Talk:Relation (mathematics)

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02-May-09: The page should be converted into a short overview of the topic, as already named appropriately, to describe "Relation (mathematics)". -Wikid77 (talk) 03:48, 2 May 2009 (UTC)[reply]

02-May-09: I suggest merging small sub-articles, such as "directed relation", into the article "relation (mathematics)" but rewritten as a short general introduction to relations, with small sections to explain various types of mathematical relations, and links to the larger sub-articles. If the page "relation (mathematics)" were treated as merely a disambiguation page, then each small sub-article (remaining stubs for 2 years) would have no place to be merged. Instead, the page "relation (mathematics)" should be rewritten as a short introduction to relations, with small sections to explain various types of mathematical relations (allowing text to be merged onto that page). -Wikid77 (talk) 03:48, 2 May 2009 (UTC)[reply]

I found two such articles and disposed of them differently. The problem I see with your approach is that it would deal almost exclusively with binary relations. The current article is a tiny content fork of finitary relation and binary relation, and we should solve this problem, rather than making it even worse. --Hans Adler (talk) 23:41, 14 May 2009 (UTC)[reply]

I just looked at the history of the current article. I wasn't aware that you had only recently taken the redirect and turned it into this list. Since this is a fresh content fork, I am reverting this bold move now, per WP:BRD. --Hans Adler (talk) 13:53, 15 May 2009 (UTC)[reply]

Just to clarify in case you are not aware: All but two of the links in your list are to articles on properties that a binary relation may or may not have. The remaining links were for binary relations in general, and for the more general finitary relations. In mathematics, "relation" almost always refers to finitary relations, but most of the incoming links (except for those from Template:Logic, which I just updated) will always be for binary relations.

I think I agree with your idea to write an article about properties of binary relations. But it should be a sub-article of Binary relation, probably the "main article" for Binary relation#Relations over a set. --Hans Adler (talk) 14:07, 15 May 2009 (UTC)[reply]

This page, "Relation (mathematics)", got a fresh start as a redirect on 2009-04-21, when the article at "Relation (mathematics)" was moved to "Finitary relation" (replacing a redirect there). On 2009-05-02, someone made it into an article stub (though it highly resembled a disambiguation page). On 2009-05-15, someone changed it back to a redirect, but to "Binary relation". A better target MIGHT be Relation (disambiguation)#Mathematics, but one would have to check first whether that change would work with the 166 articles that link to "Relation (mathematics)" – whether they refer to "binary relation" specifically, "mathematical relation" in general, or even something else. This is a little beyond me because there are many "relation" articles and I don't know their taxonomy.

Similar articles that also redirect to "Relation (mathematics)" include "Mathematical relation", "Mathematical relationship", "MathematicalRelation", "Relation (math)", "Relation (mathematics)", "Relation symbol", "Relational mathematics", and "Correspondence (mathematics)". (I haven't counted uses of these.) (Some might want different targets.) - A876 (talk) 19:10, 11 June 2019 (UTC)[reply]

This article originates from a copy of Binary relation, and is intended to become a more introductory version of the latter. Here are some suggestions:

1. Start the (lead of the) article with explanations about homogeneous binary relations (without using that name explicitly) since they are the most popular subclass of relations.
2. Add a section "Generalizations" at the article end to mention (1) non-homogeneous binary relations and (2) n-ary relations for n≠2. I guess, 1-2 examples, and a {{main}} link is sufficient for each of them.
3. Thin out the present stuff by removing advanced issues, like Relation_(mathematics)#Matrix representation, Relation_(mathematics)#Sets versus classes, Relation_(mathematics)#Enumeration, and the less popular properties of homogeneous binary relations (Coreflexive, [Left|Right|] quasi-reflexive, Antitransitive, Co-transitive, Quasitransitive, Transitivity of incomparability, [Left|Right|] Euclidean, Set-like)
4. Add more examples, e.g. from https://en.wikipedia.org/w/index.php?title=Draft:Correspondence_(mathematics)&oldid=1010046066 (I didn't check if there are useful ones). To my experience, family relations is a good example domain for non-methematicians.
5. I'd suggest tothe order Relation_(mathematics)#Definition, Relation_(mathematics)#Properties (concerns homogeneous binary relations), Relation_(mathematics)#Special types of binary relations (concerns all binary relations), Relation_(mathematics)#Operations on binary relations (concerns all binary relations), Relation_(mathematics)#Operations (concerns homogeneous binary relations), without any particular "Examples" section - plenty of examples (from different application areas) should be given in each of the previous sections, instead.

