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Template:Maths rating

reverse inclusion

The article Inclusion (set theory) redirects here.

I think it would be a good idea to also redirect 'Reverse inclusion' to here, and explain that term.

For example:

 The relationship of one set being a superset of another is called 'reverse inclusion'.

(but is this the correct definition?)

Moved paragraphs from Power set, and question

I moved the last two paragraphs here from Power set, since they seem more appropriate here.

By the way, is it equivalent to say that X ⊂ Y iff X contains no elements that are not members of Y? It seems essentially the same, and is more obviously satisfied by the empty set.

- Stuart

Yes, that's equivalent. --AxelBoldt

Merge with proper subset, and superset

Given that this is an encyclopedia and not a dictionary, is any purpose served by having 3 articles, subset, proper subset, and superset? If there's something interesting to say about one of these that doesn't fit in with another, then yes, but I doubt that this is the case. If there are no objections, I'll be combining the articles here and redirecting the others (and maybe the yet nonexistent proper superset too). -- Toby Bartels, Sunday, May 19, 2002

good idea! - iwnbap

quick edit

Corrected the inline definitions in the top section. A previous author/editor indicated that X being a subset of Y is the equivalent of 'X includes Y', and that X being a superset of Y is the equivalent of 'X is included in Y'. Actually, these are exactly opposite, and my edit reflects that. (unsigned comment by anon: 00:22, Sep 16, 2004)

Any set is a subset of itself, but not a proper subset?

I am a novice, but my sense is that Apostol (pg501 of vol II of Calculus 2nd ed) would disagree with this: "Any set is a subset of itself, but not a proper subset" (Unsigned comment by Swissdude 19:10, Apr 22, 2005)

Both statements follow immediately from the definitions, which are standard. What does Apostol say on pg501? Paul August 04:06, Apr 23, 2005 (UTC)

Proper subsets don't always have fewer elements

Likewise, if A ⊂ B, then |A| < |B|.

This is not necessarily true with infinite sets. The set of odd integers is a proper subset of the set of integers, but the two have the same number of elements (in the sense of a bijection). -- Myria 23:03, 16 November 2005 (UTC)

You arr correct. I've now fixed this. Paul August 02:56, 17 November 2005 (UTC)

Standard English usage?

Would it be inappropriate to add to the introduction a quick blurb about the use of the words "subset" and "superset" in common(ish) English? I just tried linking to this article from Finale (computer program), but the current text is more confusing than helpful to laymen.

Perhaps something like the following after the first paragraph:

In common English, the words "subset" and "superset" can describe some real-life sets of things. For example, when considering their features, the computer program Photoshop Elements can be described as a "subset" of the more full-featured program Photoshop. Likewise, Photoshop is a "superset" of Photoshop Elements.

I'm not being bold here, however, because I'm hesitant to disrupt the immediately mathematical flow of this article. Someone who's more familiar around here, please advise. SFT | Talk 05:11, 13 June 2006 (UTC)


I just re-wrote the start of the article. I pulled out the very high-level introductory information and made that the intro paragraph, and then moved everything else down into sections. This not only conforms to the style guidelines, but is much easier for non-mathemeticians to read (at least in my case). If I accidentally changed the actual meat of the article in some way, please feel free to correct any mistakes, but please don't return the page to the state where someone visiting superset has to troll through three paragraphs of math-speak to even see the word. Thanks! -Harmil 14:43, 26 July 2006 (UTC)

PS: On that note, I would suggest moving this page to Inclusion (math), as that's really the topic, not "subset". -Harmil 14:44, 26 July 2006 (UTC)

