Talk:Relation (mathematics)/Archive

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I think this article should be moved to relation (mathematics), for the same reason for which the ill-named mathematical group was moved to group (mathematics). Michael Hardy 20:12, 1 Mar 2004 (UTC)

Agree. Isomorphic 20:15, 1 Mar 2004 (UTC)
- Done. -- Walt Pohl 18:20, 16 Mar 2004 (UTC)

Inconsistent

First a relation is defined as a tuple with its graph as the last element. Then later the relation is used in a way its graph could be used in the latex formulas. -MarSch 15:25, 20 Apr 2005 (UTC)

The "inconsistency" has to do with the fact that, as mentioned in the article, a relation is often identified with its graph. I've edited the article so as to minimize the inconsistency in the article. But, unfortunately, there is an inherent inconsistentcy in standard usage. Paul August ☎ 20:47, Apr 20, 2005 (UTC)

Varied & sundried articles on relations & relatives

JA: I'm presently working on co-ordinating numerous articles and sub-articles that have to do with the logical and mathematical aspects of relations, relative terms, predicates, propositions, boolean functions, and so on. Not for the fun of it, but because intelligent discussion of these topics has simply become impossible of late, especially between the communities of mathematical, philosophical, and programming folks. The big cloud at the beginning of talking about any given thing is whether it's a "signlike" thing or an "objectlike" thing. For instance, "predicate" -- signoid or objectoid!? -- take your pick, Gimli. Here I've found that the strategic discipline used in computer science is very beneficial for cross-cultural communication, namely, to append the word "name" or "expression" to any substantive or adjective that you intend to be taken as a name or expression. For example, we can use the short form "predicate" for the formal object, and use the longer forms "predicate name", "predicate expression", "predicate formula", etc. for the semiotic-syntactic entity that denotes the corresponding object. So I'll adopt that as clarifying principle to begin. Jon Awbrey 15:36, 16 January 2006 (UTC)

JA: Bad example. We could do that for "predicate", but I will try to avoid applying the rule there, mostly just for historical reasons. A better example would be "proposition", where we can let that term denote the abstract, formal, or mathematical object, even in some contexts identifying it as a boolean-valued function, while using "propositional expression" for any signlike thing that denotes the proposition proper.

JA: I am concerned about the accessibility and utility axes of encyclopedia articles, by which I mean making useful ideas useful to readers who might have a use for them. In this connection, one of the other expositional problems that I would like to mull over for a while is the distinction between the relation and the graph of the relation. On one hand I understand the importance of getting this distinction across one way or another -- on the other hand, I know from a lot of adverse experience that a definition that begins "a k-place relation is a (k+1)-tuple ..." will have already lost 90% of the readership that is most in need of understanding what a k-place relation really is. What is wanted here is something like the arrow notation for functions f : X -> Y, that subtly forces us to regard the source X and the target Y as inherent parts of what f is. (I use to use a concept of "relational arrow", but that again is for later.) So I will think a bit on how to ease the exposition here. Jon Awbrey 18:08, 16 January 2006 (UTC)

Ramping up to relations in general

JA: I'll be trying to write several levels of exposition that ramp up in stages to a moderately general concept of relations. I'm not sure I can get to "relations without arity" in this article, but will keep it in the back of my mind as I go. I made a few cosmetic changes, subbing k for n mostly just because I always save n for some sort of ambient dimension rather than just the number of variables in a particular predicate form. I would also use the term "finitary" instead of "k-ary" if nobody thinks anybody would take it as meaning "computable". Jon Awbrey 06:26, 17 January 2006 (UTC)

JA: Well, off to dreamland for now, but tomorrow I will try to tackle that protean polymorphism issue, as I think that is one of the things that is most likely to drive a novice to despair. Jon Awbrey 06:40, 17 January 2006 (UTC)

JA: I will also make an effort to remove the reference to k's and n's from the thrust of the definition, relegating it to a parameter (that may or may not be defined for a given relation of the most general sort). This may take a few tries. Jon Awbrey 13:36, 17 January 2006 (UTC)

Which concept nests in which?

