Talk:Relation (mathematics)

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For older discussion, see Talk:Relation (mathematics)/Archive. Mangojuice 15:29, 30 January 2006 (UTC)

[edit] A cheeky but honest opinion

The was little indication in the lay-intro that a n-place relation, unlike a function with n arguments, maps only to the values TRUE and FALSE. I would imagine that the formal definition is more concrete in its extensional form, however in this case the concept is somewhat obscured from the layman, like myself. The comparison and the contrast with the concept of a function is not brought clearly to the fore.

If I am wrong, then please set me right - but clarification would be most enlightening. Oh and feel free to edit this away.

;-DanielB

[edit] I have requested intervention

I copy postings I have recently made to Oleg Alexandrov.

[edit] Relation (mathematics)

Dear Oleg,

I would not say that things are looking up for the relation (mathematics) article. The actual situation is that I (and I think Arthur Rubin, though I can't speak for him) are reluctant to start an edit war like the recent one in function (mathematics). But I will start one shortly if Awbrey does not listen to us. Please read the discussion. The definition of relation given in the intro is incorrect and does not agree with the formal definition given subsequently (which is correct for one style). Respectful consideration must be given to the definition which identifies the relation with its graph, which remains the dominant definition of this concept (and which is also simpler and so should be given first). For content rather than style (it is a draft and it needs examples and certainly some rewriting) see my User: Randall Holmes/Sandbox/relation (mathematics), in which both styles of definition are discussed from a neutral point of view. Randall Holmes 17:00, 24 January 2006 (UTC)

[edit] I request mediation on relation (mathematics)

Dear Oleg,

I request mediation on relation (mathematics). Jon Awbrey [...cut out severe language...RH] needs to listen to others, and he needs to be disabused of the idea that he can impose a program of his on the whole field of relations and related entities. Randall Holmes 17:48, 24 January 2006 (UTC)

I suggest going back to the version of 22 December 2005, supported by the current set of references, though. The additions to the technical discussion seem to me to be of little value to the exposition. Charles Matthews 19:07, 24 January 2006 (UTC)

I am amenable to that, though I would like a slight rephrase to indicate that some mathematicians really do identify the relation with its graph (it is not just a facon de parler). Randall Holmes 19:59, 24 January 2006 (UTC)

Sure - I meant only that that old text seems a better jumping-off point than the current text. Charles Matthews 20:10, 24 January 2006 (UTC)

My preference would be to return to the Dec 22 version, and to add some decent baby examples, of which there are presently none, i.e. the kind of things one sees in a first course on "discrete mathematics". The current revision is a bit weird. Sorry, Jon, I think your edits complicate the topic unecessarily. Dmharvey 21:40, 24 January 2006 (UTC)

And I stick with my comment at #The rewrite of this article by Jon Awbrey, several sections above, which agrees with what Charles and Dmharvey say. Oleg Alexandrov (talk) 01:46, 25 January 2006 (UTC)

JA: Dear Oleg, For the record, I did nothing to change the substance of the definition that was already present in this article on 22 December 2005. I repeat once again that I am not the one seeking to modify that definition. After reading that definition already in place and spending three days getting accustomed to it, I realized that it really did express the concept of relation that is most commonly used in mathematics. And anyone who actually takes the time to read it will see that it does not say that a relation is just a subset of a cartesian product. What I did realize is that the audience that I was at that time meaning to refer to a standard definition of mathematical relations would simply not be able to make use of it without doing something to make it more accessible to them. What I tried to do was nothing more than a bit of conceptual chunking and explanation of the more formidable parts of the definition that was there. If I did not succeed, no biggie, that is hardly a new experience, but I did give it a good faith try. Being asked to add to add examples, I promptly set about writing up examples, and immediately got heckled for needing several sittings to write them up. Jon Awbrey 03:44, 25 January 2006 (UTC)

As I said before, I am willing to return to something close to the text of 22 Dec (exactly to that, to begin with -- it would be an improvement). I stick to my assertion that a search of definitions found on the web, in books, and for that matter a poll of people I stop in the halls in the math department finds that most mathematicians define a relation as a set of ordered pairs. A strong minority define it in the way given in the article. I don't believe that the 22 Dec version would have to be rephrased at too much length to give a reasonable (brief) account of both definitions. This would still involve more space being given to the more complicated definition because it takes longer to explain :-) Randall Holmes 06:02, 25 January 2006 (UTC)

[edit] Examples?

Awbrey's new examples section does not give any examples! (It brings in a class of examples but doesn't exhibit one of them.) I don't think that the particular philosophical musings appearing in this section actually belong in an encyclopedia article on this subject. Randall Holmes 20:07, 24 January 2006 (UTC)

[edit] Literature on relations

[edit] Poizat (2000)

• 1. Poizat, B., A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Moses Klein (trans.), Springer-Verlag, New York, NY, 2000.

• If E is a set and m a positive integer, we call a subset R of E^m an m-ary relation with universe E. If the m-tuple = (a₁, …, a_m) in E belongs to R, we say that it satisfies the relation R; otherwise, does not satisfy R. The integer m is called the arity of the relation. (Poizat, p. 1).

• JA: This is definition of m-ary relation with universe E, and not a definition of m-ary relation. Jon Awbrey 04:34, 25 January 2006 (UTC)

[edit] Chang & Keisler (1973)

• 2. Chang, C.C., and Keisler, H.J., Model Theory, North-Holland, Amsterdam, Netherlands, 1973.

• An n-ary relation over X is a subset of Xⁿ, and a function is a binary many-one relation. (Chang & Keisler, p. 494).

• JA: This is a definition of n-ary relation over X, and not a definition of n-ary relation. Jon Awbrey 05:20, 25 January 2006 (UTC)

How do you imagine that this is a problem for me? Note that if X is a subset of Y, all relations with universe X are also relations with universe Y under this definition. I have no problem with "relation on X" as a subtype of "relation" in general (in fact, of course, being sane, I know that such subconcepts of general concepts are important). Relations with the same graph are identified under the definitions you cite here no matter what universe they are considered under. I myself might start a definition of binary relation with the sentence "A binary relation from A to B is a subset of ". This is perfectly consistent with the definition "A binary relation (unqualified) is a set of ordered pairs". Think about it. (In CS terms, types do not have to be disjoint). Randall Holmes 06:28, 25 January 2006 (UTC)

[edit] Styazhkin (1969) on De Morgan

• 3. Styazhkin, N.I., History of Mathematical Logic from Leibniz to Peano, MIT Press, Cambridge, MA, 1969.

• De Morgan's algebra of relations included the following six fundamental operations:

1. (the logical sum of the relations M and N);
2. (the logical product of these relations);
3. (the operation of obtaining the complementary relation to N; literally not-N; De Morgan called n the contrary relation);
4. (the operation of obtaining the converse relation or the converse of N);
5. (the operation of generating the relative sum of the relations M and N);
6. (the operation of generating the relative product of the relations M and N or the "composition" of those relations). (Styazhkin, p. 165).

• JA: The complement of a relation (no. 3) makes no sense if the relation is taken out of the context of a product domain. Jon Awbrey 06:45, 25 January 2006 (UTC)

De Morgan was working before modern set theory, so this is an entirely anachronistic objection. One specifies one's product domain in a given context and uses the relative complement with respect to the product domain. Again, how do you imagine that this is a problem for me? I have already remarked in published text on Wikipedia that I recognize that some concepts must be relativized to an intended domain if this is not incorporated into the definition of relation: one must say "onto Y" rather than "onto", for example (and Cieselski in a current "set theory for mathematicians" book makes exactly the same point). It's also not true that the complement makes no sense, necessarily; some of us work in NBG or Morse-Kelley set theory. Moreover, since my preferred set theory NFU has a universal set, I really have no problem with complements of relations at all :-) Randall Holmes 06:43, 25 January 2006 (UTC)

This is a dangerous argument: should I then appeal to Boole and Venn to support the position that every set A must actually have the structure (A,V), where V is the "intended universe", so that we can support the complement operation on such "framed sets"? This takes us back to type theory, and mathematicians have rejected type theory as the foundation of mathematics (and this is one of my reasons for (personally) rejecting Definition II: it amounts to using the theory of functions of Church's simple type theory in the context of untyped set theory: why?). [not to say that I don't like type theory; but everything in its place]

Are you aware that I work in alternative set theory and automated theorem proving? I think about nonstandard implementations of mathematical concepts all the time. [Look at my home page. Consider the implementation of "function" described in the slides for the talk titled "Mathematics in Three Types".] My motivation is not that the definitions you are presenting are not ones I'm used to; I have principled reasons for supporting particular definitions. But I'm not even acting as an advocate for definitions because I like them; I'm advocating presenting the definitions in use; both of them. I am entirely willing to discuss the pitfalls (of both) too. This is called "neutral point of view". Randall Holmes 06:50, 25 January 2006 (UTC)

You might enjoy the application of relation algebra to the finite axiomatization of NFU in my book. (This is a sincere invitation -- and it carries a very strong hint). Randall Holmes 06:52, 25 January 2006 (UTC)

[edit] Ullman (1980)

• 4. Ullman, J.D., Principles of Database Systems, Computer Science Press, Rockville, MD, 1980.

