole" is important because it attempts to distinquish the set {1, 2, 3} from the three numbers 1, 2, and 3. For example, it is one thing not three things.
• I'm not sure adding the word "entities" is particulary useful.

• And I changed the third sentence from: The members of a set are called elements. back to: The objects of a set are called elements or members. This is better I think because here we are trying to "define" (again informally) the terms "element" and "member" (both frequently used mathematical synonyms) using the more primitive term "object", the term used in the pevious sentence.

Paul August ☎ 12:38, July 12, 2005 (UTC)

I think these changes are good, although really "a collection" is "one thing". If possible I'd prefer some wording that makes it clearer that, for example, the set containing the empty set is different from the empty set. A good analogy is a bag or box with things in it.

I'd also probably say "objects in or composing a set", to be more specific. Deco 22:16, 13 July 2005 (UTC)

"A set is a collection of objects considered as a whole." I think we should somewhere mention in the text that this is in fact saying a set is a set, and we can't get around that, because the concept of set is so basic a thing. The sentence still gives a good intuition about it. 85.156.185.105 10:46, 22 August 2006 (UTC)

[edit] Thanks

This is a very clear page for beginners like myself, so I just wanted to thank everyone who's worked on it. Nice work folks. Lucidish 16:53, 15 August 2005 (UTC)

[edit] "see link"

I kind of dislike the style and usually wish to keep this kind of style to linking as minimal as possible as it is dissonant/disrupts reading style and doesn't flow too well in my opinion, but I wonder why it was used. Would one consider it acceptable to merely integrate it with the entire article? Ie. rather than discussing briefly about empty sets in one section, use something like

Main article: empty set

instead?

[edit] SECTION: Cardinality of a set

This section takes the controversial stance that dragons do not exist, although there are many persons who believe that this isn't the case. Isn't it biased and a case of original research to include this personal opinion about the existence of dragons on an otherwise fine and upstanding page? 71.248.217.223 07:54, 12 November 2005 (UTC)

OK I've removed the reference to "living dagons". Paul August ☎ 11:33, 12 November 2005 (UTC)

I don't argue that the dragons don't exsit. One can prove that the set of living dragons is something we denote but don't worry too much about it as you are given choice to believe whether dragons exist or not, and anyway all this set and cardinality thing is an abstract math theory which would not influence the well-being of any more or less respectable dragon even for a moment. Oleg Alexandrov (talk) 17:43, 12 November 2005 (UTC)

We could simply use pink elephants or some other less controversial nonexistent creature. The objection is still silly though. Deco 21:46, 12 November 2005 (UTC)

[edit] Unordered?

Twice Fresheneesz, has added the qualifier "unordered" to the first sentence, which I've twice removed. I think that adding this is unnecessary, and can be misleading since ordered lists are also sets. Paul August ☎ 19:44, 21 May 2006 (UTC)

I suspect that the impetus behind Fresh's additions is the ongoing discussion he and I are having at talk:quadratic equation, where Fresh has a problem with my suggestion that the solution set of the quadratic equation be represented as an ordered set, which is denoted by the use of subscripts. However, I don't think the addition to this article was appropriate. -lethe ^{talk +} 19:48, 21 May 2006 (UTC)

[edit] Improper Subsets

Can we get a clarification on Improper Subsets? —The preceding unsigned comment was added by MLeg11 (talk • contribs) 15:03, 10 September 2006 (UTC)

That's not really a term anyone uses much. If you say "A is a subset of B", then A might or might not be equal to B. If you say "A is a proper subset of B", then A is definitely not equal to B. That's the only difference between "subset" and "proper subset". There's simply no need for a term "improper subset", and it isn't used. --Trovatore 19:06, 10 September 2006 (UTC)

Oh, I might also mention—because this is the sort of point on which people sometimes get confused—that if I say "A is a subset of B", I am not asserting that I don't know whether A is equal to B. I may know full well, and simply not be saying. This is not usually because I want to be difficult. More commonly, it's obvious from context whether or not A equals B, and there's just no need for me to repeat information that's clear to both of us. --Trovatore 20:10, 10 September 2006 (UTC)

[edit] Contradiction / clarification?

Under the "Definition" header, the article reads :

A set, unlike a multiset, cannot contain two or more identical elements.

However, under "Description", the article reads :

Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list, so {6, 11} = {11, 6} = {11, 11, 6, 11}.

The use of {11, 11, 6, 11} may be confusing to the casual reader, who has just been told that a set never has identical elements. Perhaps someone who writes better math-prose than I can edit the article to clarify why this notation is valid?

Best, -- Docether 15:55, 22 March 2007 (UTC)

It is not valid. I see a contradiction here as well and deleted the contradicting part (the repetitions). Thanks a lot for pointing at this! — Ocolon 17:44, 22 March 2007 (UTC)

There is no contradiction; I added a sentence to the article already to try to explain what is going on. Although the set itself "can only contain each element once", the set builder notation can list it as many times as desired. So once an element is listed once as an element, you can ignore it if it is listed again. For example, the set

{ pq in N : p is even and q is either 1 or a prime}

only includes the number 4 once, not twice, despite the fact that there are two different ways to write 4 in the form pq specified. CMummert · talk 18:46, 22 March 2007 (UTC)

Okay. Thank you for the lesson. :-) — Ocolon 18:50, 22 March 2007 (UTC)