If relations have frames, then a binary relation is a specific case of the more general concept of k-ary relation. However, if a relation is identified with its graph, then k-ary relations are special cases of binary relations (in fact, a (k+1)-ary relation is a species of k-ary relation...) I am myself an unregenerate advocate of simplicity: the relation is its graph. Adding the frame to the relation as a component is analogous to sticking type-labels to things: objects should not have to carry type labels around with them, as the context in which they appear should provide cues sufficient to tell what type we are currently assigning to them. This is not a proposal to change the article (the usage described is unfortunately common), just a philosophical comment... Randall Holmes 23:46, 20 January 2006 (UTC)

JA: Randall, before we get started, look at the history, and where I came in, so that I don't have to take undeserved blame and/or credit for things that I did not contribute:

Relation 22 December 2005

Did I mention you? I was simply talking about the general difference between the two styles of defining a relation. I guess the "frame" terminology is yours (?), but the general practice of incorporating domains of relations as components is just that -- a general (though not universal) practice. No reference to any individual was intended. I have already figured out from the function (mathematics) war that it is difficult to tell when someone advocates a position and when they are simply trying to edit something already present into a better form... Randall Holmes 00:24, 23 January 2006 (UTC)

I now note (as I say at more length below) that your definition is actually apparently nonstandard in a way I had not noticed. It is not the same concept (or to be precise, not the same set theoretical implementation of the concept) that was defined in the original article! Randall Holmes 04:20, 23 January 2006 (UTC)

JA: In casual discussion, I always speak of a k-adic relation as being something of the form L c X_1 x ... x X_k, and never run into trouble myself -- not until somebody asks a question that's symptomatic of the fact that something I took for granted in so writing is really not understood in so reading. And this is the factor of context, frame, setting, or whatchamacallit that is reflected in specifying the cartesian product X_1 x ... x X_k.

JA: And if we make that context explicit, what we get is the definition of a k-adic relation as a pair (L, X_1 x ... x X_k) where L c X_1 x ... x X_k. (Notice that the relation itself is not yet name in this formulation.) That is probably how I'd express it if I started from scratch today. (As it happens, this is actually a notion that I explored during one of my decade-ent Wanderjahren all through the 80's under the heading of "relational arrows" and "intermediate categories", but that's an arrow of another quiver.)

JA: I was also tempted to call the subset the "figure" and the sequence of domains, or product over it the "ground", making the relation a "gestalt, I guess, but that would have imposed too radical a novelty on what's in place, plus reversing the F and the G, so I resisted that impulse. Like it or not "graph" is standard, and "frame" is pretty close to standard usage, as in "frame of reference". So I think that's about as good as I can do, as far as non-demolition continuity of improvement goes.

I actually like your terminology; but I prefer to identify relations with their graphs. Randall Holmes 00:24, 23 January 2006 (UTC)

The rewrite of this article by Jon Awbrey

Hi Jon. Thank you for your work on that article. Now, I am not an expert on the thing, but from reading what you wrote, it looks to me that the current version of the article is so much more complicated and hard to follow than the original version before your rewrite.

I would suggest the article flow in a different manner. First one defines what a relation is. In spite of all the text you wrote in that article, and all those definitions, a relation is no big deal, it is just a subset of a cartezian product. So that could be made clear.

Then, one could give a very simple example. After that, you could enter into all those complications, which are frame, arity, adicity, and all that.

In short, I would suggest you keep the old introduction, then give examples, then write all that which is the current introduction you wrote. Wonder what people think. Oleg Alexandrov (talk) 17:00, 22 January 2006 (UTC)

JA: Hi Oleg. I am still working on this article and will take your comments into account over the next few days, weeks? This is a fundamental and important topic, and it will be incumbent on us to make it both accurate and accessible to whomever might have a need for clarity about the concept. In the meantime, please review the entire history of the article and especially what's gone before on the discussion page, as some of what you say — for instance, "a relation is no big deal, it is just a subset of a cartezian product" — seems to reflect a lack of appreciation for both the substantive and the expository problems at hand. Time for lunch, so I'll comment more fully later in the day. Jon Awbrey 17:12, 22 January 2006 (UTC)

Yeah, but you don't need to start with the complicated stuff right away, that's my point. Start simple, give example, move on to complications later. :) Oleg Alexandrov (talk) 17:20, 22 January 2006 (UTC)

And more

JA: I get too confused by trying to follow 1-sided conversations at user talk pages, so I copy your remarks here.

OA: You wrote at talk:relation (mathematics):
JA: I'm presently working on co-ordinating numerous articles and sub-articles that have to do with the logical and mathematical aspects of relations, relative terms, predicates, propositions, boolean functions, and so on. Not for the fun of it, but because intelligent discussion of these topics has simply become impossible of late, especially between the communities of mathematical, philosophical, and programming folks.

OA: Well, I would suggest you don't rush. Contributions are always welcome of course, but I would like to say that from my experience grand schemes of things usually fail, especially if undertaken by somewhat new users. That is, maybe some caution, slowing the pace, and lot of discussion may be in order. :) Oleg Alexandrov (talk) 17:04, 22 January 2006 (UTC)

JA: I'm in no rush if you aren't. Though born again but lately as a WikiPeon, I am an old veteran of Relations. Jon Awbrey 17:28, 22 January 2006 (UTC)

OA: Great sir! Oleg Alexandrov (talk) 17:37, 22 January 2006 (UTC)

Discussion of Relation 2006

JA: Here is where I came in:

Relation (mathematics) — 22 December 2005

In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as "=" and "<" which occur in statements such as "5 < 6" or "2 + 2 = 4". It is also the fundamental notion in the relational model for databases.