 CITY       | STATE      |        POP
------------o------------o------------
 San Diego  | Texas      |       4490
 Miami      | Oklahoma   |      13880
 Pittsburg  | Iowa       |        509
            |            |
        Fig. 3.1.  A Relation

• JA: I am presently just collecting examples of usage, but the point about "loose talk" is the divergence between what people say and what they mean, say, on critically examining the implications of what they say. Jon Awbrey 13:18, 25 January 2006 (UTC)

• What exactly is the point of this example? This is the definition I favor (a relation is a set of ordered pairs). Note that any set of ordered pairs is a subset of a product domain. "A relation is a subset of the Cartesian product of one or more domains" is synonymous with "A relation is a set of n-tuples" (and on the usual definition all n-tuples are ordered pairs). There are no sets which are not covered by some product domain (and this continues to be true if one extends to classes). Randall Holmes 13:05, 25 January 2006 (UTC)

To clarify why I am unmoved by these examples, here is a definition by Randall Holmes:

Where A and B are sets, we define a relation graph from A to B  as a 
subset of .  According to one school of thought, a relation is   
defined as a relation graph (and so any set of ordered pairs is a relation).  According to 
another school of thought, a relation is defined as an ordered triple (A,B,G),
where G is a relation graph from A to B.

Of course, this is a definition of binary relation (which subsumes all relations under Definition I but not under Definition II). Note that I do start out by mentioning product domains; but if the product domain is not a feature of the function as an object (as it is not here or in your definitions of this section) then "relation" in general reduces to "set of tuples". Randall Holmes 13:26, 25 January 2006 (UTC)

Did you read my remarks on "loose talk" above? I'm every bit as annoyed by it as you are, and it is prevalent (both among users of Definition I and users of Definition II): each group wants to have their cake and eat it, too. Randall Holmes 13:28, 25 January 2006 (UTC)

[edit] Mostow, Sampson, Meyer (1963)

• JA: My college roommate used to call me "Packrat". He said it was because I saved all my textbooks instead of selling them back at the end of the term. Here is a standard bit of liturgy from my sophomore/junior 'Abstract Algebra' book, used at UM A², circa 1970.

• 5. Mostow, G.D., Sampson, J.H., Meyer, J.-P., Fundamental Structures of Algebra, McGraw-Hill, New York, NY, 1963.

• JA: Et sic deinceps ... Jon Awbrey 20:30, 3 February 2006 (UTC)

[edit] Minsky & Papert (1969)

• JA: Honestly, folks, I don't lay awake at night making this stuff up. I have people who do that for me. Jon Awbrey 17:34, 8 February 2006 (UTC)

• 6. Minsky, M.L., and Papert, S.A., Perceptrons, An Introduction to Computational Geometry, MIT Press, Cambridge, MA, 1969. Expanded edition, 1988.

[edit] Outside opinion on this article

I haven't been a contributor to this article, yet, but I was thinking of working on it, because I can barely understand how what is written here is related to the ordinary notion of a relation... and I'm a professor who teaches discrete math! I got the point eventually, however, I have some remarks: If you look at what links here, a significant number of them are specifically talking about binary relations (for example see Set, Partial function, Topology glossary; some articles, like binary relation or Charles Peerce refer, accurately, to the generalization here, but it appears in the text as something other than "relation"). We could correct those links to go to binary relation, but this would require constant attention, because often people refer to the concept of a binary relation when they say "relation." This article does not do enough to inform the reader that very often "relation" refers to "binary relation." This leads me to propose that this article be renamed to something like "k-ary relation," and move the binary relation article to this title, perhaps with additional prominence of the reference to k-ary relations. I suspect this would be reasonable, as even in the definition here you call the object a 'k-ary relation' rather than a 'relation.'

Second, this article reads like definitions from a graduate text. Particular phrases that pop out at me include the first sentence and "It is literally the case that to change this setting is to change the relation that is being defined." It seems to me that the formal definition of a k-ary relation is actually much clearer if we just put it first, but to make it more clear, I might give the formal definition of a binary relation first, so we can see how simply it generalized. The notions of "frame" and "graph" are obviously less important than the definition and are not, despite what the text says, necessary to understand the definition. As a result, this article is not really accessible to most people who might arrive here.

Third, this article should be expanded to be more informative about things like the history of k-ary relations, and more importantly, the importance of k-ary relations: what areas of mathematics make use of k-ary relations? Are there types of k-ary relations akin to symmetric, reflexive, antisymmetric, etc. relations?

I have limited expertise on the third point, but I can help a lot with the second, and I think the first is very important. I'll be watching the talk page here for your replies, JA and RH. Mangojuice 13:57, 25 January 2006 (UTC)

We'll see what develops. I think that both of the two of us that you have addressed think that the issues you mention are important (as do the other participants). Randall Holmes 14:50, 25 January 2006 (UTC)

It is interesting to note that under the definition I recommend (which remains most common), "k-ary relation" is a special case of "binary relation". Randall Holmes 14:52, 25 January 2006 (UTC)

[edit] O Lost

• JA: Mangojuice, I also have a bent or a bias toward discrete math, especially graph theory, being brought up in the MiGhTy school founded by Harary, Palmer, Robinson, et al. So binary relations are especial favorites, partly because they are so readily visualized as graphs, bigraphs, digraphs, and their ilk. We already have an article on binary relations, and this one is supposed to be about the natural generalization of that. How general is general? Well, k-adic (k-ary) relations are the next step up. Some people like to (try to) picture k-adic relations as hypergraphs, but the way I see it, and maybe it's just the keen-ness of my mind's eyes, all the benefit of visual representation is lost in the offing. Jon Awbrey 15:26, 25 January 2006 (UTC)

• JA: There's a POV in philosophical logic (sometimes a paper of Quine is cited) that says that all relations reduce to binary relations. Get it? Everything's just a set, right? So all relations are really about the binary relation of membership. Speaking of something lost in the offing, there's something lost there. But whatever it is, it's not the way that mathematicians in their benighted state actually think, and we are talking about standards and practices here. Jon Awbrey 15:26, 25 January 2006 (UTC)

[edit] Anabasis

• JA: As a person with some training in logic and mathematics who is also interested in the history and philosophy of logic and mathematics, and whose so-called career cast him in among a diversity of research practitioners and philosophical interlocutors from a panoply of not of necessity mathematical fields, I frequently find myself trying to explain, say, what some 17th through 20th Century light in logic or mathematics knew about this or that item of mind at the time that he or she knew it. The people that I find myself talking to are intelligent and sohisticated people, the lion's share of them far more subtle than I in their overall thought, but a fair majority of them have had the log-&-math switch turned off at some point in their lives, by what I guess has more to do with faults of nurture than faults of nature.

• JA: That is the sandal that brought me to these shores on this latest occasion, if you can imagine yourself wearing it, and though a few days of earnest cogitation assured me of the accuracy and the aptness of the definition that I found there, on or about the solstice last, and though the math-half of my wit still skips according to its custom on past the foliage at the base of the cliff, up to the bolded head of the real Definition, I know that I've left my fellow castaways still drowning in the surf. And I still feel bad about that. Jon Awbrey 14:36, 25 January 2006 (UTC)

[edit] A theoretical presentation, looking at valid motivations for both definitions (looking for common ground, and not to go in the article!)

First, I'm going to talk about "binary relation", not "n-ary relation". The issues are exactly the same, and it's much easier to talk about just the binary case.

Go back to the abstraction being implemented: this is a two-place predicate, such as "x loves y" or x < y.

The underlying issue between the two approaches is whether a predicate has domain and codomain as intrinsic features. Another, very important underlying feature is that the implementation of the concept is (usually) being carried out in set theory. It is important to note that set theory (unlike many application domains) is a one-sorted theory (or at best a two-sorted theory if there are said to be classes and "set" is not a subtype of "class").

In first-order logic as most commonly presented, there is a single domain of discourse; this domain of discourse would then be domain and codomain to all relations in a theory with that domain of discourse. In multi-sorted first-order logic, every two place predicate does have a domain and a codomain as intrinsic features, taken from the available domains of discourse in the current theory. Moreover, multi-sorted logic (a logic with types) is appropriate in practice for most application areas within mathematics. But remember that set theory (in which every application area is ultimately supposed to be implemented) is a one-sorted theory.

For a set theorist, then (and for a mathematician contemplating the implementation of mathematics in set theory in general) it is entirely appropriate to define a (binary) relation as a set of ordered pairs. It is perfectly compatible with this view to remark that a subset of the cartesian product of A and B implements a relation with domain A and codomain B; as long as one notes that concepts which essentially involve domain and codomain (such as "surjective function") must be qualified with an indication of the intended domain and codomain if no such indication is incorporated into the set representing the relation.

For a mathematician who is not a set theorist (and whose viewpoint might be better coded in a type theory) the domain and codomain of a given function might be regarded as vital intrinsic features of the function. For such a mathematician, a relation from A to B might better incorporate the domain and codomain explicitly: it is then a tuple (A,B,G) where G is a subset of the cartesian product of A and B. Further, the viewpoint of category theory includes a category of sets and relations, in which the sets are the objects of the category and the relations are morphisms between the objects: the relations are naturally organized into morphism classes between pairs of objects, and might then naturally be supposed to be labelled with their domain and codomain (this may not be the right category theoretic terminology). From a logical standpoint, such a mathematician could be taken to be working in a multi-sorted logic (which is not the case for a set theorist considering the set theoretical implementation as a piece of set theory; there is an incongruity of viewpoint).