Pinging the participants from Talk:Binary_relation#Merge_with_Heterogeneous_relation: @Rgdboer, D.Lazard, and TakuyaMurata: - Jochen Burghardt (talk) 16:15, 15 June 2021 (UTC)[reply]

@Jochen Burghardt: I also noticed that finitary relations and ternary relations aren't mentioned anywhere in this article. Should these topics be described in this article as well? Jarble (talk) 14:11, 11 September 2021 (UTC)[reply]

@Jarble: I'd suggest to mention them briefly under "Generalizations". - Jochen Burghardt (talk) 16:51, 11 September 2021 (UTC)[reply]

Since you reminded me to this article, I started to implement the above suggestions, with the most easy task, viz. the deletions (item 3). - Jochen Burghardt (talk) 17:03, 11 September 2021 (UTC)[reply]

If we find an issue in this article, should we make a fix to both it and the Binary_relation? Or focus improvements on one? Davidvandebunte (talk) 16:37, 2 November 2022 (UTC)[reply]

@Davidvandebunte: If the fix is in a text that is about some advanced issue, it would be sufficient to apply it in Binary relation. If it is clearly introductory, please apply it in both articles. If in doubt, I suggest you apply it in Binary relation, and mark the corresponding text here (Relation (mathematics) as {{dubious}}, or something similar. - Jochen Burghardt (talk) 08:26, 3 November 2022 (UTC)[reply]

I'm not sure disguising the properties of binary relations as properties of relations in general works. I was having a hard time trying to read that section with the generalization to _finitary_ relations in mind. I think it would be better to restrict that section to actual properties of _generalized_ relations or omit it entirely. 193.157.230.202 (talk) 15:34, 5 January 2023 (UTC)[reply]

This article is intended for beginners. If you are concerned about finitary relations, you'd better read binary relation, homogeneous relation, or, most adequately, finitary relation. - Jochen Burghardt (talk) 09:43, 6 January 2023 (UTC)[reply]

I wasn't concerned for myself as much as for the general reader. the article poses as an introduction to relations in general but goes then on to list properties binary relations may have as if any relation might have them, which can be misleading. 193.157.143.148 (talk) 14:11, 10 January 2023 (UTC)[reply]

The different notions are distinguished in [note 1]. - Jochen Burghardt (talk) 16:19, 10 January 2023 (UTC)[reply]

yes, I found the introduction fairly transparent about the distinction. the other sections, however, are essentially a transcription of the article on binary relations, aren't they? perhaps it would be enough to add a few "binary" attributes or point to [note 1] in a few more places. 193.157.143.148 (talk) 08:41, 11 January 2023 (UTC)[reply]

The article originated as a split-off from binary relation, see the beginning of this talk section. I started with a copy, then tried to simplify to achieve a beginners-level article. My feeling is that for the lead I arrived at that goal, but for the rest I'm still not sure how to proceed (apart from looking for more examples). Your above remark seems to confirm my impression. - Jochen Burghardt (talk) 09:27, 11 January 2023 (UTC)[reply]

the paragraph title introducing the definition of a serial relation points to an article stating that serial relations are homogeneous relations with such and such property. however, the note says that the definition can actually be generalized to heterogeneous relations, too, which does seem intuitive from the example given. there seems to be some kind of contradiction. are there any experts there who can clarify this? 193.157.230.202 (talk) 13:52, 5 January 2023 (UTC)[reply]

Relations in mathematics refer to n-ary tuples over heterogeneous domains. If the topic of this article covers only binary homogeneous relations, mentioning other types only in passing at the Generalizations section, where is the description of the most general definition?