Definition

A relation over the sets X₁, ..., X_n is an (n + 1)-tuple R=(X₁, ..., X_n, G(R)) where G(R) is a subset of X₁ × ... × X_n (the Cartesian product of these sets). If X=X₁=X₂=...=X_n, R is simply called a relation over X. G(R) is called the graph of R and, similar to the case of binary relations, R is often identified with its graph.

An n-ary predicate is a truth-valued function of n variables.

Jon Awbrey 23:23, 22 January 2006 (UTC)

JA: Despite its deceptive simplicity, the first sentence is nothing like a definition of "relation" in mathematics, but merely states one of the things of which a relation in mathematics is supposed to be a generalization. Without some hint of the intended direction of generalization, since there are always many directions of possible generalization for any significant concept, that remark provides very little service to the reader. It's more like a "build your own definition" (BYOD) pot-luck party kit". The multitude of use-mention confusions in the examples given next pretty much guarantee that only a reader who already understands what a relation is better than the writer of this "definition" would be able to negotiate what the writer — no doubt a distributed committee of writers — was trying to say in the first place. Jon Awbrey 23:44, 22 January 2006 (UTC)

As I said in other places, I find this definition perfectly acceptable. Next thing to do is give an example of a nontrivial relation. After that, you may delve into all those complications you now put front and center in the introduction. Oleg Alexandrov (talk) 01:34, 23 January 2006 (UTC)

JA: Are we still talking about that first sentence? That you are personally content with it does not suffice as a criterion of definition. I can think of no mathematical source that would even dignify it as a definition in form, much less a definition of relations in general. It is at best a "description", something that one might say is true of relations, but a vague description at that, and simply not definitive of their substance. If you have some sort of stylistic preference for beginning WP articles that way, fine, I guess, but please do not misinform readers or distort what is a universal mathematical practice by calling it a definition. That may be okay for "defining" Doctor Seuss or Doctor Who, but it does not qualify as a definition in mathematics. Jon Awbrey 04:02, 23 January 2006 (UTC)

I think that the text in the box above is much better than the current text. The opening sentence is not a definition, but it does reveal what kind of thing a relation is supposed to be. The second paragraph gives the exact mathematical definition (not my favorite one, but a commonly accepted one). In the current version, one has to read rather farther before getting the whole story in any form. Also, the definition given here is nonstandard (and the text is muddy enough that I didn't really see it until now, though I have looked at it before): the first component of the k-ary relation is here said to be the cartesian product of its domains rather than the k-tuple of its domains, and this is not the way it is usually done. This, at least, is probably not acceptable, any more than it would be accepable for me to follow my preference and stipulate that the relation is its graph, period. [I could change my mind about this if given concrete references to works which define relation this way] Randall Holmes 04:15, 23 January 2006 (UTC)

It is very encouraging to start an article with an informal description. This is a general purpose encyclopedia, so let's keep things simple. Good reads are by the way the math style manual and Wikipedia:Make technical articles accessible. Oleg Alexandrov (talk) 04:46, 23 January 2006 (UTC)

JA: I am perfectly happy starting an article with an informal description, so long as you don't, by commission or omission, mislead the reader by advertising it as something else. My current version, as of 2 minutes ago, begins with an informal wedge that advises the reader of its nature as such. The main problem with the article as I found it on or about the solstice last year was neither the informal level of the introduction nor the sophistication of the bolded definition, where only the bold would be emboldened to go — it was simply that there was no way of getting from one to the other, no escalator between them, as it were. I know very well the sort of glazed eye that would turn away from a definition that begins "a k-ary relation is a (k+1)-tuple ...", going away with only what was in that first paragraph as somehow a sufficient idea of "what's a relation". That is the problem that I've been working on — perfect success is no doubt unrealistic to expect, but I will keep at it a bit longer. Jon Awbrey 05:14, 23 January 2006 (UTC)

The definition using the cartesian product has a technical defect: the whole purpose of adding extra components over and above the graph is to be able to extract the domains from the relation, and with the definition using the cartesian product of the domains, one cannot determine the domains if one of them happens to be empty. This is a weird limiting case, true, but the best definitions also work in weird limiting cases. Randall Holmes 05:25, 23 January 2006 (UTC)

JA: The relation is defined as L = (F(L), G(L)), so there is no problem about extracting the domains from the relation, since the relation L determines F(L). Jon Awbrey 05:54, 23 January 2006 (UTC)