Finally, there is a middle ground: it is not necessarily the case that types in a type theory are disjoint. This is most easily understood using a CS analogy: objects of different types can carry type labels around with them (in which case the types are disjoint) or distinctions of type can be completely determined from context (in which case objects of different types may actually be implemented by the same object from an external standpoint). It is perfectly possible to have a "typed" standpoint (so one regards "relation from A to B" as a different sort of mathematical object when the choice of A and B is varied) and nonetheless view a relation from A to B as being implemented by a subset of ; the fact that some relation types are embedded in others then needs to be managed in context (one says "onto B" of a surjective function from A to B rather than just "onto", since the latter locution is ambiguous). Randall Holmes 15:21, 25 January 2006 (UTC)

the point being that both definitions have reasonable motivation. Since both occur in practice, and confusions result from not noticing the difference, both should be presented (and appropriate cautions given against formal errors arising from confusing them). Randall Holmes 15:26, 25 January 2006 (UTC)

[edit] coming from the RfC

I saw the RfC so I came over to see what the fuss is about. Before I read this article and talk page, I was kind of wondering how there could be a massive disagreement on the definition of a relation. Now I know ;-)

Caveats: I haven't worked on this article, in fact I hadn't read it before I was pointed here by the RfC. I am a working research mathematician, but not in anything foundational; my research is on PDE and SDE. Allow this to color your impressions of my comments as you will. ;-)

My impressions: I think the article, as it stands, is pretty inappropriate as an article in Wikipedia. I don't think that topics should be introduced in all generality at first, ever, but that generality should be built up after examples. This should be doubly true for a Wikipedia article. I find it inconceivable that any layman and/or undergrad (much less a high-school student) could read this and get anything out of it. My eyes glazed over sometime in the first few sentences, and I read this kind of thing for a living! Now, I am not saying that this article is wrong, or even bad, for certain contexts. I'm sure the authors put a lot of work into it and I don't want to criticize the text as itself. This would be a quite appropriate definition for a graduate text on set theory or computer science. But assuming I read the policies correctly, the purpose of any article on Wikipedia is that non-experts can find it accessible, not that one can generate the most general possible definition of a topic in the most concise way.

My suggestions: I'd suggest that the article be organized completely differently. Start with the definition of binary relation and send a link to the article on binary relation (FWIW, I read that article and think it is in a much more appropriate style for this community). Then give the definition of a k-tuple relation, and then give several concrete examples. (The examples section as it stands now is not a collection of examples by any means!) Then, finally, at the end, talk about this notion of frame and the issues on whether it needs to be specified.

And since I'm making suggestions, I'm of course willing to contribute some labor to this once concensus is reached.

--Deville 15:01, 25 January 2006 (UTC)

• JA: Deville, I don't know how, but if you know how to do it, you could well begin by making up a (DIYD)² template for this and the many similar "Attention" pages that I've seen hereabouts. Jon Awbrey 17:30, 25 January 2006 (UTC)

I've carried out the revert+addition of reference section. Charles Matthews 15:19, 25 January 2006 (UTC)

• JA: The work that I started on Examples was waylaid by the current acts of intervention. I'll get bak tuit here or elsewhere when the dust settles. Jon Awbrey 16:04, 25 January 2006 (UTC)

[edit] added brief alternative definition

As per (endless) discussion, added a description of the older definition. I don't think a discussion of the contrast between them is needed in this article; the difference becomes important for the first time in the discussion of functions. Randall Holmes 16:51, 25 January 2006 (UTC)

[edit] Issue clarification

Setting aside my 3 general comments, I want to try to think about the actual disputed issue, but I'm a little confused. Let me try to summarize:

The way I understand it is, we could make one of two definitions, either the "framed" one or the "unframed" one. The "unframed" one says that a k-ary relation is a subset of X_1 \times X_2 \times ... \times X_k, that is, a "graph" only. The "framed" one says that the relation must inherently be tied to its "frame", that is, the sets X_1, ... X_k. The functional difference between these two definitions is, effectively, that a k-ary relation, if defined "unframed," may actually be a relation not only on (X_1, ... , X_k) but also on (X'_1, ..., X'_k) where for each i, X_i \subset X'_i (or indeed, in some cases, where the sets shrink), whereas a k-ary relation if defined "framed" is only a relation for that particular (X_1, ..., X_k), and the relations on larger sets are related via some obvious transformation. To make this specific with an example, consider the 3-ary relation R on (Z, Z, Q) defined by (a, b, c) \in G(R) if a/b=c, and the 3-ary relation R' on (Z, Z, R) defined by (a, b, c) \in G(R') if a/b=c. Note that R and R' have the same graph, as long as we think of Q as included in R in the normal way. In the "framed" definition, R and R' are actually different relations (though you could define R' via an inclusion mapping from R), whereas in the "unframed" definition, R and R' are actually the same relation. Am I on target here, or am I missing something? Mangojuice 19:25, 25 January 2006 (UTC)

That's not about the definition, really, but about some defined equality of relations. I say framed is standard practice in mathematics, and this is relation (mathematics), so we go at it from that end. Charles Matthews 19:40, 25 January 2006 (UTC)

a literature search suggests otherwise: I find the unframed definition much more frequently. But note that I am content to have the older definition as an alternative and go with the framed one as the official first-line definition. Randall Holmes 19:49, 25 January 2006 (UTC)

just to clarify, are you saing that the definition is frequently, specifically unframed, or are you saying that many definitions don't address framed vs. unframed at all, but give a definition that is basically the unframed one? Mangojuice 19:56, 25 January 2006 (UTC)

now that I think about it, that's a bit of an unfair question. I would expect even a textbook that actively embraces the unframed notion wouldn't take the bother to explicitly reject the framed notion. :) Mangojuice 20:10, 25 January 2006 (UTC)

70 percent of sources in my unscientific survey give the unframed definition (they may say "A relation from A to B is a subset of the cartesian product of A and B", but that is the unframed definition: a relation is still the same as its graph under this definition). The remaining 30 percent mostly define "graph of a relation" as the set of ordered pairs and do not say what the relation itself is at all. Just one of the books I looked at actually gave the framed definition. Now one should look at "surjection": this is a tale of shame. Too many books which officially use the unframed definition then cite "surjective" or "onto" as an unqualified property of functions; this is an error, as under a definition which identifies function and graph, one must say "surjection from A to B or "onto B". I have a very recent set theory book which does this correctly. Randall Holmes 21:33, 25 January 2006 (UTC)

[edit] Renaming things?

• FYI, in case anyone is reading this page and not Talk:Relation (mathematics), I have proposed that we consider moving Binary relation to Relation (mathematics) and moving what is currently in Relation (mathematics) to n-ary relation. See the discussion at Talk:Relation (mathematics); in the interest of consolidating, let's have the discussion over there. Mangojuice 19:08, 25 January 2006 (UTC)

You have breached this in a confusing way, by copying text from binary relation to here. Please don't do that kind of thing. And I'm opposed to the moves. Charles Matthews 22:08, 25 January 2006 (UTC)

• JA: Mangojuice, I haven't been messing with the binary relation article -- sufficient unto the day and all -- I only scanned it one or twice, but it seemed more or less okay for the time being. So I'll copy your remark and answer it here. I was getting to this point in my reply above, but got distracted.

• JA: The question was: How general is general? And I mentioned a few things about the finite-place case of k-adic (k-ary) relations -- I use k for the arity of the predicate or relation that I have in mind, always saving n for whatever ambient dimensionality (TBA) comes along, and I'm thinking on analogy with differential forms and such, where there's always an (n choose k) parameter in the works.

• JA: Where was I? Right, general. Well, the finite dim case is nowhere near enough for some folks, count me in on a bad day, as countable and uncountable situations quickly arise. Even more bizarre, it's getting less and less unheard of these days to wrestle with "relations without arity", that is, you can have all different lengths of tuples, with maybe no upper bound on their fiendish tuplicity. What is that? Well, if you stop to think about it, which I made the mistake of doing once a couple of decadent decades back, it's the moral equivalent of a "formal language" in the comb-&-comp sense of a subset of a kleene star of a (finite) alphabet, L c A*. Obviously we don't want to put this kind of air-headedness up front, but we do need to keep it in mind, as some folks out there are really into this stuff. Jon Awbrey 19:40, 25 January 2006 (UTC)

• Since a pointer to binary relation occurs in the very first line of this article, I don't see that there is a problem. I agree with Mangojuice that most uses of relation in mathematics actually refer to binary relations (and under my archaic definition all relations (of finite arity) are binary relations anyway -- for artificial reasons, though :-) but I think that the reader should rapidly be able to find the right concept. I think there is a major divide between the finite and infinite/vague arity cases; the set theoretical implementation is quite different, since we have to use functions to implement infinite "tuples", and moreover the connection with predicates in logic (at least, the logic of languages we can actually write) is lost (though I suppose that finite vague arity might have some applications in logic). [I do know that there is infinitary logic, but it is a different enterprise]. Randall Holmes 19:53, 25 January 2006 (UTC)

• I suppose that those interested in n-ary relations would be directed there from binary relation under this proposal; no one is going to look for that title (which I think is an argument against it). Randall Holmes 19:58, 25 January 2006 (UTC)

The other possibility would be a merge of the two articles. In the merged version, we could appropriately talk about binary relations first and foremost, and then in the same article discuss the generalization of a relation. Now that I think about it, I like that idea better... it has two disadvantages though: (1) it requires more work, and (2) it would be important that the k-ary relation stuff doesn't become so bulky that it overwhelms the bulk of the binary relation stuff. I agree that if we do my proposal we'd have to pick the right title, but I think n-ary relation would do fine; we can put up redirects for any other reasonable titles. Mangojuice 20:08, 25 January 2006 (UTC)

Note that the k-ary article is now quite short. But I still don't see the need. Randall Holmes 20:38, 25 January 2006 (UTC)

So are you neutral on the idea or do you actively object? Mangojuice 20:52, 25 January 2006 (UTC)

I guess that I would have to say that I am actively opposed. A single article could have been written in the beginning, but merging them now would create a lot of trouble. Randall Holmes 21:23, 26 January 2006 (UTC)

[edit] A new name for some old ways of relating

• JA: Okay, so let's try to read, as if for the very first time, this new-old or old-new definition of a relation:

• JA: Of course, where it says "commoner in older sources" it means "old but not too old" — you have to hit the era just right to find folks dreaming that they ever talked this way.

