Talk:Function (mathematics)/Archive 9

relation
In mathematics, the word "relation" has a technical meaning, and so to avoid misunderstanding we should not use the word in its non-technical sense in the lead. (A relation is a set of ordered pairs. A function is a relation with the property that if (x,y) and (x,z) are ordered pairs in the set, then y=z.) Rick Norwood (talk) 13:11, 29 March 2012 (UTC)


 * Um, why are we having this discussion again? "Correspondence" also has a technical meaning but we are using that word in the lead; whereas Isheden would like to replace correspondence with relation, I have suggested to use relationship or relates...Selfstudier (talk) 13:16, 29 March 2012 (UTC)


 * I assume you are referring to the discussion above initiated by Isheden in response to a comment by Woodstone, is that right?Selfstudier (talk) 13:26, 29 March 2012 (UTC)

Yes, that's right. I prefer correspondence to relationship because that's what most of the sources I've looked at say, and also for the reason given above. Rick Norwood (talk) 13:29, 29 March 2012 (UTC)


 * The French "correspondance" means "relation" in a mathematical sense;; see http://fr.wikipedia.org/wiki/Relation_binaire - Une telle relation binaire est parfois appelée correspondance entre les deux ensembles.
 * I have a sneaking suspicion that this has been a source of much confusion over the years as Bourbaki in his 1939 used "correspondance" in reference to the ordered pair definition.. — Preceding unsigned comment added by Selfstudier (talk • contribs) 13:36, 29 March 2012 (UTC)


 * And the most common of the rule type definitions makes use of the term "rule of correspondence" ie a normal English usage.Selfstudier (talk) 13:42, 29 March 2012 (UTC)

This business of rule version versus ordered pair just won't go away, although the ordered pair version in our lead is more or less correct (output set = image set), the rule version is wrong, it should say that the output set is a subset of the codomain (not typically specified in the ordered pair definition)Selfstudier (talk) 14:00, 29 March 2012 (UTC)

It seems to me the reason why we are continually coming back to the lead (and in particular, the first paragraph of it) is because it is a truly miserable compromise; we are trying to cater for two (equally valid) definitions of function without mentioning one of them at all (settling instead for a watered down version) and without actually properly defining either because at the same time we are trying to make the lead accessible(to a 12 year old)Selfstudier (talk) 14:07, 29 March 2012 (UTC)


 * If by two "equally valid" definitions you mean "rule" and "ordered pairs", they are not equally valid, as has been pointed out, with references. The rule definition is something used for Freshmen, even though it is wrong, the ordered pair definition is used by all professional mathematicians. Rick Norwood (talk) 14:47, 29 March 2012 (UTC)


 * "Rule" is the watered down version of the triple, a valid function definition used by professional mathematicians. And rule is not wrong, it is simply the most common way to introduce functions informally before, at some time later on, giving one or possibly both of the formal definitions. Your implication that the ordered pair version is the only version is completely wrong.Selfstudier (talk) 15:01, 29 March 2012 (UTC)


 * Well, I am confused by that. Surely the "ordered pair" definition and the "triple" definition are very similar, the only difference being that the "triple" definition explictly identifies the domain and codomain of the function as well as the set of ordered pairs. Gandalf61 (talk) 15:13, 29 March 2012 (UTC)


 * That's right, you can think of function either as triple or as special kind of relation (the way the ordered pair is usually presented) and the remaining difference is the codomain specification. All this arose because of the argument over "rule" versions of function definitions which always include the codomain and this was (at least in part) why there was so much argument over this. Historically, what happened was that the relation/ordered pair was introduced into the education system as part of the NewMath experiment to students of an earlier age and although this experiment is now realized to have been a failure (the students could not assimilate the definition) there are still many today who were taught that way and think of it as the ONLY way....Selfstudier (talk) 15:30, 29 March 2012 (UTC)

http://abstractmath.org/MM/MMFuncSpec.htm#funcdef refers to "two nonequivalent definitions in common use" then gives both of them describing the ordered pair as "less strict" and the triple as "stricter". To be honest, someone just wanting to learn about functions might well be better off going there...Selfstudier (talk) 18:22, 29 March 2012 (UTC)

I propose,since Codomain has the situation properly explained for the most part, if the lead of that article is considered OK, we can just transfer most of it over here and take the rule stuff down into the main article. That seems preferable to giving everyone the (wrong) idea that there is only a relation/function definition.Selfstudier (talk) 14:46, 30 March 2012 (UTC)

As for rule, we now have at least two authoritative sources which provide a more than adequate explanation of what "rule of assignment/correspondence/association" actually means and it does in fact have the meaning that one would expect it to have(some also say rule OR process which is another attempted way around it).Selfstudier (talk) 14:59, 30 March 2012 (UTC)


 * I would appreciate a reminder of which two of the many sources cited above you are refering to. Rick Norwood (talk) 15:09, 30 March 2012 (UTC)


 * The following is from George B. Thomas Jr 1960 Calculus and Analytic Geometry 3rd Edition, Addison-Wesley Publishing Company, Inc, Reading MA, LCCCN: 60-5015. Previous editions: 1950, 1953, pages 16-22: (this continues for several paras above in the section "What they were teaching during all this" posted by Bill and here is another http://books.google.es/books?id=krW_SbmTe9UC&pg=PA8&lpg=PA9&ots=UegKNVLOGx&dq=%22function+is+an+ordered+pair%22 Selfstudier (talk) 15:35, 30 March 2012 (UTC)

I will just copy paste the lead of Codomain in here so it's easy to just see:

"In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image.

The codomain is part of the modern definition of a function f as a triple (X, Y, F), with F a subset of the Cartesian product X × Y. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.

An older definition of functions that does not include a codomain is also widely used.[1] For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, F). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.[2][3][4][5][6]"

Selfstudier (talk) 18:53, 30 March 2012 (UTC)


 * Ah, now I understand. I never have had a problem with codomain.  It was the use of "rule" that I had a problem.  Sorry about the confusion.  Rick Norwood (talk) 19:12, 30 March 2012 (UTC)


 * My fault for mixing up the two things together....Selfstudier (talk) 19:27, 30 March 2012 (UTC)


 * I'm not sure I understand your proposal fully. Naturally, this lead focuses on the codomain of the function which may not be the appropriate focus in this article. What parts would you like to transfer to the lead of this article? We also have to be careful not to introduce too many technical terms (codomain, image, Cartesian product, surjective) early on in this lead. Isheden (talk) 20:57, 30 March 2012 (UTC)


 * Well, we can just use the lead as a base for the lead here, if technical terms are OK in that article then they must also be OK in this article OR are we saying that the technical terms in the codomain article should be removed as well?
 * So, for example a straight cut and rearrange on the above codomain lead gives, for example:


 * "In mathematics, the modern definition of a function f is a triple (X, Y, F), with F a subset of the Cartesian product X × Y. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain.
 * An older definition of functions that does not include a codomain is also widely used.[1] For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, F). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.[2][3][4][5][6]"


 * I'm not saying that is exactly what it should be but at least it is now correct....Selfstudier (talk) 21:51, 30 March 2012 (UTC)

So what I am saying is we replace the first paragraph of the lead with something more like the above and take the informal rule based definition down to the section intuitive (or informal) definition (referenced to provide an explanation of what rule means in this context). Now you might want to make the relation-ordered pair definition more explicit and we might want to tone down some of the technalities (but not if it is simply going to result in pointless semantic discussions about what particular words mean what and when). Selfstudier (talk) 22:01, 30 March 2012 (UTC)


 * While I don't know how to get the data, I would be willing to bet that this article has at least a thousand times more hits than Codomain. Codomain is already a technical term, though I think even there the definition you cite is too technical.  I really don't want to use the "can a 12 year old read it" test, but I do want an intelligent adult with no background in mathematics to be able to get an idea of what functions are.  This would mean leaving technical terms for later in the article.  Anyone who can understand the sentence you quote above already has a good idea of what a function is.  Rick Norwood (talk) 11:55, 31 March 2012 (UTC)


 * Well, look at para 3 of the lead, it has the following


 * "The set of all inputs for a function is called its domain, and the set of all outputs its range or image. Formal definitions of functions specify the set of inputs (the domain), the set of paired inputs and outputs (known as the graph), and a set in which the outputs are constrained to fall (known as the codomain)."


 * All I am really suggesting is that you take these sentences (or something similar) up to para 1 from para 3.
 * Or are you saying that para 3 is wrong?

Selfstudier (talk) 12:03, 31 March 2012 (UTC)


 * On finding page views, I can help there. if you click on the 'view history' tab for an article and look at the top you'll see a number of links on a line starting with 'External tools', the last one says 'Page view statistics'. Just click on that. Function (mathematics) was looked at 74 thousand times and codomain 4 thousand times in the last month. Dmcq (talk) 12:08, 31 March 2012 (UTC)

As it stands the structure of the lead is to give a semi formal definition of the graph (without saying that's what it is) followed by an informal(and wrong because the image is used instead of the codomain)definition and then in para 3 another sentence purporting to explain a different formal definition of function which then uses the technical terms you are complaining of - why should these terms be not OK in para1 but fine in para 3? Selfstudier (talk) 12:14, 31 March 2012 (UTC)

OK, I rejigged the lead so you can just see it (just revert it if you don't like it and we can discuss it more)Selfstudier (talk) 12:30, 31 March 2012 (UTC)

"It is particularly important for the first section (the "lead" section, above the table of contents) to be accessible to a broad readership. Readers need to be able to tell what an article is about, and whether they are reading the correct article, even if they don't already know the topic in detail." (From Make technical articles understandable) If the lead were wrong, I would agree that it should be changed. I don't think it is wrong to leave the codomain out of the first paragraph. It just doesn't try to put in all the details up front. Rick Norwood (talk) 12:33, 31 March 2012 (UTC)


 * Sorry, I don't understand you, you seem to be viewing para 1 as being part of the lead but para 3 as not being part of the lead; all three paragraphs constitute the lead.Selfstudier (talk) 12:37, 31 March 2012 (UTC)

We need a first sentence that a layperson can read.
Note the section above titled "This is the most convoluted, abstruse definition anybody ever conceived." The person who created that section was commenting on a lead much less technical than the current lead. Based on the Wikipedia guideline I quoted above, I'm going to try once again to make the lead readable. As it currently stands, if a person does not already know what function means, this article is not going to be any help at all. Rick Norwood (talk) 12:41, 31 March 2012 (UTC)


 * I don't disagree with you in principle which is why I have said all along that the problem begins as soon as you say "In mathematics, a function is....." (apparently a Wikipedia requirement). In my view, there is no satisfactory way to complete a sentence starting like that without giving a proper definition.
 * Is it not possible to say something like "if you find the following explanation too technical please go to the "intuitive" section?