Evidence that the old definition is still dominant

I've been doing web searches on the definition of relation and the definition of n-ary relation. It appears that I'm being too diffident about my preference. With the exception of this article in Wikipedia, the dominant definition of a relation from A to B on the web is simply "subset of $A \times B$ ; I have yet to find a definition with the additional adornment of domain and codomain other than the one in this article. Similarly, the web consensus definition of n-ary relation with domains A_i is "subset of the cartesian product of the A_i's". I will keep looking, and I'm sure I'll find definitions adorned with domain and codomain, but this definition is a technical refinement (motivated by category theory originally, I believe) and could perhaps better be presented as a technical refinement... Randall Holmes 05:48, 23 January 2006 (UTC)

JA: I have already commented on this particular style of loose talk, so please see above. I talk loosely, too, at times, so what? And we've already seen with respect to the Funk Mat mess what reduces to a "Who needs WP when we already have Webster" argument.

It's not loose talk. It's sometimes necessary in the context of set theory. Look at a clean set of axioms for Von Neumann-Bernays-Gödel set theory without choice, in which the axiom of replacement uses a function rather than a property as in ZF. (I'll need to fix that article.) Arthur Rubin | (talk) 06:16, 23 January 2006 (UTC)

It is as a matter of hard historical fact the original and perhaps still the dominant formal definition of relation. It is not "loose talk". The definition adorned with domain and codomain is a relative novelty. I'm well aware that it is getting more and more common (for reasons which I consider misguided, but that in itself shouldn't affect what appears in the encyclopedia); but evidence that the older definition remains dominant is of interest. Randall Holmes 13:26, 23 January 2006 (UTC)

Discussion of Relation by a Logician User:Arthur Rubin

This is where I came in.

Relation (mathematics) — 20 January 2005

A relation is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a binary relation, such as the binary relation of 'equality' that is denoted by the sign "=" in a statement like "5 + 7 = 12" or the binary relation of 'order' that is denoted by the sign "<" in a statement like "5 < 12".

In order to approach the mathematical definition of a relation, it is useful to introduce a few preliminary notions that can serve as stepping stones to the general idea.

A relation in mathematics is defined as an object that has its existence as such within a definite context or setting. It is literally the case that to change this setting is to change the relation that is being defined. The particular type of context that is needed here is formalized as an encompasing collection from which the elements of the relation in question are chosen.

A relation L is defined by specifying two mathematical objects as its constituent parts:

The first part is called the frame of L, written frame(L) or F(L).

The second part is called the graph of L, written graph(L) or G(L).

In the special case of a finitary relation, for concreteness a k-place relation, the concepts of frame and graph are defined as follows:

The frame of L is specified by giving a sequence of k sets, X₁,…, X_k, called the domains of the relation L, and taking the frame of L to be their set-theoretic product or cartesian product F(L) = X₁ × … × X_k.

The graph of L is given by specifying a subset of this cartesian product, and taking the graph of L to be this subset, G(L) ⊆ F(L) = X₁ × … × X_k.

Strictly speaking, then, the relation L consists of a couple of things, L = (F(L), G(L)), but it is customary in loose speech to use the single name L in a systematically equivocal fashion, taking it to denote either the couple L = (F(L), G(L)) or the graph G(L). There is usually no confusion about this so long as the frame of the relation can be gathered from context.

I, too think think the version of 22 December is better. To avoid edit conflicts with JA, may I propose yet another alternate version of the introduction, although I'm not happy with the first paragraph, based on the version of 20 January referred to above.

In it's most abstract sense, a k-ary relation (where k is a positive integer) on sets X₁, …, X_k represents a property that may or may not hold between elements x₁ of X₁, …, x_k of X_k.

For a more precise definition, it is useful to introduce a few preliminary notions.

A relation L is defined by specifying two mathematical objects as its constituent parts:

The first part is called the frame of L, written frame(L) or F(L).

The second part is called the graph of L, written graph(L) or G(L), which is an arbitrary subset of the product domain, constructed from the frame.

Comment I can define a finitary relation without using the concept of function, but for an infinitary relation, the concept of function without a frame is required, or more advanced set theory than normally used to construct the frame of the domain function. I'll provide a formal definition of a framed relation using an unframed function, and we'll see whether there is any way of saving it, later.

The frame of L is specified by a function X from an arbitrary set A (called the -arity of the the relation), which, for any α in A, chooses a set X_α, called a domain.
The product domain is the formal product of the domains; the collection of all functions f on A such that, for all α in A, f(α) is in X_α.

In the special case of a finitary relation, for concreteness a k-place relation, the concepts of frame and product domain are defined as follows:

The frame of L is specified by giving a sequence of k sets, X₁,…, X_k, called the domains of the relation L.
The product domain is their set-theoretic product or cartesian product F(L) = X₁ × … × X_k.

Infinite arity relations, Relations without arity, etc.