Hi Harmil. I think your edits, for the most part, improved the article. However I have rewritten the first sentence. I think it is better to establish context first, thus I have started the sentence, with the stock phrase: "In mathematics, especially in set theory, …". I also reworded the rest of the sentence, since saying the "terms … are relationships" is incorrect — terms aren't relationships. Finally I added the meaning of inclusion.
While there is some sense in your suggestion of moving the article, I think for several reasons the article should probably remain here.
Paul August 17:01, 26 July 2006 (UTC)
What are the reasons? It doesn't actually appear to be an article about the specific topic of a subset, but of the general topic of inclusion. Why would the title of the article be subset? -Harmil 20:38, 26 July 2006 (UTC)
Why do you say that this article is not about subsets? In what way is it not? It makes no sense to say the article is about inclusion and not about the concept of subset, since the two ideas are inextricably related. Any article about one will necessarilly be about the other. It is like saying our article on divisor is really about divisibility instead. The article is about both "subset" and "subsetness" (i.e. inclusion). Now in theory the article could be reasonably titled "subset" or "inclusion". However I really think that "subset" is better, (for the same reason I think that "divisor" is better). The noun "subset" is more concrete than "inclusion", occurs much more often (and thus easier to link), and an article oriented toward the concept of subset, rather than inclusion, is what the reader will expect when they follow the link: subset. Paul August 03:03, 27 July 2006 (UTC)

Wrong symbol for proper superset?

Under definitions, is the wrong symbol used to show that B is a proper superset of A? Set theory was a long time ago for me, but that looks wrong - it appears to be the same symbol as used for proper subset. Pawl Kennedy 15:56, 9 October 2006 (UTC)

Having heard no discussion, I was bold and made the change, but the symbol size doesn't match the rest of the text of the paragraph; someone more familiar with special characters can edit this. Pawl 18:01, 17 October 2006 (UTC)

Wikipedia mathematicians have previously discussed subset symbols, and adopted a convention for use in our articles. However, that is an internal decision, and we do not discuss such matters in articles themselves. In this article, we quite properly document the choice of conventions in published mathematics, without advocacy.
FYI, I keep a personal page with a huge table of mathematical characters available for copy-and-paste. The main omission is the category of special alphabet characters, such as fraktur or script or blackboard bold, though a handful are there. --KSmrqT 05:58, 19 October 2006 (UTC)

Kindly don't BLANK

I revamped the definition to include formal definitions, as well as English renditions of those notations, making the definition both rigorous and accessible. At least that is the intention. It can certainly be improved upon and tweaked - however large scale reversion based on personal preference is not acceptable. No attempt was made to explain why the reversion was better other than "IMHO". Care to explain the why? - Vedexent (talk) - 01:14, 14 November 2006 (UTC)

Not speaking for Paul here -- but there is no special presumption in favor of making articles longer. It is legitimate to remove text on the grounds that, while there's nothing wrong in it, it doesn't improve the article. Otherwise any hope of delivering a tight, focused narrative would be out the window.
Whether or not your changes did improve the article is, of course, a matter that can be discussed. --Trovatore 01:28, 14 November 2006 (UTC)
And I guess that is what I would like here - a discussion of what best suits the article. That at least is objective and non-partisan. I didn't so much object to the removal of my edit per se - although obviously I would prefer it - it was the arbitrary nature of the reversion without explanation, based apparently on personal preference.
The reason I edited it the way I did is as follows: The English-only rendition attempted to do two separate things. It attempts to be a rigorous definition, and it attempts to be accessible to the general public. This is the approach you'll find in many math texts, but it tends to be less than ideal as English is not designed to deliver rigorous mathematical definitions. What you usually end up with is something that one can "puzzle out" in English, and derive something a little more rigorous from. In other words, it tends to be a hybrid that does neither task very well.
In this case, I tried to split the semi-rigorous, semi-accessible definitions in rigorous notation and accessible English, while preserving the points made in the original text (I'm a big one for not losing article information).
To me, this accomplishes both "tasks" more accurately. - Vedexent (talk) - 01:41, 14 November 2006 (UTC)
Y'all just don't 'get it, do ya? "I agree with Paul" is not a reason or argument based on function or structure, or even English. It's personal preference. Truth through aesthetics and democracy? Lots of luck there. - Vedexent (talk) - 01:44, 14 November 2006 (UTC)
Vedexent, try dialing it back a little. Yes, the evaluation is subjective; it has to be. How a reader responds to the article is also subjective, and improving that response is our goal. "Democracy" kind of sort of -- adhocracy is the more accurate term here (one I quite like). --Trovatore 01:52, 14 November 2006 (UTC)