Selfstudier (talk) 12:50, 31 March 2012 (UTC)


 * Really, you want something like para 2 to be para 1...Selfstudier (talk) 12:53, 31 March 2012 (UTC)


 * OK, I looked at your rejig, what you are doing is essentially transferring a cut down version of the "intuitive" section up to para 1 but don't forget that we still have the requirement that the lead be a summary of what is in the article...Selfstudier (talk) 13:05, 31 March 2012 (UTC)


 * Oh, you are doing it in bits, I will wait until you have finished...Selfstudier (talk) 13:08, 31 March 2012 (UTC)


 * Thanks. Let me explain what I am trying to do.  Keep in mind that my calculus students, who are in the top five percent of all college students as far mathematical knowledge is concerned, often do not know that "f(x)" is read "f of x" not "f times x".  I want a layperson to be able to read at least the first paragraph, and get some rough, basic idea of what functions are all about and that understanding what a function is is essential for further progress in understanding mathematics.  Many people will probably stop there.  In each paragraph, I have provided a little more information, for those who want to know more.  But we can't get everything in the lead.  It is a brief summary of what is in the article, but cannot mention everything in the article, or it becomes cluttered. Rick Norwood (talk) 13:13, 31 March 2012 (UTC)


 * As I said, I have no problem with your basic approach which is more reflective of how the concept is actually taught (to someone who doesn't know). Much easier when you dispense with the "is"...:-) — Preceding unsigned comment added by Selfstudier (talk • contribs) 13:16, 31 March 2012 (UTC)


 * I definitely think major changes like this should be discussed first. That's the way we've worked in the past. The current lead paragraph is really not very encyclopedic. A typical reader will probably already know that $$f(x)=2x$$ is a function, but reads this article in order to understand what is actually meant by the concept of a function in mathematics. Why would you assume that the reader of this article is not interested in an accurate description of a function, albeit without unnecessary technical jargon that can reasonably be assumed to be unknown to someone who doesn't already know what a function is? Wikipedia is not a textbook and the purpose is to present facts, not to teach the subject matter. Isheden (talk) 16:15, 31 March 2012 (UTC)
 * I agree about the need for discussion but I'm not sure that I agree that the lead is attempting to teach function (which as I am sure you are aware is a long slow process) but more an attempt to deal with the "12 year old" criteria; as we have seen up to now it is quite difficult and perhaps impossible to meet all the various requirements of accessibility, mathematically accurate etc and now as well "encyclopedic" whatever that means in this context.
 * In any case, I think it is not set in stone and I have made one or two amendments to it since it first went up, maybe there is a way we can get into an agreeable form from here?

Selfstudier (talk) 18:09, 31 March 2012 (UTC)
 * If we really are to throw away the results of months of intense discussion, I don't think this is the way to do it. Instead, I would like to propose going back to this old version of the lead: Isheden (talk) 18:58, 31 March 2012 (UTC)


 * Hum, I just had the first paragraph something similar to that (ie it included words like codomain, http://en.wikipedia.org/w/index.php?title=Function_%28mathematics%29&diff=prev&oldid=484838103) and that was why Rick tossed it out:-)

Selfstudier (talk) 19:05, 31 March 2012 (UTC)


 * The sentence you're referring to was added with this edit shortly after you and Dmcq had been working on the section Definition. I don't think that the person in question was commenting on the lead, since he or she was explicitly referring to the definition. Since then, the Definition section has undergone further revision and I'm not sure whether that comment is still valid. Isheden (talk) 16:34, 31 March 2012 (UTC)

I completely agree with Rick that the first paragrah in particular should be made as accessible as we can while maintaining enough technical correctness not to mislead the reader or say anything that would be embarrassing to say in a room full of mathematicians. &mdash; Carl (CBM · talk) 19:20, 31 March 2012 (UTC)

Some new edits
I made some edits that were previously agreed upon (the Bourbaki stuff) but I forgot to fill in the "reason for edit" - apologies, I will try and remember to do that:-)

Could someone have a look and see if I did the references properly?

Selfstudier (talk) 18:13, 31 March 2012 (UTC)

"f(x) = 2x" is not a function
The lede was edited today to include the following sentence:
 * For example, f(x) = 2x is a function.

That is nonsense; "f(x) = 2x" is an expression. It can be used to define a function, of course; it can be used to define many functions. For example it gives one function from integers to integers, another from reals to reals, another from points in $$\mathbb{Z}/5\mathbb{Z}$$ to itself, and so on for any setting in which 2x is a term. Whatever we want to say that a function is, we cannot say that 2x is a function. This very issue - that an expression is not a function - was discussed in detail just a few weeks ago in the section "function is a rule" above.

Unfortunately, the new text was not easy for me to fix with a small edit, as I have tried to do in the past, so I am going to revert it to the immediately previous version from March 23. It would be much easier if those who wish to change the lede would get agreement here first. I am sure that the editor who wrote the sentence I am worried about is aware of the distinction, and that a short discussion here could have resolved things. On the other hand the edit that added that sentence was a response to a different edit today that made the lede too technical. Everyone here knows that the lede is going to need to be a compromise between accessibility and technical correctness. &mdash; Carl (CBM · talk)


 * Now you have reverted to a version which is also wrong (outputs instead of codomain, see above discussion)Selfstudier (talk) 19:30, 31 March 2012 (UTC)
 * "Codomain" is unacceptable jargon, at far too high a level for the target audience. —David Eppstein (talk) 19:33, 31 March 2012 (UTC)


 * Codomain is used in para 3 of the lead so why cannot it be used in para 1 of the lead? (also please see above discussion).
 * I am not concerned about whether or not the word codomain is used but I am concerned that the existing para 1 is wrong.

Selfstudier (talk) 20:06, 31 March 2012 (UTC)


 * Could you think of a way of improving the accuracy of para 1 without using jargon? We discussed above that the best way of illustrating that this would be with a good example. The function $$f(x) = x^2$$ from R to R was suggested, but rejected because the target audience may not be comfortable with the set of real numbers. I still think the picture that is presently in the section Definition could be useful in the lead. Isheden (talk) 20:15, 31 March 2012 (UTC)


 * (edit conflict) By "wrong" you mean that the sentence "a function is a correspondence from a set of inputs to a set of outputs" implies that the "set of outputs" is the range of the function, and therefore doesn't include the specification of the codomain, which may be larger than the set of outputs, right? To me that is a technicality that we can safely skip in the first paragraph, since it is covered in the fourth, and "wrong" is an oversimplification as big as the one you're complaining about (for instance, "outdated" or "out of fashion" might be equally accurate in place of "wrong"). As for why words that are in later paragraphs shouldn't be in earlier ones: see Make technical articles understandable and Manual of Style (lead section). —David Eppstein (talk) 20:19, 31 March 2012 (UTC)


 * I explained this already (twice) in the discussion above, it is the SECOND sentence that is wrong not the first, a rule based definition is domain, codomain and rule (think about it).
 * Apart from this, if you look at the diff for the edit that I made, it explained what codomain was ie as in para 3, "the set in which the outputs are constrained to fall" and it seems to me that explain codomain without jargonSelfstudier (talk) 22:04, 31 March 2012 (UTC)


 * I think real number is just about okay as the term is introduced at about the same time and certainly will save a lot of trouble here. I did a small edit to the lead of real number to try an simplify it a little. Dmcq (talk) 21:33, 31 March 2012 (UTC)


 * Um, you think real number is simpler to understand than codomain, is that right?Selfstudier (talk) 22:06, 31 March 2012 (UTC)


 * Oh I see, well I prefer codomain mentioned at the end as at present but I think we can get the first paragraph right by simply saying 'to a set of possible output values', I'll stick that in since everyone seems to be changing it again and see what people think. Dmcq (talk) 22:28, 31 March 2012 (UTC)


 * I don't see the point of reverting the possible outputs edit, it gives the basic idea of codomain without being explicit. Also I think the second sentence is too repetitive and I don't like the unique in it, it can easily give the impression that the same output value can't be used more than once. Dmcq (talk) 22:41, 31 March 2012 (UTC)


 * This is the version I would edit what's there to:
 * In mathematics, a function is a correspondence between a set of inputs and a set of possible outputs that associates each input with exactly one output. Informally, a function can be thought of as a rule that assigns an output  to each input. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x").
 * I don't see what the revert saying 'This is just the ordered pair again' means. Dmcq (talk) 22:48, 31 March 2012 (UTC)


 * I undid my undo and I am see this "Informally, a function can be thought of as a rule[2] that assigns to each input a unique output"...which is just the ordered pair again?Selfstudier (talk) 22:50, 31 March 2012 (UTC)

Now it says "Informally, a function can be thought of as a rule[2] that assigns an output to each input." so you might as well not bother to include this sentence at all as it is completely without any meaning. I would like to go back to my earlier proposal which is to include the proper triple in the lead paragraph (without jargon) and leave the rule based (ie a watered down triple) discussion for the "Intuitive section. Selfstudier (talk) 23:02, 31 March 2012 (UTC)


 * This seems to be going around in circles, with the same few points being made over and over. I strongly believe that the first sentence, at least, should be something a layperson can read and understand, completely free of jargon.  I like "A function has an input and an output."  It is a true statement.  It captures the essence of "function".  It contains no jargon.  And it avoids the common but false statement that a function is a rule, so that it is not necessary for those who go on to higher mathematics to be untaught what they have been taught.  But if you find that unacceptable, please suggest a better first sentence that it would be possible for a layperson to read and understand.  As the lead stands at the moment, nobody who does not already have a substantial amount of mathematics under their belt will be able to read it.  And I see no point in editing it, because it will probably be totally different tomorrow.  Rick Norwood (talk) 05:46, 1 April 2012 (UTC)


 * I do wish we would just give up going on about this rule business, this has now been completely dealt with; it is not the informal definition that is the problem, it is the way that the function concept is taught that is the problem, as is confirmed by many studies.Selfstudier (talk) 09:12, 1 April 2012 (UTC)


 * (edit conflict) I rather have the impression that the attempt to explain the concept of a function without oversimplifying but also without introducing unnecessary terminology has been able to produce a fairly stable lead. Now the heavily debated "rule of correspondence" has been replaced by a set of ordered pairs which seems to be accepted by most editors. Regarding the first sentence, my proposal is still "A function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output." Isheden (talk) 09:14, 1 April 2012 (UTC)


 * I think you can change that back, that only came about because Dmcq was attempting to deal with my complaint about para 2.Selfstudier (talk) 09:23, 1 April 2012 (UTC)

Isheden, I can see what you are doing in the first para and I have no problem with it save that it seems to me that you can reword it in a slightly different way to avoid the repetition....Selfstudier (talk) 09:45, 1 April 2012 (UTC)


 * Hmm, now the first sentence reads like a cross between the ordered pair definition and the triple. We are trying to cover the fact that there are two valid definitions in use, no?Selfstudier (talk) 09:55, 1 April 2012 (UTC)


 * I changed it a bit so that we have two distinct definitions and to try and avoid repetition of the unique aspect...Selfstudier (talk) 10:06, 1 April 2012 (UTC)

Suggestion
The lead currently reads:

"In mathematics, a function is a correspondence between a set of inputs and a set of outputs that associates each input with exactly one output. More formally, a function is defined by a set of inputs, a set containing the outputs and a set of input-output pairs (the correspondence just referred to). The output of a function f corresponding to an input x is denoted by f(x) (read "f of x")."