JA: Really folks, this is not a game of aleph-one-upmanship. I am not trying to impress anyone. I am trying to provide the reader with a workable, moderately general and powerful concept of relations. I am quite familiar with these supersized sorts of generalizations, and I have taken pains that my treatment leaves the space open to them, but without distracting the novice reader with their mention at the outset. Later, Dude. Jon Awbrey 06:14, 23 January 2006 (UTC)

Your definition is wrong. The frame is not the product domain, but the sequence of domains. Arthur Rubin | (talk) 06:19, 23 January 2006 (UTC)

JA: If you take the time to read the revision history and the play-by-play commentary that I maintained on the talk page when I started this -- does anybody ever do that, I wonder? -- you will see that I began by defining the frame as the sequence of domains, but put it aside for the current purpose simply because it adds an extra complication that impedes the reader at this point in the game. I just don't think that it's helpful to raise the AC spectre at this point, for the current ends. Jon Awbrey 06:30, 23 January 2006 (UTC)

I'm willing to defer the definition for infinitary operations to the body of the article, provided that you accept the fact that the frame is the sequence of domains, rather than the product of the domains. Arthur Rubin | (talk) 06:24, 23 January 2006 (UTC)

The issue is not AC. If one of the domains is empty, it is impossible to recover the domains from your form of the frame. Randall Holmes 13:23, 23 January 2006 (UTC)

Further, the definition "a relation is a set of ordered pairs" is not "loose talk". It is the exact definition of relation which has been dominant for most of the century, and which a web search on the quoted string "a relation is" (for example) suggests is still dominant. I am a mathematicial logician and set theorist; I generally know when I am talking loosely. Randall Holmes 13:23, 23 January 2006 (UTC)

Try the web searches "a relation is a set of ordered pairs" versus "a relation is a tuple" or "a relation is a triple". The results are instructive (and I have tried various variations). Then go read mathematical sources (I'm sufficiently interested to do a library search in QA today). I think at the very least the unadorned definition should be given equal time (as I do in function (set theory). Randall Holmes 13:32, 23 January 2006 (UTC)

a concrete example of the error in Awbrey's definition

Under Awbrey's definition, the unique relation with first domain A, second domain the empty set, and third domain C is exactly the same object as the unique relation with first domain A, second domain B, and third domain the empty set, because $A \times \emptyset \times C = A \times B \times \emptyset$ : the frames of these two relations are the same set (and they both have the empty graph). Thus it is impossible to determine the domains of these relations from Awbrey's definition. The whole point of this kind of definition is that it should be possible to extract the domains from the relation. So Awbrey's definition is an unsatisfactory middle ground between the correct "category-theoretic" definition (which will draw all distinctions of this kind) and the classical definition (which would of course identify these empty relations but also makes many other identifications). If there is to be a frame at all, it must be the tuple of the domains, not their cartesian product. Randall Holmes 14:48, 23 January 2006 (UTC)

Logic & Rhetoric

JA: Ladies & Gentlemen of De Jure, I know that it's difficult to balance the complementary claims of logic and rhetoric, where I always invoke the latter in its classical good sense of 'forms of discussion that are considerate toward their interpreters', but we have to maintain an equanimous perspective in this work. If you can tell me a good reason why we should bewilder the reader at this early point with punctilios about cartesian products over empty or infinite sequences of proper classes with possibly empty sets among those sequences, then I'm perfectly happy to listen to your reasons. But frankly some oracles are talking out of both sides of their mouths in this regard, quibbling about the very mention of sets in one place and quibbling about the slighting of proper classes and trifling borderline cases in another. Jon Awbrey 14:56, 23 January 2006 (UTC)

Actual mathematical errors should be avoided (we don't need to discuss the reasons for using the tuple of domains instead of the cartesian product of domains, we just need to use the tuple -- if we are to use a "frame" at all). I think that the definition of the infinitary product should be deferred to the body of the article (late), and I was the one who introduced proper classes, and I was convinced by the other members of the jury that this should be moved to the body of the article and not mentioned in the intro. Randall Holmes 15:12, 23 January 2006 (UTC)

JA: Things like the empty algebra, empty geometry, empty graph, empty group, ad nullum infinitum, all got their 15 minutes of fame at the outset of every graduate math class — no, not that kind of class — that I ever took, and there is actually a notorious paper by these two notables: Harary, F. and Read, R. "Is the Null Graph a Pointless Concept?", Graphs and Combinatorics Conference, George Washington University, Springer-Verlag, New York, 1973. So what say you all if we simply deploy the usual hedge and stipulate "nonempty" at the appropriate loco loci? Jon Awbrey

RH: I do not understand what the problem is with replacing the cartesian product of the domains with the tuple and never saying anything about the limiting case in the article: it is just that we then know that the definition has full generality. There is no particular advantage of the cartesian product as a "frame" over the tuple of domains that I can see (of course I am biased since I regard adding a frame at all as almost pure loss). Randall Holmes 17:16, 23 January 2006 (UTC)