• The era during which people talked this way began a long time ago -- and continues. But I am not promoting this definition (in the article), merely reporting that it is used (which is true and important for an encyclopedia article). Randall Holmes 20:41, 25 January 2006 (UTC)
• I could have said (and supported) "the commonest definition of relation continues to be..." but that would be perceived as hostile. I'm not interested in promoting this definition, merely in ensuring that it appears. Did you read what I said above about it (in the section on looking for common ground?) Randall Holmes 20:43, 25 January 2006 (UTC)

• JA: But this is mathematics, where a truly good-beautiful idea just does not have an expiry date, so a more important question to ask of a definition would probably be: What the heck does it define, in the first place?

• There is fashion in mathematics as in other fields, and it is at least formally possible that some fashions may be pernicious. Randall Holmes 20:49, 25 January 2006 (UTC)

• JA: Does the above definition define "relation"? How do we tell, anyway? Well, "lege, lege, lege, ora, relege, labora et invenies" is the usual rule here, And on the re*reading part of that sloggin' we may chance to see that the new rite does not define a relation either. It just cant get over the hedge with which it first set out to qualify itself, to wit, "a relation over" something or other.

• JA: Moral of the story time. If you want to form a notion of a relation detached from context, you have to do a lot or a little more work, abstracting from datasets of more concretely embedded, mathily not mafially speaking, situated relations of the form "relation over this", "relation over that", and so on.
• JA: Jon Awbrey 20:36, 25 January 2006 (UTC)

• I say that, over and over. But it is generally a little extra work; in a context where it is a lot of extra work, one is always free to use the other definition! Randall Holmes 20:45, 25 January 2006 (UTC)

• Oh, I see, you're not saying what I thought. My previous remark was based in understanding you be saying (as I do) that to use the context free version requires one to qualify various predicates of the relation which do depend on intended domain and codomain. You appear to be saying on the second reading that I can't really be defining relation in this way because I chose to define "relation from A to B" first. But I am. For one thing, the general case of relation occurs as an instance of this definition as well as an abstraction from it, if the universal set or class occurs: "relation from V to V" is "binary relation". For another, just read my book, pp. 29-31 (the pagination might be slightly different in the online version). "Domain" and "range" are there defined as the results of operations on relations, which are defined pristinely as sets of pairs. In the function section you can see how I handle "one-to-one", "onto" and "bijection". I'm really quite remorselessly consistent. The only reason that I defined "relation from A to B" first was to make it easy to frame the other definition as an alternative! Randall Holmes 21:05, 25 January 2006 (UTC)

• JA: But we're not talking about the bottle and vintage of axiomatic set theory that U or I or the Mathematician on the Moon has acquired a taste for. We're talking about the boys, er, persons in the hall, what you might call T.A. Mits, the average mathematician in the study, T.C. Mits' less wealthy sister. I already know what T.A. uses for set theory, its her own, her native brand. Jon Awbrey 21:22, 25 January 2006 (UTC)

• JA: The point is that this definition don't hunt. It stays in the hedge that it started from and does nothing that the other doesn't do, except for the confusing people about what's being defined part. And that's a bad thing. Jon Awbrey 20:52, 25 January 2006 (UTC)

• Independently of whether it is good or bad, it is still in use (and in fact, as far as I can tell, still more common than the framed definition). That is enough to settle the question as to whether it should be mentioned in an encyclopedia article. Your assertions about what it does and doesn't do are simply wrong. Randall Holmes 21:07, 25 January 2006 (UTC)

• As an outsider on this debate, let me offer the following. The core of the issue is whether the unframed definition is actually an alternate definition, or just a less precise one. What I'm getting at is, I agree that many books would not be so precise as to define frames when defining relations, but that doesn't mean that they (1) reject the idea, or (2) (equivalently) equate a relation with its graph even when the context is vague. If some significant source actually do expicitly reject the idea, it is "used." But if not, then mentioning the unframed definition as an alternative suggests that there's some disagreement on the point. If in fact virtually all sources that don't give frames don't reject the idea there wouldn't be a disagreement. The difference I'm bringing up is whether the unframed definition is a minority POV vs a fringe POV. Keep in mind, it's not just what the source SAYS, it's the POV they have. Mangojuice 21:18, 25 January 2006 (UTC)

• They are definitely different definitions. Moreover, the unframed definition is dominant, in the sense that it is used far more (try a literature search): it is the majority position, not the minority, and it is the historical implementation of relation in set theory. The new definition is gaining ground, but it still does not hold the field. Randall Holmes 21:24, 25 January 2006 (UTC)

to be more precise: a source that says "A relation is a set of ordered pairs" repudiates the framed definition, because under the framed definition a relation is not a set of ordered pairs. It is as simple as that. Randall Holmes 21:26, 25 January 2006 (UTC)

• JA: This crosses over from "loose" to "sloppy". I find no respectable sources (or targets) that write such a sloppy def as this sort of out-of-context 1-liner. Jon Awbrey 21:38, 25 January 2006 (UTC)

As a precise example -- consider set-class axiomatic set theory (such as Von Neumann-Bernays-Gödel set theory or Morse-Kelly set theory) using the axiom of replacement:

If F is a (class) function whose domain is equal to a set, then its range is equal to a set.

or the alternative formulation

If F is a (class) function and x is a set then F[x] is a set.

This makes no sense if we're talking about framed functions. Arthur Rubin | (talk) 21:35, 25 January 2006 (UTC)

• When I am feeling uncharitable, I think that the whole issue arises from the mathematical community accepting one bad definition: this is to parse "a bijection is a one-to-one and onto function from A to B" as "a bijection is a (one to one and onto function) from A to B rather than the (IMHO correct) "a bijection is a one-to-one and onto (function from A to B). "Surjective" is not a property of functions: functions have unique ranges, but not unique codomains. Without this one misconception, the issue would not arise. Randall Holmes 21:44, 25 January 2006 (UTC)

[edit] Re: Spect, Retro: Spect, Pro: Spect

• JA: Yes, the word "respectable" was ill-chosen. I retract that. Halmos and the appendix of Kelley were my first two brushes with set theory, and I get all teary-eyed just looking into them again. But I respect all sorts of people for all sorts of reasons that have nothing to with how much use I can make of their ways of thinking. What's in NST is just no longer adequate for my day-to-day needs. And putting aside the fact that I probably derive a different moral from Kelley, what's in that appendix is a beautiful luxury that hangs on the wall in its golden, er, unframed state and has never helped me with any of the real math problems that I have ever had. As much joy as (axiomatic) sets can be, I know as a practical matter that, in a pinch, I always trust to my own naivete, and not some other book's naivete. I do not try to say why, I am just being honest about the way it is.

• JA: But we're supposed be addressing the sorts of material that is found in "discrete maths", remember? At least for the first hundred kbytes or so. Jon Awbrey 22:34, 25 January 2006 (UTC)

• • So you want me to count occurrences of the sentence "A relation is a set of ordered pairs" in the discrete math texts? It's a common definition there, too. Randall Holmes 22:47, 25 January 2006 (UTC)

[edit] A relation is a set of ordered pairs: batting 1000 so far. Many many respectable references.

JA said above: "JA: This crosses over from "loose" to "sloppy". I find no respectable sources (or targets) that write such a sloppy def as this sort of out-of-context 1-liner. Jon Awbrey 21:38, 25 January 2006 (UTC)"

• Sorry, Jon, it's common. [...unkind rhetorical question deleted...RH] Jean Rubin, Set Theory for the Working Mathematician, p. 51. It's written in symbols, but that's what it says. Randall Holmes 21:48, 25 January 2006 (UTC)

• Words, too: "A relation, then, will be defined as a class of ordered pairs". Randall Holmes 21:50, 25 January 2006 (UTC)

• Jech, Thomas, Set Theory (a major authority), p. 8: "An n-ary relation R is a set of n-tuples". [...unkind rhetorical question deleted...RH]Randall Holmes 21:53, 25 January 2006 (UTC)

• Drake and Singh, Intermediate Set Theory, p. 215: "Rel r for r is a relation, i.e., a set of ordered pairs." Do you want more? I've got a shelf full. (this is 1996). Randall Holmes 21:55, 25 January 2006 (UTC)

• Halmos, Naive set theory, p. 26: "We hereby define a relation as a set of ordered pairs". Old but good. [...Unkind rhetorical question deleted...RH]

• Quine, W. v. O., Mathematical Logic, p. 198, "it is thus convenient to think of each relation as a class of pairs of elements". Calling some people unrespectable rebounds on you.

• Handbook of Mathematical Logic, Jon Barwise, ed., in an article by Shoenfeld, p. 329, "Set theory deals not only with sets, but also with relations and functions, which are sets of ordered pairs".