I suggest that you show that to a person with no training in mathematics and ask them what they make of it. I have no problem with the first sentence. But the second sentence will be unintelligible to the non-mathematician. I think any reference to "pairs" needs to be better motivated, and further down in the lead, and that the second sentence needs to be an example of a function.

Comments?

Rick Norwood (talk) 11:24, 1 April 2012 (UTC)


 * The "correspondence" is the pairs? So if sentence one is comprehensible then so is sentence two. Sentence one is describing the pairing process, why is this not sufficient?

Selfstudier (talk) 11:31, 1 April 2012 (UTC)

Try the experiment I suggested above. Rick Norwood (talk) 11:40, 1 April 2012 (UTC)


 * I tend to agree with you about an example but not as sentence two because you will find that you are not able to give a proper/meaningful example without a prior explanation; we have had your fx = 2x and we have had x^2 both unsuccessfully up to now.
 * Since we have an example of sorts in para 2, I think another (or perhaps a different) example could go there, possibly together with a graph (ordered pair) and showing the codomain and domainSelfstudier (talk) 11:42, 1 April 2012 (UTC)

I think we have to take into account the difference between teaching it and giving a factual account of it although I appreciate the line between the two is somewhat arbitrary. Do I think that you can learn the function concept just from this article, certainly not; then again, is there are any single article from which you could learn the function concept, again the answer is no. We do know that this is a confusing concept that takes a long time to absorb fully and so it should not be a surprise if people "don't get it" by reading a couple sentences and looking at an example on here. Selfstudier (talk) 12:05, 1 April 2012 (UTC)

I just showed it to my partner who not only does not know any math but doesn't want to either, I had to explain that a set was a collection, that a correspondence meant pairing and after that her only meaningful comment was "what's f?" Selfstudier (talk) 12:12, 1 April 2012 (UTC)


 * I think that establishes that the second sentence does not belong in the lead. One point you keep returning to is that the article should not "teach" what a function is.  No.  But it should explain what a function is in language a layperson can understand.


 * You say the example f(x) = 2x is "unsuccessful", but all that has actually been demonstrated is that a couple of people don't like it. Others do.  I think you will find that virtually all math books, from the most elementary to the most advanced, would have no trouble at all with saying that f(x) = 2x is a function.


 * I won't make any changes in the article until some of the others weigh in, but you might try this out on your partner:


 * "A "function", in mathematics, has an input and an output. For example, we can write f(x) = x + 3.  The name of the function is f.  The name of the input is x.  The output is gotten by adding 3 to the input.  If the input is 5, then the output is 5 + 3 = 8, and we can write f(5) = 8."


 * Comments?


 * Rick Norwood (talk) 12:20, 1 April 2012 (UTC)


 * Sorry, I disagree, it shows that people (my partner anyway) doesn't understand any of sentence one, two or three without explanation(and I am not at all surprised about this, the mythical 12 year old would probably do better).
 * I agree that many educators would happily say f(x) = 2x is a function BUT (as we keep being informed) we are not educating we are simply being factual ie you can have (somewhere) a sentence saying "Many educators (wrongly) would say...."
 * I already explained my partner doesn't want to know about math, I showed her that and she just grimaced (because it has formulas and numbers in it)

Selfstudier (talk) 12:50, 1 April 2012 (UTC)

http://simple.wikipedia.org/wiki/Function_%28mathematics%29

"In mathematics, a function is the idea that one quantity (called the input) completely determines another quantity, often called the output. Modern mathematics defines functions using sets. A function can then be seen as a rule. With this rule, each element of the set of inputs will be assigned one element of the set of outputs. Even though the underlying structures are sets, mathematicians talk about domain and codomain of a function. Each element of the domain will be associated with one element of the codomain.

An example of an elementary function that acts on the real numbers as both the domain and the codomain could be f(x) = 2x. Every real number will be assigned a real number that is twice as big. In this case, f(5) = 10.

Mathematics knows other objects than ordinary numbers. While elementary functions may act on numbers, functions can also be maps. They may act on algebraic structures like groups, or geometric ones such as manifolds. A function that acts on the sets {A,B,C} and {1,2,3} and that associates A with 1, B with 2, and C with 3 is an example of a non-elementary function.

It is sometimes said that f(x) is the image of x, and x is the pre-image of f(x). Also, it is said that the object x is transformed or mapped into f(x). A function is another way of representing y in formulae like y = x+2, which would be f(x) = x+2"


 * -) — Preceding unsigned comment added by Selfstudier (talk • contribs) 13:52, 1 April 2012 (UTC)

Not sure what simple.wikipedia guidelines are, understandable by a 6 year old? A ways to go, methinks...Selfstudier (talk) 14:01, 1 April 2012 (UTC)


 * I quoted the Wikipedia guidelines above, but here they are again. "It is particularly important for the first section (the "lead" section, above the table of contents) to be accessible to a broad readership. Readers need to be able to tell what an article is about, and whether they are reading the correct article, even if they don't already know the topic in detail."


 * In your suggested rewrite above, the first sentence is fine. You lose your audience with the second sentence.  One difference between the "broad readership" above and your partner is that the person who comes to this article wants to know something about functions.  I think the article should tell them something about functions.


 * I'm still waiting (and will continue to wait until some time Monday) for others to weigh in on this discussion.


 * Rick Norwood (talk) 14:23, 1 April 2012 (UTC)

Lol, I am copying that simple.wikipedia article (plus a link to it) here for information, it is not written by me or suggested by me. it is supposed to be a simpler version of wikipedia but is happily doing all the things we have labelled as being "wrong". Apropos of nothing, [http://lmathdl.maa.org/t_and_l/sampler/rs_9.html 9. Key Aspects of Knowing and Learning the Concept of Function] is worth a read..Selfstudier (talk) 15:04, 1 April 2012 (UTC)

Maybe you could write your idea about the lead over on that site instead and then we could refer people having trouble with the lead here over to it (you know, a tag of some description saying "Read about function on SimpleWikipedia!"; after all, I assume that is one of it's purposes)Selfstudier (talk) 15:16, 1 April 2012 (UTC)


 * Opinion My opinion on this is that the tone and level in that citation is appropriate for Simple Wikipedia, which incidentally is where it was taken from. In this article I think we can assume that the reader is seriously interested in finding out what a function is. The guideline WP:TECHNICAL explicitly states "Encyclopedia articles should not "tell lies to children" in the sense of giving readers an easy path to the feeling that they understand something at the price that what they then understand is wrong." I fear that is what will happen if we say f(x) = 2x is a function in the lead. Isheden (talk) 15:28, 1 April 2012 (UTC)


 * Selfstudier: I have been polite to you throughout this discussion, and I see nothing gained by your rudeness. Of course, the MAA article is a good one, and a link to it would be useful. But that does not solve the problem of making this a better article. The idea that laypersons should go to SimpleWikipedia is insulting. The point about writing for a borad readership, quoted above, does not suggest writing for people who have trouble with reading, but rather writing for intelligent people who have trouble with technical subjects. Your willingness to dismiss such people as needing SimpleWikipedia is unacceptable under Wikipedia guidelines.


 * I don't understand, codomain is no longer in the first paragraph, nor is domain, triple or any of those others, all that is left is set (an undefined concept in any case).
 * I am sorry if you thought I was being rude, who do you think it is that reads SimpleWikipedia? Are you saying that SimpleWikipedia does not need to exist?
 * The point I was trying to make is that SimpleWikipedia, supposedly a simpler version of Wikipedia, is happily using codomain, domain, etc in its lead and we are not Selfstudier (talk) —Preceding undated comment added 16:03, 1 April 2012 (UTC).


 * Isheden: I agree with what you say up to the point where you reject the statement that F(x) = 2x is a function. I'll provide some quotes if you give me a few minutes. Rick Norwood (talk) 15:32, 1 April 2012 (UTC)
 * This is commented upon in the first section of the article: "In that case, we would describe f as a real-valued function of a real variable. It is not enough to say "f is a function" without specifying the domain and the codomain, unless these are known from the context, (...) however it is standard to take the largest possible subset of R as the domain (...) and R as the codomain." Isheden (talk) 15:42, 1 April 2012 (UTC)


 * Isheden: here are the examples I promised above:


 * Examples, picked at random from my bookshelf:


 * Keryszig, Advanced Engineering Mathematics, p. 2 "given functions of x ... for example y' = cos(x)"
 * Munkres, Topology, p 14, "In calculus, a function is often given by a simple formula such as f(x) = 3x^2 + 2." He goes on to say, "As one goes further in mathematics, however, one needs to be more precise about what a function is. Mathematicians think of functions in the way we just described, but the definition they use is more exact."
 * Lomen & Lovelock, Differential Equations, p. 2, "Example 1.2 The Natural Logarithm Function".


 * Do I really need to go on? Given that all of these authors (and I suspect virtually every other author who uses the word function, since I found these so easily) talk about polynomial functions, trigonometric functions, and logarithm functions without mentioning the domain and co-domain.  But you insist that in this simple introductory article we make the very first paragraph incomprehensible to lay readers by going into domain and codomain before the reader has any chance to understand what a function is.


 * While we're waiting for others to weigh in, here are a few comments already made that oppose introducing mathematical jargon in the first paragraph:


 * "Codomain" is unacceptable jargon, at far too high a level for the target audience. —David Eppstein (talk) 19:33, 31 March 2012 (UTC)


 * By "wrong" you mean that the sentence "a function is a correspondence from a set of inputs to a set of outputs" implies that the "set of outputs" is the range of the function, and therefore doesn't include the specification of the codomain, which may be larger than the set of outputs, right? To me that is a technicality that we can safely skip in the first paragraph, since it is covered in the fourth, and "wrong" is an oversimplification as big as the one you're complaining about (for instance, "outdated" or "out of fashion" might be equally accurate in place of "wrong"). As for why words that are in later paragraphs shouldn't be in earlier ones: see Wikipedia:Make technical articles understandable and Wikipedia:Manual of Style (lead section)#Introductory text. —David Eppstein (talk) 20:19, 31 March 2012 (UTC)


 * I completely agree with Rick that the first paragrah in particular should be made as accessible as we can while maintaining enough technical correctness not to mislead the reader or say anything that would be embarrassing to say in a room full of mathematicians. &mdash; Carl (CBM · talk) 19:20, 31 March 2012 (UTC)


 * Rick Norwood (talk) 15:51, 1 April 2012 (UTC)


 * If you feel that this article could be simplified without losing technical correctness, I would like to invite you to revise the first section of the article (with the intuitive description), and after that make an attempt to summarize the main ideas in a new lead paragraph and propose it here on the talk page for discussion. Isheden (talk) 16:52, 1 April 2012 (UTC)

Thank you. I've done this several times in the past, I'm willing to do it again. But, fair warning, I'm going to give an example that does not mention domain and codomain. I strongly believe that needs to wait until the basic idea is in place.