JA: This is naturally what I wrote write out of the box:

Relation (mathematics) — 17 December 2005

A relation $L$ is defined by specifying two mathematical objects as its constituent parts:

The first part is called the frame of $L$ , written $frame\,(L)$ or $F (L).$

The second part is called the graph of $L$ , written $graph\,(L)$ or $G (L).$

In the special case of a finitary relation, for concreteness a k-place relation, the concepts of its frame and its graph are defined as follows:

The frame of $L$ is specified by giving a sequence of $k$ sets, $X_1, \ldots , X_k$ , called the domains of the relation $L,$ with the understanding that their set-theoretic or cartesian product $X_1 \times \ldots \times X_k$ is to be taken. Under these circumstances, then, a reference to the frame of $L$ can mean either the sequence of sets or their cartesian product, since it's the same information for all practical purposes in this context.

JA: The reason that I revised this to use the product rather than the sequence was simply that the first version put a couple of distracting rhetorical kinks in discussion that I judged to be more impediment to first time readers than they were worth. There is a limit to how far a complex subject can be simplified, of course, but I'm accustomed to stepwise refinement, and we can always add yet another lamina later.

"loose talk"

In this note, I give concrete examples of "loose talk", taken from discrete math texts (I've seen examples on web sites while doing my recent survey, too). It is loose talk to define a relation as a set of ordered pairs, define the domain as the set of first coordinates of elements of the relation, define the range as the set of second coordinates of the relation, define a function as a special case of relation in the usual way, and then say that a function is surjective if it maps onto the whole of its range. It is also loose talk to define a relation as a tuple (A,B,G) and then identify xRy with "(x,y) is an element of R". This does not make it loose talk to define a relation as a set of ordered pairs: the way I deal with this in discrete math class, when students are given the simple definition of relation and function and the (now wrong) definition of "surjective", is to point out both possible solutions: I say that one can (but I don't) deal with this by defining a function as a triple containing its domain and intended range (I do not say "codomain"; it is never in the book), but one also can (as I do) fix the problem by saying "onto B" rather than simply "onto": one defines "surjective function from A to B rather than "surjective function". This works without the baroque complication of the structure of relations and functions entailed by the "category-theoretic definition".

An additional comment: I have never seen an undergraduate discrete math book actually get this completely right, in either style. This is annoying, because it isn't really that hard.

There is a powerful philosophical reason to avoid the category theoretical definition. Mathematical definitions should have as few obviously artificial features as possible. In the definition of relation as a set of ordered pairs, the only artificial feature is the detail of the definition of ordered pair (which can be ignored: it is as it were encapsulated). Consider the definition of function as conditioned by these two definitions of relation, and the definition of function application:

a function is a set of ordered pairs no two of which have the same first coordinate.

versus

a function is an ordered triple whose third coordinate is a set of ordered pairs all of which have first coordinate in the first coordinate of the function, second coordinate in the second coordinate of the function, and in which no two first coordinates of elements are the same.

and

f(x), where f is a function, is the unique y (if there is one) such that (x,y) belongs to f

versus

f(x), where f is a function, is the unique y (if there is one) such that (x,y) belongs to the third coordinate of f.

The complication of the definition of "surjective" in the classical approach is much cheaper in terms of philosophical (or even pedagogical) bother than the complication of the definition of every relation and function in the category theoretical approach. Randall Holmes 15:09, 23 January 2006 (UTC)

Epigraphs

JA: The following is what is generally called an "epigraph":

	All rising to Great Place is by a Winding Staire
	Francis Bacon, Essays, Civil and Moral (1625)

JA: It is hardly "inappropriate", but to the contrary quite aptly telegraphs one of the themes that is more formally developed in the text. It is customary to employ such epigraphs even in the most severe of mathematical treatises, even especially in the most severe of mathematical treatises, to relieve the very severity thereof. Jon Awbrey 17:06, 23 January 2006 (UTC)

RH: an epigraph is a personal reflection on the content (an authorial touch) and an encyclopedia article is impersonal. Randall Holmes 17:24, 23 January 2006 (UTC)

JA: Not of necessity purely personal, as any reading of semi-literate math lit amply demonstates. There's a clear and apt logical relationship between the epigraph and the more severe issues of the text. I see nothing about writing encyclopedia articles per se that dictates being dull. Jon Awbrey 18:18, 23 January 2006 (UTC)

There is a difference between a mathematical treatise or a textbook (where epigrams are appropriate) and an encyclopedia article. They are written with different purposes, and in different styles. An encyclopedia is a reference work, it is not primarily intended to be didactic. Paul August ☎ 22:01, 24 January 2006 (UTC)