Set theory texts are easily accessible to me. I also have a shelf full of undergraduate discrete math texts (I teach the stuff). Some of them, but only a minority, now use the framed definition. Would you like more page references? Randall Holmes 22:11, 25 January 2006 (UTC)

My library search was not of set theory titles: I looked at set theoretical preliminaries in books on a wide range of mathematical subjects. 70 percent of them (and I avoided the old ratty ones) give this definition (though they sometimes define "relation from A to B"). Randall Holmes 22:13, 25 January 2006 (UTC)

There is a false dichotomy here which people have tried (Awbrey and others) more than once. Axiomatic set theory...is set theory. There is no other set theory. There are some other systems, but they tend to define relations...as sets of ordered pairs! Randall Holmes 22:17, 25 January 2006 (UTC)

All the logic books on my shelves (four, and I'm only counting current ones) define "n-ary relation on A" as "subset of Aⁿ". You seem to derive some sense of justification from this. However, as soon as one says "binary relation" in general, one is presumably talking about objects which are subsets of some A². It is a theorem of set theory that for any B, there is an A such that iff...here it comes...B is a set of ordered pairs... Randall Holmes 22:35, 25 January 2006 (UTC)

I found one example of the framed definition on my shelf so far: it is given (and well-motivated) in K. Joshi, General Topology, 1983.

Is there supposed to be a consersation above? Why does it look like only one person talking? I'm confused. -lethe ^talk 04:00, 26 January 2006 (UTC)

This note is a rant, addressed to JA whose silly statement is quoted above. The gaps were occupied by going to the shelf in my office to get another book... Randall Holmes 21:29, 26 January 2006 (UTC)

There is only one person talking; this talk page has been changing so fast, sometimes people write their comments in multiple chunks. Mangojuice 13:09, 26 January 2006 (UTC)

[edit] Predicates as boolean valued functions

I have two problems with that. First, is that true? That's not the way I heard it, when I learned me some mathematical logic. As I learned it, a predicate is just a symbol used in a first (or higher) order language, and in fact it is a structure, which assigns a semantics to the language, which you might call truth assignment, I guess. But anyway, as far as I know, a predicate is not a function. Secondly, it strikes me as quite odd to define a predicate as an n-ary truth valued function as if that were somehow different than an n-ary relation. They're the same, as surely as any map X→2 is just a subset of X, up to notation. -lethe ^talk 04:20, 26 January 2006 (UTC)

• JA: I corrected the heading to something that makes sense. There is a quite common variation in usage — the word to the hip is polymorphism, perverse or otherwise — that I was in the progress of explaining before the ceiling flew away. Maybe I'll get bak tuit when the chat rheum quits humming so NP-hard. It takes a standard brand of categorical factorization diagram by which the predicate p : X -> B, where B = {0, 1}, factors through the k-dimensional coordinate space B^k. I have some old snapshots from Asciiland that I will dig up, or maybe somebody here already has one that will do the trick? Jon Awbrey 13:58, 26 January 2006 (UTC)

[edit] Merging proposal (please read)

Hey -- I propose merging this article with binary relation. I have mentioned this idea, and the idea of moving articles around, a couple of times but not gotten direct responses. Randall said he "didn't see the need," but I don't take that as opposition, it seems neutral to me. JA hasn't said anything, and I'm certainly interested in outside opinions. Again, my reason for this is that many times people will come here looking for an article on binary relations, and this article barely mentions them. What's more, I would say that the term "relation" much more often refers to a binary relation than a k-ary relation, while people rarely use the term "binary relation" to refer to binary relations. The more accurate naming would be to have the article on binary relations here and the article on k-ary relations elsewhere.

For clarity, what I propose is that the article under Relation (mathematics) talk first and foremost about binary relations, and introduce k-ary relations in a section, but still define it and give an example. This would be a merge of this article into binary relation. The current article is half as long as the binary relation article, so it would certainly have to be cut down in order to merge it. I propose that we make binary relation redirect here, but make a new main page for k-ary relation where we can have the full text of this article and go into more detail. Mangojuice 13:19, 26 January 2006 (UTC)

• JA: Strong objection. You have apparently just missed the whole point of the article and the level of generality that is appropriate to it. Jon Awbrey 14:04, 26 January 2006 (UTC)

* I totally DO get the idea that this article is about the generalization of a relation. What I'm saying is that, while that definitely is something that belongs on Wikipedia, and in fact probably does deserve its own page, it shouldn't be the "Relation (mathematics)" page, because common usage refers to _binary_ relations, specifically. I'm going to compile a list of links to this page and what they mean, so you can see the evidence here. Mangojuice 14:24, 26 January 2006 (UTC)

• JA: You needn't bother, I am quite familiar with all of the usages, abusages, along with their histories. The circumstance that "relation" defaults to "binary relation" in some people's minds is a historical anomaly that arose in part on account of the fact that (>2)-adic relations are vastly more complex than 2-adic relations and that Frege and Russell in particular gave them very short schrift. Not so with the Peirce-Schröder line of work, however. Jon Awbrey 14:58, 26 January 2006 (UTC)

* Well, I did anyway. Check out User talk:Mangojuice/relation (mathematics). I don't mean to argue that it may be better to use "relation" to refer generally to relations, while using the term "binary relation" specifically to refer to binary ones. However, look at WP:NPOV. That's a point of view, and my evidence is proof that it's not a majority POV. If you look, about half the pages that (1) use the term relation correctly at all and (2) don't list both links in a list -- about half refer to binary relations, and the ones that don't are mostly set theory topics. It's important that wikipedia be NPOV: and it's important in naming as well as in article presentation. (I wrote this earlier. Mangojuice 15:41, 26 January 2006 (UTC))

Mangojuice: On reconsideration, there's no reason this article couldn't be general without being uninformative about binary relations. I'm working on a merged version now that I hope will be satisfactory; just read it imagining that there is no article on "binary relation"... actually I think we can do this without needing a separate article for n-ary relations. I'm working on it in my user space and will post a link here when I'm done with the draft. Mangojuice 15:45, 26 January 2006 (UTC)

Neutral in principle, but not in practice: don't rock the boat!!! Randall Holmes 21:31, 26 January 2006 (UTC)

[edit] Roads to mathematics — democratic, dreaming, royal

• JA: Now that a couple of new roads to mathematics have been paved by the good intentions of the WikiWaffle chain gangs, there will be a vote on the Riemann Hypothesis Tuesday next, and we'll put that Bad Boy to bed for good. Jon Awbrey 15:26, 26 January 2006 (UTC)

I strongly advise you not to escalate a point about how to write one (1) article. Charles Matthews 15:30, 26 January 2006 (UTC)

• The metaphor resonates with me. To my colleagues I quote Plato's (or Euclid's) "There is no royal road to geometry" -- then follow it up with my own "There is no democratic road to the calculus, either (alas)". Randall Holmes 21:33, 26 January 2006 (UTC)

i.e., Mathematics is an aristocratic activity. Live with it. Randall Holmes 21:33, 26 January 2006 (UTC)

• JA: I was partly thinking of Freud's description of dreams as the "royal road to the unconscious". I wish we had expressways in mathematics, but it just doesn't seem likely anywhile soon. The association, free or else, to Plato is probably an allusion to the motto about geometry over the door, but the bit about "no royal road" is supposedly Euclid talking to Ptolemy₁.

For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. (Heath, "Euclid and the Traditions about Him", in Euclid, Elements, Thomas L. Heath (ed.)) — Perseus at Tufts

• JA: I also post here a note that I put at Talk:Topology in case set theorists and topologists no longer hang out at the same intersections:

• JA: I've been noticing this widespread misunderstanding here in WikioPolis as to the general meaning of the word "elementary" in mathematics. It all goes back to Euclid's Stoicheia (singular Stoicheion), a Greek word for many things, from the shadow of a gnomon on a sundial, to letters and numbers, to steppen stones and steps of a stairway to Platos's Heaven, to the stars, but in geometry having the specific senses glossed below:

Stoicheion. 3. the elements of proof, e.g. in general reasoning the prôtoi sullogismoi, Arist.Metaph.1014b1; in Geometry, the propositions whose proof is involved in the proof of other propositions, ib.998a26, 1014a36; title of geometrical works by Hippocrates of Chios, Leon, Theudios, and Euclid, Procl. in Euc.pp.66,67,68F.: hence applied to whatever is one, small, and capable of many uses, Arist.Metaph.1014b3; to whatever is most universal, e.g. the unit and the point, ib.6; the line and the circle, Id.Top.158b35; the topos (argument applicable to a variety of subjects), ib.120b13, al., Rh.1358a35, al.; stoicheia ta genê legousi tines Id.Metaph.1014b10; (Liddell & Scott, A Greek-English Lexicon)

• JA: Incidental Muse — Loreena McKennitt, Elemental Jon Awbrey 14:45, 1 February 2006 (UTC)

[edit] My merged version draft

Check here to see my first draft of a merged version between this and binary relation. I was aiming to be general, yet informative about the specifica case of binary relations and their importance. I included the "Sets vs. Classes" section of the binary relation article (which certainly belongs more in a general article anyway) and I grabbed the references and bibliography (which, BTW, probably need to be cut down somewhat) from this article, and generally tried to merge definitions. I added a bunch of examples of my own ... I shied away from using "toy" examples as existed in both articles... though I think the examples section could use some work, and partially repeats stuff in the intro. Some examples (for instance, the example of functions) really work better in the combined article than they would in either separate one. Comments, please! Mangojuice 19:43, 26 January 2006 (UTC)

Actually, the sets/classes issue belongs (near the end, not in the intro) in an elementary binary relations article. The fact that equality, membership, and subsets are not (set) relations unless restricted does need to come in somewhere. It is not really needed in the n-ary relations article (nor is it needed in the function article) because there are no commonly used functions or n-ary relations which are likely to cause confusion. Randall Holmes 21:37, 26 January 2006 (UTC)