Proposed first paragraph of the lead:

The concept of a function is fundamental to all modern mathematics. The essential property of a function is that for every input, there is one and only one output. An example of a function is f(x) = x + 2. The name of the function is f, the name of the input is x, and to get the output, we add 2 to the input. Thus, if the input is 3, the output is 5, and we write f(3) = 5. The expression f(x) is read "f of x". This same function is often written y = f(x), where y is the name of the output.

Rick Norwood (talk) 17:42, 1 April 2012 (UTC)-
 * Fails WP:LEADSENTENCE. —David Eppstein (talk) 17:56, 1 April 2012 (UTC)


 * This is not just giving an example, this is rewriting the lead altogether (the first paragraph of it anyway); what Isheden was saying was first to revise the "intuitive description" section and only then summarize the main ideas in a new lead paragraph (which I took to mean that the lead should reflect the article contents).
 * Also I still don't understand your complaint exactly, the existing first paragraph does not use the words domain,codomain etc.

Selfstudier (talk) 18:00, 1 April 2012 (UTC)

David Eppstein: Wikipedia has numerous guidelines, which I try to follow. In previous first sentences I have suggested, I've always begun with a defintion. This time, I thought I would try following the guideline mentioned above, "Readers need to be able to tell what an article is about, and whether they are reading the correct article, even if they don't already know the topic in detail." However, I have no objection to putting the definition in the first sentence.
 * It's more than just not having the definition in the first sentence (although I think it should be in the first sentence). There is no definition at all in your entire suggested replacement paragraph. It describes properties of functions but doesn't say what the objects are that have those properties. It's las if you wrote about hands, saying that they generally have five fingers and can hold and catch things, but neglected to mention that they're appendages of the human body. —David Eppstein (talk) 20:08, 1 April 2012 (UTC)

Selfstudier: My "complaint", exactly, is that nobody who doesn't already know what a function is can read the first paragraph as it currently stands.

That said, I am willing to work with anyone who genuinely wants to make this a better article, so I will continue to work on the lead here. Maybe if we give an informal definition in the first sentence, then an example, and separate the mathematical definition into a paragraph by itself, which the non-mathematically inclined reader can skip over, that would be acceptable to everyone.


 * In mathematics, a function is a correspondence between a set of inputs and a set of outputs that associates with each input exactly one output.  For example, a function might be given by f(x) = x + 2.  The name of the function is f, the name of the input is x, and to get the output, we add two to the input.  (The expression f(x) is read "f of x".)  If the input is 3, then the output is 5, and we can write f(3) = 5.


 * A formal, mathematical definition of a function is that a function has a set of inputs, a set in which the outputs are constrained to fall, and a set of paired inputs and outputs, with the property that for a given input, there is one and only one output.


 * The concept of a function is fundamental to all modern mathematics.

Rick Norwood (talk) 19:59, 1 April 2012 (UTC)


 * Well, the first sentence is a definition, what I don't get is why that one is supposedly understandable and the other one, which is merely an elaboration of (includes)the first, isn't; but that it then becomes comprehensible by virtue of the rest of the new paragraph 1, I don't think that follows, in fact all that has been done is to convey the idea of function as rule (formula) without consideration of any other metaphor (note that I am not saying that it is not a valid example,it is perfectly fine, just not deserving of such prominence in the lead,just an example).
 * It would be better to discuss as many metaphors as possible and this is in fact done in paragraph 3 of the lead and I would be tempted myself to try and make para 3 into para 2 somehow.
 * I agree with inclusion of the statement about the function concept being fundamental,even as the first sentence.(I know it is the first sentence of the "intuitive description" section, not entirely sure what it is doing there).
 * Did we ever discuss the idea of just pointing people to the "intuitive section"?

Selfstudier (talk) 22:14, 1 April 2012 (UTC)


 * You do have a point with the statement above "I have no problem with the first sentence. But the second sentence will be unintelligible to the non-mathematician. I think any reference to "pairs" needs to be better motivated, and further down in the lead, and that the second sentence needs to be an example of a function." We just have to be careful not to say anything that would be embarrassing and keep a tone that is encyclopedic. Here is another suggestion for the first paragraph:
 * In mathematics, a function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output. An example of such a relation is f(x) = x2, which relates an input x to its square, which are both real numbers. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x").
 * Then, the fourth paragraph could read:
 * In modern mathematics, a function is defined by its set of inputs, a set containing its outputs, and a set of input-output pairs describing the relation between inputs and outputs. The set of inputs is known as the domain, the set of paired inputs and outputs as the graph, and the set containing the outputs as the codomain. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.
 * Isheden (talk) 05: Nor30, 2 April 2012 (UTC)


 * If I look at the latest version as amended by David:


 * "In mathematics, a function[1] is a correspondence between a set of inputs and a set of outputs that associates each input with exactly one output. More formally, a function is defined by its set of inputs, a set containing its outputs, and a set of input-output pairs describing the correspondence between inputs and outputs."


 * the second sentence reads very similar to the first and so it seems to me that we ought to be able to have one sentence that incorporates both.


 * I do not agree that ordered pairs (graph, input-output pairs) are less understandable than real numbers, the first concept is taught well before the second so the proposed new version is in fact less accessible. Nor do I believe that this new version will result in a better understanding (or motivation) of ordered pair by the time the reader gets to paragraph 4; if they don't have it at para 1 they won't have it at para 4 either.


 * I will try to think of a suitable sentence to replace the current first and second sentences. If we can do that and we have para 4 as well, then one or more examples may be added at any point in an attempt to elucidate the definition.

Selfstudier (talk) 10:04, 2 April 2012 (UTC)

OK, I came up with the following sentence (which trys to avoid unnecessary repetition as well as reference back) and have amended the lead to reflect it, I hope that it is acceptable

"In mathematics, a function[1] is defined by its set of inputs, a set containing its outputs, and a correspondence between the inputs and the outputs that associates each input with exactly one output." Selfstudier (talk) 10:55, 2 April 2012 (UTC)


 * There is no way to prove that one sentence is clearer than another, but I've taught mathematics at every level from remedial to the graduate for many years, and for what its worth I think readers will understand Isheden's version much more easily than Selfstudier's version. I'm sorry, Selfstudier, but we've had a lot of discussion, and I think since at least Isheden and I have come to an agreement, it is time to move on. Rick Norwood (talk) 12:45, 2 April 2012 (UTC)


 * Fortunately, we are not teaching mathematics, so I have now, just like you and Isheden, placed a proposal on the table and explained what I think is wrong with yours, with Isheden's (which is different yet again) and with what was there before my latest amendment. So I'm afraid that it is not yet time to move on.
 * If you or anyone else would like to revert my current amendment would you please revert it to the version that was last accepted by most editors.

Selfstudier (talk) 13:01, 2 April 2012 (UTC)


 * While we are not teaching mathematics, I think most editors agree that the intended audience for this article is not primarily looking for a mathematically exact definition in the first sentence, but for a simple, yet reasonably accurate description of the concept of a function. Therefore, I don't think the sentence should begin with "a function is defined by". Also, it seems that most editors agree that it would be helpful with a good example early on in the lead. So I don't understand the rationale for reverting to your version, which cannot be considered as accepted by the majority. Isheden (talk) 13:12, 2 April 2012 (UTC)

I really think it is time to move on. Rick Norwood (talk) 13:03, 2 April 2012 (UTC)

Isheden's version
I have cut and pasted Isheden's version into the lead. Isheden: do you have any objection to adding to your version the following sentence: If the input is 3, then the output is 9, and we can write f(3) = 9. Rick Norwood (talk) 13:02, 2 April 2012 (UTC)


 * No, I don't mind. We had a similar numerical example earlier, but it disappeared during the discussion about whether "numbers" would be enough. However, choosing a negative input may be nice, since it illustrates that an output does not correspond to only one input. Isheden (talk) 13:17, 2 April 2012 (UTC)


 * However, I think the last edits have been a bit rushed. Presently, there is no sentence that introduces the set of ordered pairs. I think we should strive for real consensus before moving on. Isheden (talk) 13:56, 2 April 2012 (UTC)

Everyone would like a consensus. I hope that is possible. I'll add "If the input is -3, then the output is 9, and we can write f(-3)=9". I'll look at the question o a sentence that introduces the set of ordered pairs, but won't make any change to that effect until we've discussed it here. Rick Norwood (talk) 14:28, 2 April 2012 (UTC)

Latest proposal
If I look at the latest version as amended by David:

"In mathematics, a function[1] is a correspondence between a set of inputs and a set of outputs that associates each input with exactly one output. More formally, a function is defined by its set of inputs, a set containing its outputs, and a set of input-output pairs describing the correspondence between inputs and outputs."

the second sentence reads very similar to the first and so it seems to me that we ought to be able to have one sentence that incorporates both.

I do not agree that ordered pairs (graph, input-output pairs) are less understandable than real numbers, the first concept is taught well before the second so the proposed new version is in fact less accessible. Nor do I believe that the new versions proposed by Isheden will result in a better understanding (or motivation) of ordered pair by the time the reader gets to paragraph 4; if they don't have it at para 1 they won't have it at para 4 either.

I have now included a suitable sentence to replace the current first and second sentences. If we can do that and we have para 4 as well, then one or more examples may be added at any point in an attempt to elucidate the definition.

If anyone would like to revert my amendment then please revert it to the version that was last agreed by a majority.

Selfstudier (talk) 13:05, 2 April 2012 (UTC)


 * Let's at least give Isheden's version a chance. Rick Norwood (talk) 13:12, 2 April 2012 (UTC)


 * I am not going to descend into an edit war with you merely because you have chosen to jump the gun on an edit that you happen to support.
 * I have also made a proposal and I judge from your actions that you are intent on ensuring is not discussed at all.
 * Now we wait....

Selfstudier (talk) 13:22, 2 April 2012 (UTC)

Since I said "at least give Isheden's version a chance", I don't understand your statement that I am jumping the gun, while you, presumably, are free to edit at will, and that I am "intent on ensuring is not discuessed at all" (just how would I do that?). Rick Norwood (talk) 14:24, 2 April 2012 (UTC)

Ongoing discussion of the lead.
Isheden suggests introducing the idea of ordered pairs. Here is the current paragraph 4:


 * The set of inputs to a function is known as the domain, the set of paired inputs and outputs as the graph, and the set in which the outputs are constrained to fall as the codomain. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

How about something like this?


 * The input and output are often expressed as an ordered pair. In the example above, we have the ordered pair <–3, 9>.  A complete definition of a particular function will give the set of inputs, called the domain, the set of paired input and outputs, called the graph, and a set in which the outputs are constrained to fall, called the codomain.  For example, we could define the function f(x) = x2 by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers .


 * Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

Rick Norwood (talk) 14:41, 2 April 2012 (UTC)


 * A newbie question: I was on-board until the example of the last sentece: Where did the "graph" go in f(x) = x2. What physical object exactly is "the graph" in the case of a continuous (all points possible in the continuum) function as opposed to discrete pairs such as a lookup table? Bill Wvbailey (talk) 15:37, 2 April 2012 (UTC)
 * It's an infinitely long (notional) lookup table. --Cyber cobra (talk) 15:52, 2 April 2012 (UTC)
 * Worse than that, almost every entry is infinitely long, so the table is an infinitely long listing of infinitely long entries that cannot be specified (see quote below0. This is not an object in the newbie intuitive sense of the word. I'd be more inclined to describe the graph as an analog machine, say a really big slide-rule, say, or a plotting device such that, fractal-like, no matter how close you squint there's always another digit. Again it's not surprising that newbies cannot grab the idea: "Where's the graph"? What is "the graph"? Here's a quote to back it up:
 * From the algorithm article:
 * "No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13] cf Boolos & Jeffrey (1974, 1999)
 * BillWvbailey (talk) 17:35, 2 April 2012 (UTC)
 * Intuitively, the graph of f(x) = x2 is the curve in the Cartesian plane (at least for a domain such as -5 ≤ x ≤ 5). Formally, it is the set of all ordered pairs (x, x2). If the domain is a countable set such as the integers, you could write a possibly infinitely long lookup-table. For an uncountable set this might not be possible. Isheden (talk) 09:24, 3 April 2012 (UTC)


 * I think the example is good. However, at the beginning of paragraph 4 I would like to propose "In modern mathematics, a function is defined by its set of inputs, a set containing its outputs, and a set of input-output pairs describing the relation between inputs and outputs." Possibly "describing the relation ..." could be skipped. Isheden (talk) 15:53, 2 April 2012 (UTC)
 * One more remark: The set of ordered pairs is just all pairs of the form ; that x and x2 are real numbers is determined by the domain and codomain definitions, respectively. Isheden (talk) 16:22, 2 April 2012 (UTC)

Maybe we should explain the difference between a graph in the sense the word is used in elementary mathematics and a graph in the sense it is used in abstract mathematics. The article graph (mathematics) which this article links to also needs work, but I hesitate to add yet another page to my watch list. How about something like the following:


 * The input and output are often expressed as an ordered pair. In the example above, we have the ordered pair <–3, 9>.  This ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function.  But no picture can exactly define every point in an infinite set.  In modern mathematics, a function is defined by its set of inputs, called the domain, a set containing the outputs, called its codomain, and the set of all paired input and outputs, called the graph.  For example, we could define the function f(x) = x2 by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers .  Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

I don't think it hurts to say "real numbers" twice, since that avoids having to say that the graph is a subset of the cross product.

Let me know what you think. Rick Norwood (talk) 12:07, 3 April 2012 (UTC)
 * It's probably good enough to insert in the lead and to let other editors work on it. Isheden (talk) 13:09, 3 April 2012 (UTC)

I made the change. Rick Norwood (talk) 14:21, 3 April 2012 (UTC)

Gandalf61's edit
Gandalf61 changed f(x) = x2 to x → x2. I kind of like this, but I reverted it, because I think we need to discuss it here, first. There are (at least) three ways we could introduce the example, f(x) = x2, x → x2, and y = x2, and we could use any one, any two, or all three. Any thoughts about how to be clear without overloading the lead? Rick Norwood (talk) 14:20, 3 April 2012 (UTC)
 * I think x → x2 has some merit because it indicates a direction from input to output that is not visible when using the other two ways and it does not introduce any extra notation. Isheden (talk) 14:28, 3 April 2012 (UTC)
 * Shouldn't it be $$x \mapsto x^2$$ with a mapsto? &mdash; Carl (CBM · talk) 15:03, 3 April 2012 (UTC)

--


 * Edit conflict: Is this logical implication? The introduction of a new sign to replace the familiar "arithmetic equivalence" (the unfortunate choice → being the familiar logical sign for implication), is a rabbit-hole we do not want to descend into (at least in the lead). The theories I'm familiar with (e.g. used by Goedel) either add arithmetic equivalence as an axiom outright, or regard it as a generalization (over all variables of a specified class, cf PM and Goedel 1931) in other words, the sign = presupposes the notion of "function" (e.g. classes of individuals of type 2 here):
 * "[21] x1 = y1 is to be regarded as defined by x2∀(x2(x1)⊃ x2(y1) as in PM (I, *13) similarly for higher types." (Goedel 1931 in van H:600, with ∀ replacing Goedel's ∏ "for all" but retaining his sign for logical equivalence "⊃")
 * Maybe a different sign, e.g. =>, or one of these from unicode: ⇛ ⇒ ⇝ ↣. The idea that a function is indeed a one-way "process" is interesting, and it is "correct", e.g. we know that the function {(3, 5), (4, 5)} is not reversible, given output sign 5 we cannot deduce the exact sign that fell into the hopper. But I'd discourage introducing something as subtle as this in the lead. BillWvbailey (talk) 15:31, 3 April 2012 (UTC)

Should we then start a new paragraph to introduce f(x) = x2 and y = x2? Or maybe put the entire example into a separate paragraph, and add the ordered pair notation. Maybe something like this:


 * In mathematics, a function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output.


 * An example is the function that squares any real number. This concept may be expressed in any one of several different ways.  We may name the input $$x$$ and write $$x \mapsto x^2$$.  We may name the function $$f$$ and write $$f(x) = x^2$$ (read "f of x equals x squared").  Or we may name the output $$y$$ and write $$y = x^2$$.  Another notation sometimes used is to write the input and output as an ordered pair, $$$$.  Suppose the input is –3.  Then the output is 9 and using each of the notations above we could write $$ -3 \mapsto 9 $$, $$f(-3) = 9$$, $$ 9 = (-3)^2 $$ or we could write the ordered pair $$ <-3, 9> $$.

If we go with this or something like it, we'll need to do a little rewriting in the last paragraph.

Rick Norwood (talk) 15:06, 3 April 2012 (UTC)


 * This is OK with me if the \mapsto is corrected. &mdash; Carl (CBM · talk) 15:08, 3 April 2012 (UTC)
 * This notation is even more attractive if it is called a rule. Otherwise one might wonder why it is not called an equation or formula instead. Isheden (talk) 15:14, 3 April 2012 (UTC)

Let's postpone discussing "rule" further until we get mapsto sorted out. I'm having trouble with a parsing error and have not sorted out what I'm doing wrong yet. Can either of you see what I'm doing wrong? Rick Norwood (talk) 15:16, 3 April 2012 (UTC)
 * I think the commas may be problematic inside the math environment. Isheden (talk) 15:17, 3 April 2012 (UTC)

Thanks. But it doesn't mind the comma in $$$$.

Note to CBM, please join the ongoing discussion. We've talked at length above about whether or not $$f(x)=x^2$$ is or is not a function, and I cited three books, and could easily cite thirty books, that call expressions like this functions. To go to great lengths to avoid calling this a function is a little bit like going to great lengths to avoid calling 2 a number. Yes, 2 is a numeral rather than a number, but everybody calls 2 a number anyway. Rick Norwood (talk) 15:22, 3 April 2012 (UTC)


 * I think I am in the ongoing discussion (unfortunately the volume is so large that it's very hard to follow everything). Of course there are some books that call those things functions - probably the calculus book I teach from says it. But those books are being sloppy in a way that we cannot afford to be sloppy in this article. In a calculus class we don't really expect students to know what functions are, and there is a lot of context because pretty much all the objects they work with are real numbers. But in general math we care much more what a function is, and we no longer have the fixed context of the reals. So we have to be more careful about things than a calculus textbook might be. We can't say that a function is a rule, or that "x^2" is a function. &mdash; Carl (CBM · talk) 15:31, 3 April 2012 (UTC)


 * At the same time I do not think we need to mention domain or codomain at all in the first paragraph - we don't have to say what a function is in that paragraph, but we should avoid making claims that are either false or would be embarrassing to make in a room full of mathematicians. &mdash; Carl (CBM · talk) 15:36, 3 April 2012 (UTC)

Thanks to whoever fixed the parsing error. Please tell me how you did it.

Actually, the examples I gave were well above the calculus level. They were from Keryszig, Advanced Engineering Mathematics, Munkres, Topology, and Lomen & Lovelock, Differential Equations. That said, I have no problem with the changes you made. And, yes, the length of these discussions can be daunting.

Rick Norwood (talk) 15:42, 3 April 2012 (UTC)
 * The problem with the math is that it used a minus symbol instead of a hyphen, but latex only accepts a hyphen and displays it as a minus symbol. This is one reason that it's better to type &minus; as "&amp;minus;"


 * The Munkres quote I see above is 'In calculus, a function is often given by a simple formula such as f(x) = 3x^2 + 2." Note that Munkres is distinguishing between the formula and the function just as I have tried to do in the article. I would tend to lump diff eq. books in with calculus because they are aimed at engineers as much as mathematicians. Many physics books still use infinitesimals(!).


 * I think the situation with the number/numeral distinction is a little different because, even though the numeral "2" is not the number 2, the numeral generally only defines one number. But the expression x2 can define many different functions, and the issue that the domain has to be specified is exactly the issue that an article like this has to be very careful to keep straight. &mdash; Carl (CBM · talk) 15:47, 3 April 2012 (UTC)

Thanks. Are we good to go with the paragraph above? Is there any way to stop the first $$x$$ from being higher than the second? Rick Norwood (talk) 15:59, 3 April 2012 (UTC)


 * Does this symbol show up for you? &#x21A6; &mdash; Carl (CBM · talk) 16:36, 3 April 2012 (UTC)

Just barely. If I wasn't looking for the arrow head, I'd mistake it for a dash. Rick Norwood (talk) 16:39, 3 April 2012 (UTC)


 * That's out, then. If we are using &lt;math> then there is no easy way to make things line up, unfortunately. Maybe we could do a hack like this: x $$\mapsto$$ x2. The rest could be in regular wikitext. &mdash; Carl (CBM · talk) 16:46, 3 April 2012 (UTC)

I think, unless someone objects, I'm going to insert the paragraph above exactly as it appears. Thanks. Rick Norwood (talk) 19:54, 3 April 2012 (UTC)
 * First, I think the section is excessive in notation. There is no need to introduce 3-4 various ways of denoting the same thing in the lead. Second, I think it's better to wait with the lead paragraph. We should not move on without consensus this time. Isheden (talk) 20:10, 3 April 2012 (UTC)

Ok, I'll wait and see if others voice an opinion. Rick Norwood (talk) 20:45, 3 April 2012 (UTC)

No comments so far, so I'll mention another option. If the paragraph above is too long for the lead, it could serve as the first paragraph of the "Notation" section. Rick Norwood (talk) 12:12, 4 April 2012 (UTC)

Unless someone objects, I'm going to go ahead and put the paragraph in the Notation section. Rick Norwood (talk) 13:36, 5 April 2012 (UTC)

Help at Simple Wikipedia
There are only a few (none?) math editors at SW so I have started work on "function" over there with the idea of understanding just what "simple" might mean in this context. Anyone not too busy here would like to help out, SW Function Selfstudier (talk) 17:46, 4 April 2012 (UTC)

Moving right along
It seems as if we've done about as much on the lead as we can get a consensus on, and I think we've done a good job. Turning to the rest of the article, I've made a few minor changes, mostly matters of style. I'll pause for comments before I do any more. Rick Norwood (talk) 12:26, 6 April 2012 (UTC)