Epitaphs

JA: If by non-didactic you mean you do not intend anybody to learn much from reading your articles, then what I have seen in my first month here tells me that you are at great risk of a smashing success. But I read all those fine sentiments at the Math Project page about making math more accessible, and I just wonder what that really means in practice, if not something that would dare to be didactic by any word. Maybe you don't see the contradiction lurking in your system there, but I doubt if I can be the only onlooker who does. Jon Awbrey 22:58, 24 January 2006 (UTC)

Well maybe I'm all wet about how all this should work, and maybe I'm going to hell in a hand basket, but I hope not ;-) Yes, of course I want people to be able to learn from reading these articles, and yes of course I want mathematics to be accessible. But the fact remains that we are still writing an encyclopedia not a textbook. A frequent problem that new editors have is that while that may have considerable expertise in the subject matter, they have (not surprisingly) little experience in encyclopedia writing style. You might want to consider if it is possible that through lack of experience (I'm assuming you haven't written for an encyclopedia before) you might not be aproaching the article in the most appropriate way. You do think that that there is a difference between encyclopedic writing and textbook writing, don't you? How would you describe it? Paul August ☎ 00:20, 25 January 2006 (UTC)

the continuing error

The definition is now not wrong in the sense of mathematical error, but wrong in the sense that it is not the definition anyone uses. This makes it inappropriate as an encyclopedia definition. This definition may have some merits (I can even see some); but it should not be the main definition in an encyclopedia article. Evidence that this is anyone's official definition of relation (in print) is required for this to stand. Randall Holmes 18:30, 23 January 2006 (UTC)

Results of literature search

Looking at books in our library, in all subjects (going down the shelf and looking for set-theoretical preliminaries sections, if any) I found the following distribution. (And I did skew against old books): Seventy-five percent give the old definition (a relation is a set of ordered pairs, and a function likewise); of the remaining twenty-five percent, some just make vague cautionary noises, some define the graph of a relation or function and do not say what a relation or function actually is, and a few give the category theoretic definition (only one gives it explicitly in my sample). I may continue this through some more shelves (to make sure there isn't a prejudice determined by which branches of mathematics I went through) but this, along with web surveys, strongly suggests that the dominant definition is "a relation is a set of ordered pairs". An encyclopedia is not a vehicle for reform of mathematics: it should report what the definition of a concept actually is. The dominant definition is the classical, simple one, at least from the evidence I can see. The category-theoretic definition has a significant following and should be explained -- after the simpler one (which is also easier to explain!) is given. Randall Holmes 22:11, 23 January 2006 (UTC)

My current (draft) thinking

One can look at my current draft if one is interested. Randall Holmes 02:28, 24 January 2006 (UTC)

Relation (mathematics)

I will eventually edit out the cartesian product frame and keep editing. I would rather that you do it. Please listen to Rubin and myself; this definition is wrong. Further, space must be given to what is in fact the dominant definition, even though you do not like it (a relation is a set of ordered pairs) [hint: this is called neutral point of view]. (Added: even in terms of category theory, your definition is wrong: one does not want to exclude the empty set from the category of sets and functions (or the category of sets and relations), where it plays a significant role). Randall Holmes 00:11, 24 January 2006 (UTC)

Take a look at my current draft. Notice that (whatever I have said in discussions about the relative merits of the two definitions) I cover them neutrally. I present the one that I actually prefer earlier not because I prefer it but because it is (1) simpler and (2) is actually the commonest definition. Randall Holmes 02:35, 24 January 2006 (UTC)

JA: [ad hom deleted] Jon Awbrey 12:34, 26 January 2006 (UTC)

JA: I apologize for the pique in what I wrote, but there is something in the substance that I want you (and others) to understand. I will work on that a while and try again later in the day. Jon Awbrey 11:40, 24 January 2006 (UTC)

I am rather more widely educated than you think (I am not going to learn anything from your article that I do not already know). I also have to watch my own postings lest some personal annoyance creep in; I will not take personal offense and I also apologize if I have given any. But you must remember that this is a mathematics article, not a computer science nor a philosophy article. Moreover, you do appear to be attempting to impose a special POV of your own, and in the long run this simply won't work. Randall Holmes 16:52, 24 January 2006 (UTC)

Further, I observe that the formal definition of relation that you give is correct (for the "category-theoretic form") and does not agree with the definition you give in the intro. Further, you say in the formal definition that you are recapitulating what is in the intro -- but you haven't. You really need to change the intro to agree with the formal definition; I don't understand why this is such a sticking point (those who have trouble with abstractions will not find the cartesian product any friendlier than the sequence of domains). You must in addition give a full and much less dismissive place to the view which identifies the relation with its graph -- or I will. Randall Holmes 16:52, 24 January 2006 (UTC)