What are you planning to do with this article? It does not seem to contain the informal example in the existing relation article (which is currently looking quite nice). It also contains a mathematical error: the connection with characteristic functions is not described correctly. I would leave things as they are. Randall Holmes 21:40, 26 January 2006 (UTC)

I take it back about characteristic functions; I wasn't reading carefully enough... Randall Holmes 21:42, 26 January 2006 (UTC)

Lest the intervening retraction make it unclear, I'm not interested in merging the existing articles; each of them seems to do its own job on its own level fairly well right now. Randall Holmes 21:45, 26 January 2006 (UTC)

[edit] Reduction of "relation" to "binary relation"

There are two completely different senses in which "relation" can be reduced to "binary relation". I have implicitly mentioned one and Jon Awbrey has explicitly mentioned another; there might be some confusion as to which is which, so I will spell out the two reductions and the contexts in which they work. I remain on the record as against merging the two articles. Randall Holmes 16:59, 27 January 2006 (UTC)

• Under the old set theoretical implementation of "relation" (a relation is a set of ordered pairs, or a relation from A to B is a subset of the cartesian product of A and B; pace remarks by Awbrey, these are the same definition), all n-ary relations are binary relations. This follows from the purely technical fact that under the usual definition of an n-tuple, all n-tuples are ordered pairs, too, so an n-ary relation (a set of n-tuples) is also a binary relation (a set of ordered pairs). This ceases to be true if one uses the framed definition of relation; it ceases to be true if one changes the definition of n-tuple, too; it is a strictly implementation-dependent (and therefore suspicious) identification. It should also be noted that the logical n-ary relation implemented by a given set of n-tuples is not the same logical relation as the binary relation implemented by the same set of n-tuples (obviously!). Randall Holmes 16:59, 27 January 2006 (UTC)

• It is a further observation (Awbrey seems to ascribe it to Quine, but it may be older) that under certain readily satisfied circumstances logic does not require the use of any n-ary relations with n>2 at all. Namely, if one has ordered pairs as a component of one's logic (a function symbol) there is no need to use any logical n-ary relation (n>2) at all, ever. The relation written can be replaced everywhere with the binary relation . If the foundations of mathematics are implemented in set theory, we of course have an ordered pair, and we have no primitive relations of arity higher than 2, nor (on the basis of this result) any reason ever to define a logical relation of arity higher than 2. In a type theory, this insight can only be applied if the type system is closed under cartesian product. The same argument also eliminates the need to consider logical functions with arity greater than 2. Randall Holmes 16:59, 27 January 2006 (UTC)

• There is a relationship between the two reductions: depending on exactly how n-tuples are defined in terms of pairs, the binary relation identified with an n-ary relation in the first paragraph may be the binary relation which replaces that n-ary relation in the second paragraph.

• I rather like this reduction: relations (and operations generally) of arity greater than 2 are syntactically inconvenient, though there are some ternary operators which are occasionally handy. I've probably just irrevocably propelled myself into the outer darkness from the JA standpoint :-) Randall Holmes 16:59, 27 January 2006 (UTC)

• JA: I will have to get back to this later, but just so nobody gets me wrong, let me just say for the (broken) record that I consider both of these putative "reductions" to be sophisms, though for different reasons in each case. Jon Awbrey 22:10, 27 January 2006 (UTC)
• RH: I indicated reasons why the first one should be taken with a grain of salt, but both are formally demonstrable results, one of set theory, and one of mathematical logic. But I didn't think you would like either of them :-) Randall Holmes 22:16, 27 January 2006 (UTC)

[edit] Another attempt at explanation

I'm trying to explain this again, because I feel like you guys aren't getting my real points. RH -- my merge proposal has absolutely nothing to do with the idea that n-ary relations can be thought of as binary relations. That's interesting to someone who theorizes about these things, but not to a lay reader, which is what wikipedia is for. I'm proposing the merge because very often people refer to "binary relations" simply as "relations," and that is an important aspect of the concept of relations in mathematics that is effectively receiving no coverage. Even the term "binary relations" is not explained here, it's just used. I really have nothing against the binary relations article: perhaps it is worth having two articles, but binary relations MUST be discussed in this one. Currently this article is about NON-binary relations. I would like it to be about relations in general. Mangojuice 19:12, 27 January 2006 (UTC)

• the case n=2 is included here, so we aren't talking about non-binary relations per se. But your point is reasonable, and I have no objection to the header you added. Randall Holmes 19:41, 27 January 2006 (UTC)

I consider the header the absolute minimum one can do to help this issue, sort of a stop gap, until we find a better solution. Mangojuice 20:55, 27 January 2006 (UTC)

• the reduction of "relation" to "binary relation" had come up in other contexts; this discussion was not really entirely about your proposal. After all, on the whole the substance of the previous section probably works in favor of your proposal rather than against it...Randall Holmes 19:44, 27 January 2006 (UTC)

Specific to the issue about the example, it really isn't explanatory to have these toy examples like the "X suspects Y likes Z" kind of thing. The problem with those is that they really do nothing to explain the *concept* of a relation, they simply help get a reader familiar with the formal definition and notation of relations. This isn't what the reader would want. It's not even helpful for students of the subject: if we don't use the same definition and notation their class uses, it's not helpful. Mangojuice 19:12, 27 January 2006 (UTC)

• I beg to differ; I think the example is quite useful -- but it isn't a panacea, just an example. What's wrong with getting used to the formal notation? Often the ability to manipulate the formal notation correctly precedes (and leads to) understanding of the concept. Randall Holmes 19:41, 27 January 2006 (UTC)

• It is probably impossible to teach the concept (for understanding, for one not well prepared to receive it) in an encyclopedia article. A formal definition with examples is the best one can do; when one starts trying to teach the concept (with an entirely unprepared audience in mind) one is no longer writing an encyclopedia article. This is not to say that one cannot learn a field by reading the encyclopedia; but this requires a great deal of preparation on the part of the reader. Randall Holmes 19:51, 27 January 2006 (UTC)

I hear you on that! I guess what I'm trying to say is that saying something like "Familiar concepts like equality, less than, greater than, and divides (evenly) are all basic examples of binary relations" helps by getting across what this whole relation thing is about. A toy example might be useful in addition; after all, we are introducing some notation and we might as well give an example of how to use it... but afterwards. Mangojuice 20:53, 27 January 2006 (UTC)

This reminds me very much of the discussion (some time ago) surrounding derivative. What ended up happening, which I think was a Good Thing, is that derivative is now a discussion of the high-school case (a real valued function of a real valued variable), and the cornucopia of other possible meanings of derivative are now summarised and categorised in derivative (generalizations), with appropriate links. The introductory paragraph for derivative does a good job of (1) not scaring off the high-school/undergrad students and (2) pointing more advanced readers to their intended destination. Mangojuice's observation is that the word "relation" behaves in a similar way to "derivative", and I think it's a good point. I'm somewhat in favour of Mangojuice's proposal, although not necessarily agreeing with her/his current draft text. (And I'm not intending to suggest that we should have two separate articles as in the derivative case.) Dmharvey 20:36, 27 January 2006 (UTC)

Exactly. But the way things stand, it's done the opposite way here: the relation (mathematics) article, which is the one people will probably link to, is the more difficult general one, while the link is to the basic one. I originally proposed (and still favor) the idea that this article be moved to something like 'n-ary relation' or 'relation (generalized)' and moving the binary relations article here: then the titles would be very much analogous to the way the same issue is handled under derivative. Mangojuice 21:02, 27 January 2006 (UTC)

[edit] Bibliography

On a different note, could we shorten the bibliography a bit? There are more references than are really necessary, and a lot of them are redlinks. Mangojuice 21:03, 27 January 2006 (UTC)

[edit] Is set theory a one or a many?

• JA: And while we're pondering insolubilia, we might as well take up the question as to whether set theory is a one or a many. When I was a child, much like ice cream, there were only three flavors of set theory: (1) naive, (2) the one where a set is an element, (3) the one where a set is a subset. Anyway that was my child's eye view of things, and of course, calling naive a "theory" is just as sloppy as any child with an ice cream cone, but face it, everybody still says that. But now you can get jalapeño set theory with blenderized M&M's™ if you like that sort of thing. So what's the deal here? Can it be that the store-bought axiomsets have yet to capture our intuitive conset? Jon Awbrey 22:30, 27 January 2006 (UTC)
• well, there were more flavors even then. How about New Foundations? There was also already the amusing theory of Ackermann. Nowadays I would say that there are 3 mathematically serviceable general approaches to set theory, embodied in ZFC and the associated theories with classes (and extensions with large cardinals, etc.), extensions of Jensen's variation NFU of New Foundations (and extensions), and (most recently) the powerful system of positive set theory due to Olivier Esser. Only one of these is actually much in service (and this is sociologically a good thing). There are other theories that cater to people with philosophical prejudices such as predicativism or constructivism. There is the variation of ZFC designed to ease the path to nonstandard analysis. I just wrote the article on alternative axiomatic set theories for the Stanford Encyclopedia of Philosophy, so I know more about this than I usually do (the article isn't up yet). Randall Holmes 23:44, 27 January 2006 (UTC)

but I would append the further remark that most systems (certainly the three I mentioned) provide essentially the same picture of the mathematical universe. It's all a matter of differing implementations of the same underlying abstraction. When I say that they provide the same picture, I am saying that there are ways to interpret the world of each of them in terms of the others; they certainly do not make the same surface assertions about sets! (the other two contenders both have a universal set!) Randall Holmes 23:48, 27 January 2006 (UTC)

by the way, this is probably not the right venue? Randall Holmes 23:50, 27 January 2006 (UTC)

[edit] Domain, codomain, domain of definition, range, source, target, etc.