A goes to 1 or triangle goes to 3.
In the lead, the example of a function whose domain was not a set of "numbers" was the familiar map that sends A to 1, B to 2, C to 3, and so on. At some point a few weeks ago, this was first supplimented and then replaced by a map that sends a triangle to 3 and a square to 4. This isn't wrong, but there is a problem with it, in that it is not clear what the domain is. All polygons? All equivalence classes of polygons? All polygons in the Euclidean plane? My inclination is to restore the more familiar A maps to 1, B maps to 2, and so on. Comments? Rick Norwood (talk) 12:19, 4 April 2012 (UTC)
 * Unless you specify the domain and the range and not leave it to "intuition" or "a priori knowledge", either image is insufficient. To specify a function you have to specify a triple, correct?: ( "source", "destination", "graph"), i.e. ( S, D, G ) like the Bourbaki folks label it (my preference). And it would seem (more tacit, a priori knowledge) that you need to specify how the "triple" is to be read, i.e. ( (S, D), G), as opposed to ( S, (D, G)). Correct? Bill Wvbailey (talk) 16:41, 4 April 2012 (UTC)
 * I put that in because the previous sentence said the sets could contain anything for instance geometric figures. If geometric figures is not a sufficiently good domain for you then it should be removed from the previous sentence as well. Its seemed silly to just stick in geometric figures and the set of letters was not talked about and seems very limiting. Dmcq (talk) 18:07, 4 April 2012 (UTC)
 * The reference to "geometric figures" bothered me, too. What does a circle map to?  How about a fractal?  I think clearly "polygon" is intended.  You say the set of letters is "not talked about".  That's why I'm talking about it here before making any changes. Rick Norwood (talk) 18:56, 4 April 2012 (UTC)
 * Well the function isn't fully defined by just a couple of examples, in fact one could have hexagon go to 5 as far as the definition of a function is concerned. Personally by geometric figure I think of something drawn by a straightedge and compass, but from the origami standpoint I'd be quite happy with just lines. Dmcq (talk) 19:43, 4 April 2012 (UTC)

So, how about my suggestion to go back to a function that sends A to 1, B to 2, C to 3, etc. Well defined domain and codomain, familiar example, no hassle. Rick Norwood (talk) 20:06, 4 April 2012 (UTC)


 * Edit conflict: :Isn't there a risk of confusion if the mapping is per any kind of rule, such as for a "geometric figure" (e.g. a 2-dimensional n-sided symbol with n-exterior sides)? Don't you want an example that contains some elements very abstract and seemingly arbitrary together with something familiar, e.g. ( { ✻, ✣, ✩, ✦, ✶}, { suits of a deck of playing cards }, { ( ✻, ♣),(✦, ♦),(✣, ♡),(✶, ♥), (✩, ♣) } ) Or maybe the musical notations { sharp ♯, flat♭, natural ♮ } mapped to  { the suits of a deck of cards: ♣ ♠ ♡ ♢ ♧ ♤ ♥ ♦ }. (  These symbols appeared okay on my browser.) BillWvbailey (talk) 20:24, 4 April 2012 (UTC)

Didn't work in my browser, sad to say. I don't think we want something exotic, just something that is not a number. Rick Norwood (talk) 21:11, 4 April 2012 (UTC)


 * I suppose we could pick selected symbols from: {Roman alphabet} to { Greek alphabet } or to { Cyrillic }. But not in alphabetical order. Or more abstractly, from { Greek alphabet } map to selected symbols from { Cyrillic alphabet }: { Π, Φ, Λ, Ψ }, { Д, Л, Ж, И }, { (Π, Ж), (Φ, Л), (Λ, И), (Ψ, Л) }. Plus the symbols are pretty and had better work on all browsers. Bill Wvbailey (talk) 22:54, 4 April 2012 (UTC)

I really think we want as simple and short an example as possible in the lead. Rick Norwood (talk) 12:09, 5 April 2012 (UTC)


 * I also didn't like letters because once you make them into mathematical objects they are practically numbers anyway. That's why I'm not so keen on things like date to day of the week either though I guess it could work from the familiarity point of view. I'd prefer something that started off a bit mathematical. How about something like triangle to area? Dmcq (talk) 13:05, 5 April 2012 (UTC)

If we are going to use triangles at all, I think polygon to number of sides is simpler than triangle to area. But there is still the problem of exactly what the domain is. I'm not sure how you would precisely define the set of all triangles or the set of all polygons, the problem being the larger space in which such object exist. You can't use Rn for any fixed n, because there will be a triangle in Rn+1 that is not in Rn, though of course it will lie in a two-dimensional subspace. Rick Norwood (talk) 13:34, 5 April 2012 (UTC)
 * I think you're just finding problems where there aren't any. After all you wouldn't even have an area in projective geometry, I think most people would be pretty happy with something like that anyway. It is just an introduction and we don't need things like n-dimensional space in it. Dmcq (talk) 13:45, 5 April 2012 (UTC)

---

Here's Halmos 1970:31 Section 8 Functions:
 * "It is easy to find examples of functions in the precise set-theoretic sense of the word in both mathematics and everyday life; all we have to look for is information, not necessarily numerical, in tabulated form. One example is a city directory; the arguments of the function are, in this case, the inhabitants of the city, and the value are their addresses."

[There may be a slight argument with this in the case of some families with two domiciles or two phones but the phone book takes care of that. There's an example in our book where a couple has a residence and a barn, but the 2nd entry is listed as "barn" under their name:
 * John and Mary Doe [address, phone-number}
 * John and Mary Doe barn [phone-number]

The domain is clear-cut: names of people in the city. BillWvbailey (talk) 14:15, 5 April 2012 (UTC)


 * Halmos is certainly a good reference, but in his day fewer people had two phones. I still don't see the problem with the function that sends A to 1, but since several people don't like that, I guess we keep looking.  Meanwhile, the way the lead is now is not bad. Rick Norwood (talk) 14:25, 5 April 2012 (UTC)
 * Shouldn't that be Joseph and Mary Doe stable rather than John and Mary Doe barn? ;-) If you want things that close to real world I'd have thought date to day of week was quicker and easier. Dmcq (talk) 14:34, 5 April 2012 (UTC)

The only problem with date to day of the week is that it has a number input, but really any of these would probably get the point across. Rick Norwood (talk) 14:52, 5 April 2012 (UTC) ---

Chess pieces (32 total) to name/shape/ID of the pieces: { pawn, knight, rook, bishop, king, queen }. Or to number of pieces { 16, 4, 2 }, or to color "K" { K, ~K }. Or type of piece { pawn, knight, rook, bishop, king, queen } to number of pieces in the set. Bill Wvbailey (talk) 15:50, 5 April 2012 (UTC)

How about individual to his/her astrological sign? I realize the type of astrology could need clarification or definition. It seems to me that most readers will be aware of both domain and range and that neither is apparently numerical.71.31.146.16 (talk) 13:28, 19 June 2012 (UTC)

Formal Definition
From the section "Formal Definition":

"we say f is a real valued function of a real variable, and the study of such functions is called real variables."

I think it should be real analysis.

--ConferAll (talk) 13:19, 5 June 2012 (UTC)


 * You are correct. Rick Norwood (talk) 14:26, 5 June 2012 (UTC)

Formal Definition II
An unsophisticated question: In the formal definition, the sets X & Y have to be ordered? So, it is not possible to have a function on an unordered set?71.31.146.16 (talk) 13:13, 19 June 2012 (UTC)


 * No, the sets X and Y do not have to be ordered. The pairs that define the function are ordered pairs, so that we can distinguish (1,2) (meaning f(1) = 2) from (2,1) (meaning f(2) = 1). But there is no need for the underlying sets to be ordered. Gandalf61 (talk) 13:52, 19 June 2012 (UTC)

This is a good question, and I'm going to add a sentence to the article to help readers avoid this confusion. Rick Norwood (talk) 12:04, 20 June 2012 (UTC)

Suggestions for improving the article
As requested by I'm giving some ideas how to improve the article. Jakob.scholbach (talk) 10:16, 25 July 2012 (UTC)
 * First of all, the article contains many things twice or thrice. This usually is a sign of weakness of an article.
 * For the moment I would not touch the history section, where apparently someone spent lots of work, even though eventually the section will need some trimming.
 * Also, the lead should only be sincerely worked on when the article is nearly finished.
 * A possible article structure is this:
 * 1. Motivation and definition:
 * 1.1 Motivation a) explain in the simplest terms an example of a function: four apples on a plate, two red ones, a green, a yellow one. The function assigns to each apple its color. b) Give the "fuzzy" definition of a function like "a fn. is a rule ...". Maybe point out that it will be clarified below. c) Give another example from the naturals to the integers, x maps to 3-x. d) Give two non-examples, where either not every element has an image or some element has two images.
 * 1.2. Formal definition: explain calmly what an ordered pair is and the cartesian product. Define a function as a subset of X x Y. The article currently has an "in the other definition a function is defined as a set of ordered pairs where each first element only occurs once." IMO, this definition should only appear as a side note since it is not widely used. Explain slowly, every notion should always be explained by at least one of the two introductory examples above.
 * 1.3 Notation. Be brief here. Postfix, infix notation should be moved somewhere later, if possible.
 * 2. Specifying a function. This section is surprisingly short. Too short. Explain a) list of values, b) graph, c) Formulas, d) Algorithms. Try to be concise and brief in the last two, this is not really specific to functions, I believe.
 * 3. Operations with functions
 * 3.1. Composition (example: the above color-of-the-apple-function composed with "I like the color"-function)
 * 3.2. Image and preimage
 * 3.3. Injective, surjective, bijectve. Illustrate the notion of cardinality using bijections.
 * 3.4. Identity, inverse function. Here might be a space for a little proof: f is bijective iff it has an inverse function.
 * 3.5. Restriction etc. Explain here why domain and codomain are part of the definition by pointing out that in/surjective can get destroyed when altering (co)domain.
 * 4. Functions of many variables. Example: addition, multiplication. Here might be a good spot for mentioning infix notation. Explain why this is actually a special case of the single-valued function. Also explain function spaces. The stuff under "currying" should also go here.
 * 5. Common examples of functions: along the lines of the see also list. Start with real-valued functions. Sums and differences etc. of such functions. Point (briefly!) at continuous, differentiable functions.
 * 6. Generalizations: many-valued functions (e.g. complex root, cx. logarithm), partly defined functions (division by zero?). Maybe a word (not more) about functors.
 * 7. History (possibly earlier in the article, but that section needs separate consideration).


 * I'd separate out the history section into something like "history of the function concept" first before doing anything like that and try and put a decent summary here.
 * I agree. I did the same for the Church-Turing thesis article -- both split out a separate sub-article History of the Church-Turing thesis and then wrote a summary for the main article. I am the author of most of this section, and I can testify to the amount of work it represents; I'd hate to see it bowdlerized. As there's more to be written, and as it is rather long as it stands, a split-out might be a good thing. Problem is, the notion has "morphed" from the 1700's to now; to write a good summary will not be a trivial undertaking. BillWvbailey (talk) 12:50, 25 July 2012 (UTC)


 * I don't see this many things twice or thrice excluding stuff in the lead which should summarize the article as well as being a quick readable intro.