JA: I've already said that the front-end version is the way it is for pedagogical-rhetorical reasons, as I started the other way and found myself slouching forward on 3 feet rather than running on 2 -- Shades of the Sphynx! Except for the scruples about null factors, it's the same information, and I already know a better way to handle those exceptions, as it's something that arises in real-life database practice all the time, there under the heading of "missing values". But the best way to handle that is in terms of what is actually a more general concept than a relation, to wit, a so-dubbed "relational complex", dubbed-so by analogy with a "simplicial complex" (q.v.) But it's become too much of a distraction to mention out of order all of the things that you are simply not foreseeing, since these talk pages are not really set up for interactive discussion, what with the edit conflicts and all. So try to have some patience. Jon Awbrey 17:16, 24 January 2006 (UTC)

This is not the place for a discussion of better implementations of mathematical concepts; it is a place for a discussion of the implementations which are actually in use. (in later sections of the article one could introduce possible variations). The actual definitions in use are Definitions I and II as in my sandbox article. These can be introduced briefly (and introducing I first actually helps to introduce II, since the object introduced in I is a component of the object introduced in II). Randall Holmes 06:08, 25 January 2006 (UTC)

Example !!!!!!!!!!!!

Things are looking up for the article. But come on people, there's got to be a simple example somewhere! :) And the intro is too big now, I moved the heading up a bit for that. Ideally, there would be the intro, then a short def, then an example, then the more serious definitions. How's that? Oleg Alexandrov (talk) 05:06, 24 January 2006 (UTC)

JA: So who's in a rush, now!? Really, Oleg, there's only so many hours in a day. Sufficient unto the complexity of the subject is the introduction thereof. I am sweating blood here because this is such a low plate in the global architectonics of logic and mathematics both. But there's a very delicate balance between article adequacy and article accessibility -- we've got cheer-leaders stumping for inaccessible cardinals or some team of that stripe, and we've got boo-leaders grousing because it's not the story of Gödel-Lockes the way they heard it as a child. Examples? -- I got a gadshillion of em. Gimme time. Jon Awbrey 05:26, 24 January 2006 (UTC)

You have no idea what my motivations are. These remarks are offensive. Randall Holmes 17:28, 24 January 2006 (UTC)

JA: Sorry for lack of clarity, but I had you in with the cheerleaders for higher cardinals. I was not speculating on the motives there, and you are correct that I should not do that for either side, but I was merely describing what you were arguing for at one point. Jon Awbrey 15:45, 26 January 2006 (UTC)

JA: And I'm going to put my epigraph back, because I think that folks really need to muse on the wisdom of it. Difficult ascents simply must be approached in multiple stages. Jon Awbrey 05:30, 24 January 2006 (UTC)

JA: Oleg and All: This article is key to many doors. It needs to have the right heft and shape to it. As to length, I think that it's best to let it develop a while and then think about what to do with the excess. There are already a number of spinoff stubs that have been prepared to take up the slack and to refine some of the many-splendored aspects of the subject in their own rights. So please hold your horses on that. It will work out in time. Jon Awbrey 12:14, 24 January 2006 (UTC)

corrected the definition of the frame

I have dealt with one of my non-negotiable points (I corrected the definition of the frame). Now negotiations might be opened on the other question: the definition of a relation which identifies it with its graph MUST be given respectful consideration -- and this is not covered by saying that in loose talk we can identify them. There is a rigorous, long-attested (in fact, still dominant) approach under which the relation really is "just" a subset of the cartesian product (admittedly, there are careless usages associated with this approach, which should be highlighted, as I do in function (set theory) and in my sandbox article). This must be said, up front where it can be seen. I suggest, again, looking at the article in my sandbox to see how this can be done (though I don't insist on using my writing, which admittedly can get convoluted). Randall Holmes 17:19, 24 January 2006 (UTC)

Sources

JA: Randall, If you have questions about sources, then ask them on the talk page. Jon Awbrey 17:32, 24 January 2006 (UTC)

No, I know what is a common usage and what is an eccentric one out of left field. Give references to justify this usage or it does not appear. It certainly should not appear first. Randall Holmes 17:40, 24 January 2006 (UTC)

JA: Take a number and sit down. I'm trying to work on Examples today, in accordance with a prior request. It's not all that important at this point in the text, but merely introduces in passing a bit of useful language, quite handy since the days of DeMorgan and Peirce, that has the added benefit of making sense for the aritiless case that some folks are so fond of, whereas "tuple" is slightly more forced. Jon Awbrey 17:54, 24 January 2006 (UTC)

Please stop using condescending language. You have already given personal offense with your remarks to Alexandrov. This article is not yours, and it is not going to end up looking exactly the way you want it. Please listen. Randall Holmes 18:04, 24 January 2006 (UTC)

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