Hi Vedexent. I'm sorry my revert upset you. I'm willing to discuss your ideas on how to improve this article. Paul August 02:41, 14 November 2006 (UTC)


THE SYMBOLS ON THIS ARTICLE ARE NOT CORRECT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! —Preceding unsigned comment added by (talk) 16:52, 7 September 2007 (UTC)

Most of the symbols used in the article don't show up correctly in IE (but they do in Firefox). Can someone replace them with the correct TeX ones? Bubba73 (talk), 20:41, 14 February 2007 (UTC)

The symbols also show up just fine in Netscape 7.2, but not in IE 7.0. — Loadmaster 14:47, 11 September 2007 (UTC)

Data Types??

'In formal definitions, sets can be either numerical or alphanumerical, including the use of symbols. Regardless of the types of data inside sets, in order for a subset to be defined, both sets must have the same data type:...'

Isn't this completely wrong? I can have a set A of all true statements and a set B of all oranges and clearly .InformationSpace 06:55, 21 March 2007 (UTC)

Ok. Nobody has said anything and this statement is just so clearly wrong, and inconsistent with definitions of subsets that I'm going to remove itInformationSpace 23:40, 21 March 2007 (UTC)

I think it's meant to refer to the fact that in order for there to be a subset, it has to be of the same "type"; if they aren't of the same form, i.e real numbers, then obviously, by definition, it isn't a subset. You didn't need to revert a huge section. 11:41, 26 March 2007 (UTC)

An important distinction: If a relation, such as , is not defined for then has no meaning. If then is defined and is false! InformationSpace 01:14, 29 March 2007 (UTC)

InformationSpace is quite right here. The passage he removed was frankly nonsense ("numerical or alphanumerical"?). --Trovatore 01:25, 29 March 2007 (UTC)

mathematical markup? Sets as lists!??

Ok. I also spotted this:

'When used in mathematical markup, two sets are generally defined as A and B, and constructed with finite or infinite lists of data -- the subset is then used as a means of conveying that one set is a "condensed" version of the other, and is shown as "belonging" or "existing within" another element, through the symbol \subset indicating via its rounded-arrow shape the superset or "parent" pointing to the subset, or "child".'

Which I'm going to remove also. It is pretty clear that the person who wrote these does not have a strong background in mathematics---perhaps with more of a 'computer science' background. As sets and subsets really 'belong' to mathematics we really need some respect for mathematics when discussing them. InformationSpace 23:56, 21 March 2007 (UTC)

Ø subset proof

Proposition 1 trivially proves that the empty set is a proper subset of every other set. But are there consistent set theories where Ø is not a subset of all (or any) set? I'm not sure how useful these theories would be, but I'm just curious. — Loadmaster 14:44, 11 September 2007 (UTC)

Includes vs contains

If find it confusing that the intro uses "is contained inside" and "containment" as words for "is a subset of", that is , also mentioning "inclusion". The rest of the article uses the "includes/is included" terminology. Isn't "is contained" a word that should be rather used for only? For instance, if we can say that {1} contains 1, then isn't it confusing to also say that it contains (is a non-strict subset of) {1}? Daniel Bonniot de Ruisselet 09:23, 2 November 2007 (UTC)

Venn Diagram

I know what a subset is, and still had a hard time understanding what the diagram is saying. Would anyone find it helpful? Oded (talk) 18:13, 4 April 2008 (UTC)

I suppose it's possible that someone would find it helpful, but I think more people would just find it confusing. Looks like someone's idiosyncratic take on the thing -- most of the time we talk about whether something is or is not in a set, not whether it's a "possible situation" for it to be in the set. I'm going to remove it. --Trovatore (talk) 07:36, 6 April 2008 (UTC)