 * I'm not sure I like motivation and definition stuck together. Why not just change 'intuitive description' into motivation - why should definition go down a level and would notation stay at its current level?


 * In the definition section there's quite a few citations which just define a function as a relation without specifying X and Y especially in logic and in theorem proving, I guess that's the same as them having natural numbers from 1 instead of zero which is also common for them, just because it is less common elsewhere doesn't mean it is something we can ignore as I have seen it in calculus textbooks.


 * A bit more on specifying a function sounds good. That's what motivates my thoughts about moving the history section to a subtopic.


 * I believe I had some problems with the ordering of the types of function section because of dependencies, I wanted to avoid referring forward. You'd need to be careful about that. Dmcq (talk) 11:43, 25 July 2012 (UTC)


 * A very fundamental concern expressed by Jakob.scholbach is


 * 3.5. Restriction etc. Explain here why domain and codomain are part of the definition by pointing out that in/surjective can get destroyed when altering (co)domain.
 * Indeed, experience with both theoretical and practical aspects of functions in a great variety of applications demonstrates that it is advantageous to include in the definition of a function only the information that is present in the corresponding set of pairs, nothing more (and obviously nothing less). Associating a codomain is overspecific under all circumstances, and the (essentially illusory) advantages of making codomains a part of the definition are always more conveniently obtained in a different way.  Also, if in the triple (X, Y, F), the symbol X by definition denotes the domain, it is redundant since the domain is specified by F (Aside: some rare conventions (horresco referens) let X  stand for a superset of the domain, which establishes symmetry but is as useless as the codomain).  Since the article has already recognized the existence of codomain-free definitions for quite some time, I made three small illustrative additions, completing some definitions to partially restore the balance.  To conclude: the burden of proof (by comparing the pros and cons) is on the editors who want to keep the codomain-laden definition as the primary one and the codomain-free definition as secondary.  Most arguments strongly favor a reversal. Boute (talk) 11:54, 25 July 2012 (UTC)
 * I can't say I agree with you. If you want to treat a set of functions together they have got to have something in common and having a common domain and codomain is the usual basic property. Otherwise one might as well just deal with an individual functional relation of a particular one. There is no point in talking about 'function' if dealing with a particular instance like a square root. As to (X,Y,F) that comes from the definition of a correspondence, besides which it destroys the whole point if you just start talking about a set of functions which can be put in the form (Y,F) where the domain of each F is common. In logic there is a point to functions without a specific domain and codomain but I think one is better off dealing with them as functional relations. Dmcq (talk) 13:25, 25 July 2012 (UTC)
 * Thanks: your comment allows illustrating what I mean by saying that you can get what you want/need much better without codomains. To treat a set of functions together, use X → Y, which designates the set of functions with domain X and range included in Y, as in the International Standard document just added to the references (for once, the standard has the best engineered definition). Note that the domain always is an essential part of the function definition.  If the function is defined by giving its graph (as a set of pairs), the domain is the set of first elements.  If the function is defined via f(x) = e where e is an expression, the domain has to be specified explicitly. Boute (talk) 14:35, 25 July 2012 (UTC)
 * @Boute: Do you have a reference at hand that covers the two definitions of a function and compares the implications of the two as far as other basic notions (composition, inverse function, surjectivity etc.) are concerned? Jakob.scholbach (talk) 15:32, 25 July 2012 (UTC)
 * I don't have a reference for a complete comparison, but the paper on generic functionals discusses the arguments in favor of the codomain-free definition and elaborates the ramifications (composition, inverse function, surjectivity and much more) of this design choice. It is a fortunate coincidence that this is also the choice made in the ISO standards. Note that Rudin in Real Analysis (p. 8) uses the same conventions.  He also talks about f : X  → Y being a function onto Y (iff the range is included in Y), not just onto!  It is not clear where the codomain-based overspecification came from; it is certainly restrictive (compare the composition definitions). Boute (talk) 16:05, 25 July 2012 (UTC)
 * I've just removed the bit you put in saying function composition involved doing in effect a database join between the two relations so it chops down the domain of the first function applied. I have never seen that being done and your citation talked in terms of functions where there was a match of the codomain and domain of the next function. At best I'd have thought for a codomain free way one would say the composition was defined if the image was included in the second functions domain, anyway what you say really does require a good citation.
 * The fact that you have "never seen that being done" is a personal issue and does not invalidate a perfectly good reference (and the superior conceptual view reflected in it). The remark "justifying" your removal is inappropriate: the reference exactly says the same thing as my addition.  Please read it, be fair, and replace the addition (with Bartle as a second reference - you want more?). Boute (talk) 19:14, 25 July 2012 (UTC)
 * The citation did not back up what was said. Please do not split up other peoples contributions unless htey are very long and there is a good reason. Dmcq (talk) 22:22, 25 July 2012 (UTC)
 * Please explain why you think that any of the the citations (Meyer, Bartle) did not back up (exactly) what was said. Boute (talk) 02:32, 26 July 2012 (UTC)
 * Thus I wonder about the definition in this ISO standards document - not that I consider it a bit of mathematics but I guess we should do something about it. Does anybody have access to that and can they say exactly what it says about functions rather than interpreting it thanks?, does it define what a function is what does it say about domain and range or whatever? I've had that document quoted a couple of other times about intervals and spherical coordinates and it is annoying not to have a clear idea where it gets its ideas from or who uses it but I didn't get the idea that it actually named anything because it was in many languages - it would be useful to have names for the different spherical coordinates. Dmcq (talk) 18:05, 25 July 2012 (UTC)
 * As this is an occasion where the ISO standards got the definitional design exactly right, nothing should be "done about it", apart from emulating it. Also, again you suggest that I am "interpreting" sources.  Never would I stoop to such intellectual dishonesty (apologies accepted). I always have the documents from which I quote at hand (black on white on paper, yes), and reproduce the content faithfully. Here is the literal quote:
 * f : A  → B, meaning: f maps A into B, remarks: the function f has domain A and range included in B.
 * I have seen many variants of the function definition over more than half a century, and some of them (the ones with least ballast, not surprisingly) stand head and shoulders over others. To the Rudin reference, I might add Bartle, who defines composition (p. 16, theorem 2.3) in the same codomain-free way as Meyer. Boute (talk) 19:14, 25 July 2012 (UTC)
 * Boute, you seem to be reiterating your point that your preferred definition is in some way superior. I'm not convinced of that: having a clean condition for the surjectivity of an application is a must in many mathematical branches. The majority, if not quasi-totality of mathematical sources does define a function using its codomain. If we want to keep this second definition in more than just a sentence or two, we need an authoritative source comparing both approaches. An apparently non peer-reviewed paper written by yourself (?) is not a good base to stand on. As far as I can see right now, this codomain-free definition should be covered only briefly. The article has many more problems right now, but this will be something to adress at some point. Jakob.scholbach (talk) 21:25, 25 July 2012 (UTC)
 * The condition for f being onto Y (Y explicit, not an attribute of f) in the codomain-free setting is as clean as any other and fulfils all purposes you mention (if you have something else in mind, please give a specific example). I disagree with the "majority" remark, which only indicates that you have seen different books (most of the dozens of books I have seen on this subject give codomain-free definitions, some don't even use the word; the only exceptions are books on category theory, some of which (e.g. Pierce) explicitly warn that the category-theoretic "function" does not have its ordinary mathematical meaning).  The paper I mentioned was peer-reviewed and cited only because you asked some information. I also mentioned other sources.  Meyer (1990) and Bartle (1964) give the same codomain-free definitions, also for composition (only difference: Meyer writes f;g for composition).  Quoting Bartle literally:
 * If f and g are functions, the composition $$\ g \circ f\ $$ is a function with
 * $$\mathcal{D}(g \circ f) = \{x \in \mathcal D(f):f(x) \in \mathcal D(g)\}\ $$ [more]
 * Notice how codomains would get in the way. Boute (talk) 02:32, 26 July 2012 (UTC)
 * I don't see how that definition from the ISO standard supports anything except the function from A to B given in this article as the first definition. It does not say anything about f itself which is what the bit following that talks about and that  there is a disagreement about. Dmcq (talk) 23:00, 25 July 2012 (UTC)
 * The ISO document says "domain A and range included in B "; a codomain-based definition would have said directly "domain A and codomain B ". Codomains are nowhere mentioned. Boute (talk) 02:32, 26 July 2012 (UTC)
 * To be clear a codomain free definition is one that says something like {<1,A>,<3,C>,<} is a complete definition of a function, say f, and the composition rule you were specifying would say that if g was {,,} then their composition gf is defined and is {<3,triangle>,}. Is that correct? Dmcq (talk) 23:36, 25 July 2012 (UTC)
 * That is indeed correct (also matches Bartle's and Meyer's definitions). Boute (talk) 02:44, 26 July 2012 (UTC)
 * I will sign off for some time (weeks, months) to do other work, but will regularly look here for further comments. Particularly interesting would be "statistics" on codomain-based versus codomain-free definitions in the mathematical literature (category theory excepted; there the statistics are clear) and the changes over, say, the past 60 years. Of course, a meaningful preference would have to be based on a systematic comparison of the pros and cons, not statistics.
 * Important afterthought (so evident that I forgot mentioning this before): any explanation of the notion of "function" should include a clear definition for function equality. Too many texts leave this implicit.  Equality is arguably the main lever in reasoning about any mathematical object. Boute (talk) 02:44, 26 July 2012 (UTC)
 * Couldn't resist briefly looking into some more literature and reporting. First book grabbed from my shelf: Apostol's Calculus, p. 54: f and g are equal iff they have the same domain and f(x) = g(x) for every x in that domain (same def in Axler's Precalculus p. 35).  No codomains are involved.  Kolmogorov and Fomin Introductory real analysis (p.5, top, codomain-free).  Although analysis texts, these sources define functions on arbitrary domains (fully general).  From CS texts: Lamport Specifying Systems p. 48: same (codomain-free) equality definition. Boute (talk) 03:31, 26 July 2012 (UTC)
 * I don't know whether you looked in the archives or not but this issue was discussed (at length, ad nauseum...)not that long ago. After much debate and research, it became clear that there are two accepted definitions of a function, one as ordered pair and the other with codomain with the first being preferred in set theory/logic. Also, current education practice (in the UK, not sure about the US) up to undergraduate is to first teach the definition that includes codomain and attention is only given to the ordered pair definition if a student's later studies take him or her in a direction where that would be useful. The article does reflect this situation (if anything it gives too much prominence to the ordered pair definition in my opinion).I do agree with you as to the definition of equality (evidently different in each case)Selfstudier (talk) 09:09, 26 July 2012 (UTC)
 * Well I've had another look at Bartle because of this composition business and it seems he used different definitions of composition in different books. I looked at Introduction to Real Analysis by him and Donald Sherbert and it says "In order to be able to do this for all f(x) we must assume the range of f is contained in the domain of g", however in his 'The Elements of Real Analysis' he clearly does do what Boute says and the composition of two functions has a signature defined by the functions themselves and their composition can be null. I'll go and stick that bit I removed back in with the reference to the one where he does the joining business. Dmcq (talk) 09:51, 26 July 2012 (UTC)
 * The ISO reference doesn't seem to say anything useful. Just because it doesn't mention codomains doesn't mean anything one way or the other and I don't believe there is a standard term codomain free. It doesn't seem to show the whole notation just the bit for values so that is some evidence but I don't believe it was supposed to be any sort of thing except a display of notation. Dmcq (talk) 10:30, 26 July 2012 (UTC)
 * Boute,I don't know if this link is helpful for you ; (sorry I have not the time to go digging out again all those references you asked me for on my talk page)Selfstudier (talk) 13:29, 26 July 2012 (UTC)
 * Very helful, thanks (Bourbaki texts are expensive).Boute (talk) 07:08, 27 July 2012 (UTC)
 * Interesting experiment for all concerned: Google on "functions f and g are equal" (quotes included). Most instructive! Boute (talk) 07:08, 27 July 2012 (UTC)
 * discusses all these issues (as an aside, note that "image" is nowadays the standard usage in the UK and that previously "range" meant codomain but now is not really used at all.Selfstudier (talk) 08:39, 27 July 2012 (UTC)
 * Right to the point! Observations about this reference: (a) Its essence is also found in Halmos Naive Set Theory pp. 30-33 (1960!).  (b) In particular, it defines a function as a set of pairs (not some triple with what is arguably ballast), which is essentially codomain-free.  Three cheers for simplicity!  (c) The name confusion image-range-codomain (or similar) is easy to cope with: in anything you write, define exactly what you mean (don't assume any "standard" terminology; there is none) and briefly warn for variants; done! Boute (talk) 11:22, 27 July 2012 (UTC)