The current article misguides readers into thinking that all relations are binary. Diego (talk) 12:36, 22 March 2023 (UTC)[reply]

Answering my own question - in the conversation above I've found that Finitary relation was initially placed at this page, and it's the topic I'm talking about. I've added it to the disambiguation hat note. Diego (talk) 13:28, 22 March 2023 (UTC)[reply]

Please add arrow diagram for first image. Yuthfghds (talk) 14:35, 24 June 2023 (UTC)[reply]

Nicholas Bourbaki used the term relation for a well-formed formula in his description of formal mathematics in The Elements of Mathematics – Theory of Sets as described at Talk:Well-formed formula#Bourbaki formulas. A relation is referred to as a property twice on page 348. The order of development, functions before products of sets, seen in Summary of Results (page 351) is contrary to the developments which starts with a product of sets, defines a relation as a subset of it, and finally a function as a type of relation. Rgdboer (talk) 02:15, 15 March 2024 (UTC)[reply]

This should be mentioned. However, it doesn't fit in this beginner-lever article. A better place is binary relation or, more generally, finitary relation. - Jochen Burghardt (talk) 12:14, 16 March 2024 (UTC)[reply]

We already have an article about binary relations. And making the jump from a binary relation to higher order relations isn't that big.

What's the point of limiting the scope of this article? Farkle Griffen (talk) 14:59, 20 August 2024 (UTC)[reply]

The idea is to have an introductory-level article relation (mathematics) and several advanced-level articles (binary relation, ...). See also the above discussion at #Restructuring towards a more introductory article. The introductory-level article starts with the most common use cases of relations, viz. homogeneous binary relations, and at its end just briefly mentions / links some generalizations. This construction has been inspired by group (mathematics) (basic) / group theory (advanced). Jochen Burghardt (talk) 06:10, 21 August 2024 (UTC)[reply]

In the Totality Properties section, the Serial relation description begins with:

For all x ∈ X, there exists some y ∈ X such that xRy. Such a relation is called a multivalued function. For example, the red and green relations in the diagram are total, but the blue one is not (as it does not relate −1 to any real number), nor is the black one (as it does not relate 2 to any real number).

The first statement is clear and correct:

   For all x ∈ X, there exists some y ∈ X such that xRy.

This is the definition of a serial relation.

The next statement about multivalued functions is misleading. There are plenty of "serial" (left-total, or total) relations that are not multivalued. In fact, the examples that follow, the red and green relations, are not multivalued at all -- every x produces only a single y. The fact that the red relation contains some x that produce the same y does not make the relation multivalued, it only makes the relation non-injective.

It is also true that the blue and black relations are not serial (as long as their domains are not restricted to x >= 0 and -1 <= x <= 1, respectively). But the blue and black relations are in fact the ones that are multivalued functions, as they produce multiple y values for each x.

Can someone explain this discrepancy? Ajackson716 (talk) 08:29, 27 October 2025 (UTC)[reply]

I'll try it: There is a narrow notion and a wide notion of multivalued function. The first notion requires ${\displaystyle f}$ to have more than one output for some input, while the second notion allows it (but doesn't require it). The lead of Multivalued function uses

• the narrow notion in its introductory sentence ("has two or more values"), but
• the wide notion in its second paragraph ("If ${\displaystyle f}$ is an ordinary function, it is a multivalued function").

Btw: this discrepancy should be fixed. The wide notion is also used at the start of section "Inverses of functions" to obtain a simple statement; using the narrow notion, one would have to say e.g. "If f : X → Y is an ordinary function, then its inverse is the ordinary or multivalued function".

Use of wide and narrow notions of the same name is somewhat common in mathematics. If an explicit distinction is needed, one often adds "properly" to indicate the narrow notion.

I believe to remeber that Douglas Hofstadter or Daniel Dennett remarked in one of their books that wide and narrow notions also occur in everyday language far more often than expected. - Jochen Burghardt (talk) 17:18, 27 October 2025 (UTC)[reply]

Thanks for another cogent explanation. Still, that entire first paragraph, with its examples of the red and green relations with respect to left-totality (which makes sense), is only harmed by the incidental inclusion of the notion of multivalued functions, of which the blue and black relations are much clearer examples. As stated it can easily be read to imply that the blue/black functions are not multivalued. I don't see a good reason to intertwine the class of multivalued functions with an example excluding clear instances of such functions in service of defining a related property. At minimum, it should be noted that while all of the relations in the graph are technically multivalued from the wide perspective, only the red and green ones are left-total (again, unless the blue and black domains are restricted to their active domains). Ajackson716 (talk) 19:37, 27 October 2025 (UTC)[reply]

What about simply deleting the sentence "Such a relation is called a multivalued function"? After all, this is a beginner's-level article (see head note), so we don't have to explain subtleties like wide and narrow meaning of "multivalued". Jochen Burghardt (talk) 21:31, 28 October 2025 (UTC)[reply]