• JA: I'm still in the process of going through as many of the pre-dusty and post-dusty books that I can find on my ultra-dusty shelves and reviewing the variorum of usages in this regard, but I am coming to the conclusion that there is no longer any choice but to accustom ourselves and our readers to the facts of life, that the usages they are a'changing, in large part it appears due to the increasing influence of category theory and computability considerations. I am desperately seeking to avoid the extremity of a word like "corange" ("corrange"?), despite the fact that it would make life so much easier for poets to finally have a word to rhyme with "orange", so here's an alternative suggestion. It would be copacetic with current practice to use "source" and "target" for the designated domains, with "domain" and "codomain" as synonyms in the 2-ary case, using "j^th image" or maybe "j^th project" for the natural projection on the j^th domain, with the synonym "range" in the appropriate context. Jon Awbrey 14:08, 28 January 2006 (UTC)

dude that is actually pretty funny. It never occurred to me that corange pseudo-rhymed with orange. I'm sure I once even heard a song called "nothing rhymes with oranges". Now I want to hear a whole poem, or a limerick or something! :-) Dmharvey 19:49, 28 January 2006 (UTC)

• JA: Quackwork Orangements Jon Awbrey 04:46, 29 January 2006 (UTC)

[edit] Not all relations are finitary, nor even have arities defined

• JA: One of the more glaring inaccuracies in the article as it currently stands is that not all relations in mathematics, as they are currently treated by those in the field, have finite arity, or even have a parameter analogous to arity defined on them. Now, you may not want dwell on this factoid at the outset of discussion, but you do not want to preclude all later discussion of it either. So a bit of careful phrasing is called for, which I had once upon a time put in place, of course. Jon Awbrey 14:18, 28 January 2006 (UTC)

• No, the scope is correct now. It is controversial whether those other things are relations at all. If they are mentioned, it should be only late in the body of the article. Randall Holmes 17:35, 28 January 2006 (UTC)

• JA: I doubt if it's necessary to mention those sorts of options in this article at all — I still don't know yet what level it will settle at — but it's not my place to close off the options either, as these things really are being taken more and more seriously of late. Jon Awbrey 17:54, 28 January 2006 (UTC)

[edit] Your note on language

Your note on language is wrong. If a mathematical definition says that something is an n-tuple, it is just that -- an n-tuple. It is the informal concept being implemented, not the formal mathematical concept, that is parcelled out into n pieces. Randall Holmes 17:35, 28 January 2006 (UTC)

• JA: Yes, I get all that. It's an informal expedient, designed to prevent some readers from closing their eyes and walking away, as I know quite well from experience that they will, and it's the easiest way possible to do this short of some de-monstrous digression into various POV's on the ontology of mathematical objects. Jon Awbrey 17:50, 28 January 2006 (UTC)

• JA: And I agree about the pleonasm of "set-theoretic product" -- I just thought I remember some people abjecting to the use of "cartesian product" earlier on? Jon Awbrey 18:00, 28 January 2006 (UTC)

• one still should not make statements that are actually false. see how you like my note on the use of n-tuples; one could add an analogy to the notion of "record" in CS datatypes, but I thought that would be more clutter.

• My personal comfort zone (when I call in mediation) has to do with statements being made which are false to mathematical fact or usage; this isn't (mostly) happening yet. But the pedegogical approach of the article is beginning to veer away from what I take others' understanding of "encyclopedia article" to be. Just a friendly reminder... Randall Holmes 20:58, 28 January 2006 (UTC)

• JA: It's not false. What you say is one POV. I have seen several others argued, and it's not worth the distraction to the reader at this point, so I leave it up in the air. My rule here is to help the reader get past a tricky wicket, not hit him/her over the head with every mallet on the field. Jon Awbrey 21:04, 28 January 2006 (UTC)

• You misunderstood me. I wasn't saying that anything in the article as it stands is false (the note on language was unequivocally false; a mathematical definition says what it says; there is some truth in what you said but it reflected a confusion of conceptual levels -- but that is not what I was talking about). I was saying that as long as content remains true, I'm not unhappy (or not too unhappy) but the way things are going may invite intervention from others. Randall Holmes 01:24, 29 January 2006 (UTC)

[edit] Embedded, included, situated relations

• JA: My sole purpose in adapting cartesian/relativistic idea of "frames of reference' to the present setting was to reduce the cognitive load on the reader from a (k+1)-tuple to a pair. Seeing as how that particular facility of language has now been expurgated from the account, there is no residual use for it. Thus, I resorted to the analogous examples in topology ("embedded"), categories and sets ("inclusion" or "included"), and several usages ranging across the recent literatures in logic, model theory, pragmatics, and cognitive psychology ("situated"). And I can't really imagine any casual readers who would be foolhardy enough to rush in where angelic dictators lay down their treads. Jon Awbrey 01:42, 29 January 2006 (UTC)

• I don't actually know what the term for this kind of definition is, but people on the talk list seemed to take naturally to "framed". What I would suggest is giving just one term for this purpose; one might ask (anyone listening?) what terms are commonly used for the contrasting forms of definition we are discussing? Giving three different terms (none of which are standard in mathematics) is confusing. Randall Holmes 02:00, 29 January 2006 (UTC)

• I'm not going to engage in an edit war about this, but I continue to think that "framed" (and just "framed") would be better terminology. It's a good metaphor. Randall Holmes 02:04, 29 January 2006 (UTC)

• JA: It was a good metaphor, but the A-frame that it was meta for got burned down in the last purge. As it is now, I think that it's helpful if the article alerts the reader to a sample of the language that he/she might actually encounter elsewhere. If it were necessary to pick one term, I guess that the most generic would be the one that reminds us of the inclusion relation and prepares us for the use of inclusion maps . Jon Awbrey

[edit] the nb

The point is that it is not a figure of speech that the framed relations are (k+1)-tuples; as defined, they really are (k+1)-tuples, and it is harmful to equivocate on that fact. What is true (and what may bother the reader) is that the tuples appear artificial; as indeed they are (it is quite clear that we don't naturally think that a relation (even a typed relation) is a tuple). This is one of the main reasons I consider this definition (and all its cousins) to be bad definitions (and especially bad in intro pedagogy)!!! What can be said (the absolutely true thing that you are trying to point out) is that this artificial device does capture a real feature of the preformal concept of "framed relation" that is being implemented: the fact that it comes in k+1 chunks. Randall Holmes 02:00, 29 January 2006 (UTC)

[edit] There's and then there's

JA: Randall, the reader advisory is doing its job as well as can be expected at that point in the text, so I count that one as "satisficed" for now. Still, a mathematical definition that says so and so is an n-tuple is not exactly the sort of thing that we would want to pin a cast in stone ontology on. Before we go about using "is" in anything that we might want to dignify as a properly "ontological" sense, as distinguished from merely a conventional or informational sense, we normally expect to see some argument for properties of invariance and so on. For example, the fact that the same information can be given as the (k+1)-tuples (X1, ..., Xk, G) or (G, X1, ..., Xk), or the pairs ((X1, ..., Xk), G) or (G, (X1, ..., Xk)), and so on, militates against reifying any one of those styles as the substance of the matter. Jon Awbrey 02:28, 29 January 2006 (UTC)

• RH: that's exactly why such definitions are bad. I never define a mathematical concept as a tuple, ever. (This sometimes leads me to frame (no pun intended) definitions in somewhat unusual ways; because I work in alternative set theory I can actually make the official definitions :-). If you define a relation in this way, you have to adopt a convention (and there is no convincing reason to pick one over the other). Once you pick a convention, you have to stick with it: that shape of tuple is a relation, forever after. That's how math works, formally. There is no kindness in giving a definition and then saying "kind of...sort of..." about it. There may be some merit in giving the definition and pointing out that we could have done it in other ways (so that the student understands that the definition really is artificial and doesn't have some inherent authority which she just doesn't get). Can you tell yet that I really think about these things? I teach philosophy of math, too :-) Randall Holmes 02:43, 29 January 2006 (UTC)
• Note that here we are actually talking about the same thing! Randall Holmes 02:48, 29 January 2006 (UTC)

• JA: Our druthers do not rule the roost here — non-reformer non-reform thyself. The persistent reader will run into these ways of talking — and if we provide them with a bit of propadeutic hermeneutic, that will be of service to them. Jon Awbrey 02:56, 29 January 2006 (UTC)

• I'm not suggesting reforming anything in the article! Randall Holmes 22:07, 29 January 2006 (UTC)

[edit] problem sentence

Gotta say that I think the intro is looking better, especially with that disclaimer (see binary relation for a more basic presentation) at the top. However, the next sentence A mathematical relation is an abstract object that captures formal aspects of a given type of concrete situation among a number of roles that things might play carries almost no information.... it's a very painful sentence. I'd rewrite it but I'm not even sure what it's trying to say. Dmharvey 11:46, 29 January 2006 (UTC)

Also, is it really necessary to say "The level of generality reflected in such a description is best approached in several stages, for example, as follows."? If that's the best way to do it, then why not just do it that way, instead of first explaining that this is what we're about to do? Dmharvey 11:48, 29 January 2006 (UTC)

[edit] Polymorphism, and plain ole perversity

• JA: I'm going to delete that remark about polymorphism for now, unless/until I can think of a non-bewildering way to explain it. Hard experience teaches that it's best to use different names, or at least different formats, for predicates (the functional kind) and sets, so I will copy the paragraph here and think on it some more. Jon Awbrey 02:36, 30 January 2006 (UTC)