Proper subsets should be explained

I realize that there once was a separate article for "proper subset" that has since been merged into this one, and proper subset now redirects here. Unfortunately the definition of "proper subset" has been lost in the process. Could somebody please add it again? I'm not enough of a mathematician to do so myself. -- Scratchy (talk) 02:08, 8 October 2008 (UTC)

It's there. Look at the second para of the "Definitions" section. --Trovatore (talk) 03:05, 8 October 2008 (UTC)

Deleted section: Properties

An edit on 2008-05-12 by removed the entire section titled "Properties". Was this a valid edit or should it be reverted? | Loadmaster (talk) 05:38, 26 January 2009 (UTC)

It was a valid edit. The paragraph was very idiosyncratic and hardly salvageable. --Hans Adler (talk) 08:33, 26 January 2009 (UTC)

Limits of subsets

It is usefull to know that A_{n} \uparrow A means A_{n} \supseteq A_{n+1} and A = \bigcup_{n=1}^{\infty}A_{n}. Simirarly A_{n} \downarrow A means A_{n+1} \supseteq A_{n} and A = \bigcap_{n=1}^{\infty}A_{n}. —Preceding unsigned comment added by Boucekv (talkcontribs) 11:24, 5 May 2011 (UTC)

partial order and power set

sorry I'm new to this, but this line got me confused

For any set S, the inclusion relation ⊆ is a partial order on the set \mathcal{P}(S) of all subsets of S (the power set of S).

it means it is a partial order of the powerset of S; or a partial order of the powerset of every subset of S; or both? the end half of the line could be just

For any set S, the inclusion relation ⊆ is a partial order on the set \mathcal{P}(S) (the power set of S). or For any set S, the inclusion relation ⊆ is a partial order on \mathcal{P}(S) (the power set) of all subsets of S.

does it make any sense? thanks — Preceding unsigned comment added by (talk) 14:46, 23 August 2012 (UTC)

The power set operator takes a single set S as input, and produces a single set consisting of all the subsets of S as output. The inclusion relation ⊆ is a partial order on the power set of S, which is the same as saying that the inclusion relation ⊆ is a partial order on the set of all subsets of S.

There is no such thing as "the power set of all subsets of S". — Preceding unsigned comment added by (talk) 23:04, 3 February 2013 (UTC)

Definition of Improper Subet

This should be included here. --Primus1x (talk) 17:37, 4 January 2014 (UTC)

There is no mathematical concept called "improper subset". --Trovatore (talk) 22:44, 4 January 2014 (UTC)
Perhaps Primus1x meant trivial subset? I've seen that term used often enough to warrant a mention here, if only to clarify the distinction between it and proper subset. Bill Cherowitzo (talk) 04:38, 5 January 2014 (UTC)
Hmm, can we find it in high-quality sources? A simple-minded search finds hits in WikiAnswers, which I think don't really count, and at least one that seems to be about a quite different concept from graph theory. I'm not sure it counts as standard terminology. --Trovatore (talk) 08:06, 5 January 2014 (UTC)
I thought you might ask so I rummaged through a number of "Intro to Proofs" texts to find some citations. My imagination must be running wild, in the 10-12 books that I looked at, only one defined a trivial subset (the empty set or the set itself) and that was in a text that I've hardly ever looked at (Foulis & Munem: After Calculus: Analysis). However, I did come across three instances of the use of improper subset (as in the expression, A is the only improper subset of itself). One of these (Smith, Eggen, St. Andre: A Transition to Advanced Mathematics) also footnoted that, "In some settings, the subset ∅ of A is also considered to be an improper subset of A." I don't know why I had the impression that "trivial subset" was used more often and I would not advocate its inclusion in this article (in its defense, it does permit one to say, "... a nontrivial subset" instead of "... a nonempty proper subset", but that is not much of a shortening). Bill Cherowitzo (talk) 18:22, 5 January 2014 (UTC)