Because a relation as above defines uniquely an n-ary predicate that holds for x₁, ..., x_n if (x₁, ..., x_n) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent:

• You should not do really this, because the described usage is common and should be presented. It will be more confusing if actual usages are not given... Randall Holmes 17:11, 3 February 2006 (UTC)

• JA: Yep, will probably put it back in a couple of days, byteing the bullet of a hopefuuly systematic ambiguity -- there's a story here to tell, or somewhere else to tell, about what happens when you try to get an partially automated reasoner to rechnen with human polysemeiosity, but never mind that now -- anyway, the question is just how to insulate it properly. And whether even to mention ploymorphism, as I know we deployed this forked tongue long before we ever forged that word as a hall pass for it. 22:10, 3 February 2006 (UTC)

[edit] Coming attractors, strange and otherwise

• JA: I'll set the bottle out to breathe a while before taking another draft, but here's a few notes to self and others on future party plans and residual dregs. Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] Bibliography

• JA: I don't know, but if the consensus is that this article should be "just the basics", then I can start moving some of the refs & bib to more specialized and theoretical articles. Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] Domain issues

(Text in progress, 30 January 2006) Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] Examples

• JA: It might be useful to develop the concrete examples a little more. The divisibility relation really belongs with binary relations, but it does seem to serve a useful purpose here. I have some 3-place examples that I can add later on. Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] History

• JA: Don't know how much people think should be here, but I can fold in a few notes and see how it goes. Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] To infinity and beyond

• JA: Right now the title is a misnomer, since the article as it is only covers finitary relations, and not even polyadic in the proper sense. But we can maybe wait and think on that. Jon Awbrey 11:24, 30 January 2006 (UTC)

[edit] Nonemptiness

Do sources stipulate that domains are nonempty? Not in my experience; after all, nothing goes wrong when they are empty, with either of the given definitions. Randall Holmes 19:38, 31 January 2006 (UTC)

• JA: I'm pretty sure that most pure and applied work that I can think of excludes empty domains, or puts them to one side very early in the game. And it's hardly more surprising than excluding a "factorization" like:

from a doomain where you were sorta hoping for a unique factorization theorem before doomsday. It probably depends on what you mean by "nothing goes wrong". I'm if not a pragmatist about that. Jon Awbrey 20:10, 31 January 2006 (UTC)

• JA: Sorry, Lethe, but one can hardly anticpate the Fx on every browser. For instance, neither of my browsers can read some of the symbols that some folks like so much, but you don't hear me complaining about it, as if they'd do it on purpose especially just to irritate me in particular, though I am sure that's why they do it, do you? Jon Awbrey 14:20, 3 February 2006 (UTC)

[edit] Notes & Queries

• Move query to end and reply. Jon Awbrey 14:42, 3 February 2006 (UTC)

• I am confused by the second definition of relation (mathematics) and the consequent second definition of function (mathematics). I understand that there really is a very good reason for this second definition, but I am clueless as to what that very good reason might be, so please explain, and please let the § on formal definitions end where the formal definitions end. Bo Jacoby 12:39, 3 February 2006 (UTC)

• JA: Bo, it's best to append new talk at the bottom, as I only ran across your question by accident. But I desparately need more coffee right now, so'll get bak 2 u in a sec. Jon Awbrey 14:42, 3 February 2006 (UTC)

• JA: Thanks for reminding me of this questying beast, as I've been meaning to get back to it under one of the headings above. I've actually begun to take a historical linguistic interest in Where and When, and now that we thinketh hem Why exactly did this chaunge in usage first begin to take place. By "Where" I joust mean in What parts of mathematicke, and so we come to the question "When". When did "We", Wel, some of "We", first start employing this "newefangel" forme of speche, anyway? But it's Freya's Day, TGIF, and many distraughts draw me off for the nonce. Jon Awbrey 18:48, 3 February 2006 (UTC)

• I don't think there's any very good reason for the second definition, and I'm a professional mathematical logician and set theorist... The first place (I think probably the only place in elementary mathematics) where the difference between the two definitions is important is in the definition of bijection. There is this cute idea that a bijection from A to B is a function which is "one-to-one" and "onto" (covers its target set). The problem is that from the graph of a function (the set of ordered pairs) one cannot tell what its intended target set is. Any function at all is "onto" its actual range. The only way to be able to identity a function as "onto" (a surjection) is to incorporate the intended target set as a component of the function. I believe that the correct approach is to define "surjection from A to B" as a function with range B. Then a function which is an injection from A to B and a (surjection from A to B) is a (bijection from A to B), and the definition of function (and so of relation) remains simple [note that it does not make sense to say that a function is a surjection or bijection without indicating the intended codomain under this usage). There are other reasons in more technical mathematics why one might want to consider types of functions ("functions from A to B") rather than all functions, but any interesting properties of "functions from A to B" can be expressed as properties with A and B as parameters; there is no reason to incorporate A and B into the function (or to incorporate domains and codomains into relations in general). However, all private opinions aside, the second definition is used and so should appear. Randall Holmes 17:09, 3 February 2006 (UTC)

Summary: A weak reason for the second definition to appear is that it is used somewhere in the literature. A strong reason for it to disappear is that it is confusing. The fair compromise is that it is moved to the bottom of the article tegether with a warning. Bo Jacoby 13:09, 8 February 2006 (UTC)

• JA: The second definition is used widely, both in the "purer" removes of mathematics and in the "impurer" realms of applications, for example, computer science, probability and statistics, systems engineering. People who continually self-report their lack of acquaintance with this and many other such facts speak only to their qualifications for determining the content of articles of general information. My teacherly advice to such people has always been that they open their minds, take some time to acquaint themselves with a novel subject matter, rather than rushing so rashly to reduce the subject matter to what is currently familiar to them. Jon Awbrey 14:06, 8 February 2006 (UTC)

• JA: Let me try to explain what I think is the motivation of the embedded, framed, or grounded version of the definition. This is not something that anybody ever explained to me, so I'm having to guess at what the underlying rationales might be. Unfortunately, the teachers that I had in mathematics almost uniformly (at random?) seemed to shy away from saying the reason why, so I'll have to leave it at my best guess.

• JA: Reason 1. I think that there's something analogous to the use of the function notation f : X -> Y that's going on here. A historical gloss from Cat.Work.Math comes to mind, so I will go dig that up and get back to it later today. Jon Awbrey 18:50, 8 February 2006 (UTC)

• I'm afraid, in spite of my actual preferences, that I have to back Awbrey on this. This definition is not merely advocated by a few eccentrics; it is widely attested, and actually even occurs in undergraduate discrete math books (where it does causes confusion!). So it must appear as an official definition. Alas. Randall Holmes 01:17, 11 February 2006 (UTC)

[edit] Typography

• JA: Maybe it's just the time of night, but that "damacles arrow over the vector notation" is making my eyes twitch, and they're not really vectors, anyway, but coordinate tuples, so I'm going to try replacing it with bold or roman bold and see how it looks. Jon Awbrey 06:28, 4 February 2006 (UTC)

[edit] Triadic vs. Ternary

Minor quibble: "Ternary relation" would be more in line with the terms unary and binary, which are both used extensively on Wikipedia. Also, a naive Google Test shows that "ternary relation" is used more often in the outside Web. Aragorn2 19:23, 19 April 2006 (UTC)

JA: Both are used quite a bit. "Monadic", "dyadic", "triadic" actually have a longer history, and they have a tradition of usage in some parts of logic and philosophy where they talk about 3-adic relations a lot. Using "binary" also creates havoc in computer science contexts, so it's often useful to call the relations "dyadic" there. There are whole literatures that will not change their ways, so it's best to get used to a mixed bag here. Jon Awbrey 19:56, 19 April 2006 (UTC)

Articles should be self consistent and indicate the existence of alternative terminology. --MarSch 11:52, 20 April 2006 (UTC)

[edit] Plenty of new characters

Unicode table Mathematical Operators ([1]) is full of relation symbols not discussed yet in the relation articles. Examples are: ≦, ≨, ≪, ≲, ≷, ≼ etc. Is someone able to discuss these symbols in the article? --Abdull 22:05, 28 May 2006 (UTC)

[edit] is a relation usually n-ary?

I know no author who thinks of n-ary relations (n≠2) without explicitely saying it, while I could cite many references where "binary relation on E" means "relation from E to E" (thus 2-ary, of course) and therefore (although less frequently and usually explained to avoid confusion), "binary relation on ExF" means "relation from ExF to ExF".

I'm not "against" n-ary relations (how could I...), but I would not start the article like "the notion of relation is a generalization of ... binary relation" but rather put "the notion of n-ary relation is a generalization of ..." and all that is about n-ary relations in one single paragraph like a "special case", maybe creating an extra page the day where there is enough to say about, and clearly state that this page mainly deals with the principal case, which are the 2-place relations ("from some set E to some set F, for which one has composition, reciprocal, injectivity, surjectivity, etc. etc.).

Those who are against, please give a reference where a "relation" is n-ary with n arbitrary, without this being explicitely mentioned.

As a side note, notice that apart from the definition, all examples given on that page are binary relations. — MFH:Talk 17:26, 13 October 2006 (UTC)

For one thing, there's a separate article on binary relation. I had suggested previously that we should move this article to another location, such as n-ary relation or something similar. It might be a good idea to make that suggestion on WP:RM, to get a wider input. Mangojuice^talk 20:26, 13 October 2006 (UTC)