I started restructuring the article along the lines above. In particular, all mentions and implications (unless I missed something) of the codomain-free definition are now in one subsection, where they should be kept. I prefer not to have this definition in the way of the standard one, in order to keep things as simple as possible. (Otherwise we need to repeat "with definition a) ..., with definition b)".) We need to avoid phrases like "a [property of a function] is often called [this and that]", since such a seemingly vague statement causes confusion on the part of the reader. Boute and others, maybe you could polish the corresponding subsection into a coherent bit of text? (Please don't make it longer, just better :)) Jakob.scholbach (talk) 12:59, 27 July 2012 (UTC)


 * Good idea. Sorry I can't contribute to editing for some time (just briefly comment).  Also, be careful not to leave the article in an intermediate state that may be confusing or contain inconsistencies.  Example: the second and third paragraph of "Formal Definition" should make clear to what extent different conventions are used: a function as a set of pairs, and a function as a triple whose third element is a set of pairs. Boute (talk) 14:42, 27 July 2012 (UTC)


 * Ordered pair definitions have a direct product A×B that the ordered pair  is contained in, where A is the domain and B the codomain. While older books sometimes use different words, the modern books and articles I read are pretty standard in using "domain", "codomain", and "image". Rick Norwood (talk) 13:06, 27 July 2012 (UTC)
 * Wish that were so. I have seen half a dozen fairly recent category theory books where one calls "domain" what the other calls "source"; similarly for "codomain" and "target". Even in some recent UK book (by top-class authors) I saw "range" where Selfstudier would use "image". Let's not overly worry about words, but just warn the readers. Boute (talk) 14:29, 27 July 2012 (UTC)

''I give up. This can't work. Reason: Bloch, Proofs and Fundamentals'' p. 131. Quote (modulo typography):
 * Definition 4.1.1. Let A and B be sets. Giving due weight and avoiding giving undue weight means that articles should not give minority views as much of, or as detailed, a description as more widely held views. Generally, the views of tiny minorities should not be included at all, except perhaps in a "see also" about those specific views, denoted f : A  → B, is a subset F of A x B such that for each a in A, there is one and only one pair in F of the form (a, b).  The set A is called the domain of f and the set B is called the codomain of f.

Why is this so horrifying? (a) "denoted f : A  → B " suggests that the function is denoted f : A  → B. OK, sloppy English, let it pass. (b) "a function f ... is a subset F of ...". The "is" can only mean f = F. Anyway, the full sentence says that a function from A to B is a subset of A x B with certain properties. (c) The domain can be derived from this subset ("for each a in A "). The codomain (as defined) cannot, unless the function is onto B (which we do not know), so it is not a well-defined property of the function (defined as the aforementioned subset). The subsequent "explanation" on the same page (131) only makes matters worse. Boute (talk) 17:23, 27 July 2012 (UTC)


 * All very well so now you need to find another author who has identified the efforts of Bloch as being wrong and cite that author. The way Wikipedia works, at the end of the day it doesn't matter what I or any other editor thinks if it cannot be suitably backed up from an independent source or is contradicted by an independent source. Fact is, as was stated in the source I gave you, many books and sources have definitions and content that others describe as being wrong, how to finally determine which is correct? Wikipedia leaves it to the reader to examine the sources and decide for themselves.Selfstudier (talk) 18:04, 27 July 2012 (UTC)
 * You are right in describing the manner in which some think Wikipedia should be managed. I studied the rationale for the many Wikipedia rules, and they make sense only for sociological, historical, etc. issues, which needs backing up by independent sources, as is also the case for journalism.  However, mathematics is something quite different.  If something is inconsistent, it suffices pointing it out, and no backup is needed.  Perhaps Wikipedia is not suited for mathematics.  On the other hand, I strongly endorse your remark that the readers should decide for themselves, but editors should be able to agree on what to provide them. Boute (talk) 18:18, 27 July 2012 (UTC)
 * It was a while ago now so I cannot remember exactly, there was an argument about how the issue of describing a function as a rule (commonplace with many authors) should be dealt with; I think Bloch was originally cited in support of it being wrong to describe a function as a rule and then, having been cited for that, was then cited for other things as well. So although I agree about editors being able to "agree on what to provide", the reality of the situation is that not all the editors are present all of the time. What tends then to happen as far as I can see is that the article slowly evolves, perhaps in fits and starts, over time and usually improves.(Math articles not that long ago were often very poor).Selfstudier (talk) 18:32, 27 July 2012 (UTC)
 * What Bloch said was a function from A to B, not just a function, was a set of pairs. There is no need to derive A if we are talking about a function from A to B because A is given. Dmcq (talk) 20:08, 27 July 2012 (UTC)
 * You correctly pointed out one of the internal discrepancies in Bloch's definition. If a function is a set of pairs with the usual unique second element property (a commonly accepted formal definition since Halmos (1960) or earlier), then that's it: the function itself contains no other information (once again: three cheers for simplicity).  The domain is by definition the set of first elements (of the pairs), and the image (some call it range) the set of second elements. Saying that A is given can only refer to the context, and creates an obligation: if one wants to claim that a function is from A to B, one has to verify that the domain is exactly A, and that the image is contained in B.  Often this verification is trivial (e.g., by construction), but that does not make it less essential conceptually.  A related issue: going via the Cartesian product A x B is correct, but is criticized regularly by math educators.  Fortunately, the concept of pair is more primitive than that of a Cartesian product (in fact, the CP is defined as a set of pairs).  So, functions (and, by the same token, relations) can be defined in an equivalent way without mentioning the CP, saving the CP for later ("Remember how we defined functions (relations)? Well, ...").  Conclusion: one can please the educators without sacrificing rigor. An opportunity. Boute (talk) 05:53, 28 July 2012 (UTC)
 * There is no internal discrepancy that I can see. Bloch explains what is happening with the definition he is used pretty clearly. One doesn't have to verify anything. His function is a triple and his function from A to B is a set of pairs. Have a read of the paragraph after the definition. Dmcq (talk) 15:25, 28 July 2012 (UTC)
 * Of course I read that paragraph but, unfortunately, it introduces more discrepancies. One example: the function written as a triple (A, B, F) can only be f (even the chaotic explanation leaves no other interpretation).  So f clearly contains the A and B information.  Yet, at the end, the author pontificates against writing only "let f be a function", arguing that it would leave the domain and codomain unknown!  Returning to the original discrepancy.  The definition says "a function f from A to B", so f is the function from A to B.  It is the subject of the sentence.  Then the definition continues saying about this subject that it is a subset F of A x B. Then I say: forget it. Of course, we all know how to explain these things correctly, but that is an entirely different issue.  Boute (talk) 20:29, 28 July 2012 (UTC)


 * An anecdote, relating to the desire to reduce all mathematics to set theory (this has some merits, not discussed here, but also risks, the topic of this anecdote). Halmos (p.22) defines an ordered sequence c b d a as the set {{a, b, c, d}, {b, c}, {b, c, d}, {c}}.  Evident at first sight.  But look how it all collapses if a = b = c = d!  Similarly, set theorists define an ordered pair by (a, b) = {{a}, {a, b}} (Halmos, p. 23).  The earlier collapse still occurs, but if you talk only about pairs (knowing the length is 2) it is harmless.  However, the definition still entails silly artefacts, such as {a} and {a, b} both being members of (a, b).  That is why Reynolds introduces the concept of primitive pair - outside set theory. Boute (talk) 05:53, 28 July 2012 (UTC)


 * Another anecdote (also directly relevant to this article) Velleman's How to prove it (2009) defines a function on p. 226 as a set of pairs, just like Halmos and many others.  In particular, defining f : A → B implies that the domain of f is A and the range included in B (exactly as in the ISO standard). Illustration: example 7 (p. 227): defining f := {(x, y) : Real x Real | y = x2} makes f a function from Real to Real ("to" meaning "into").  Note that this f is also a function from Real onto Real>=0.  The notion "onto" is defined p. 236 in a very strange way: "Suppose f : A → B".  We say that f is onto iff for all b in B there is some a in A such that f(a) = b", as if "onto" were an adjective, not a preposition, and as if, given f, the set B is also known (it is not).  Halmos (p. 31) does it linguistically and mathematically correctly, saying "f maps A onto B", mentioning B explicitly since it is not an attribute of f.  Lesson: we can't be careful enough in selecting sources. Boute (talk) 08:32, 28 July 2012 (UTC)


 * "Onto" is indeed an adjective for many mathematicians, and is the property of a function, which they define as a triple. That trend is very common in the algebraic and category-theoretic contexts. For example the inclusion $$\mathbb{N} \to \mathbb{R}$$ is not a surjection, but the identity isomorphism from $$\mathbb{N}$$ to itself is a surjection - as functions these may actually have the same graph! It is only in the set-theoretic setting where a function is identified with its graph. There are two definitions in the literature, which differ in this way, and neither is superior to the other. &mdash; Carl (CBM · talk) 17:33, 28 July 2012 (UTC)

"Onto" in the mathematical sense is not a preposition (not the same as "I get onto the streetcar."). The easiest way to see that is to replace it with its synonym, "surjective". Rick Norwood (talk) 12:30, 28 July 2012 (UTC)


 * There are two issues: (a) The mathematical inconsistency as described, which is independent of what wording one uses (onto as a preposition, onto as a synonym for surjective) because it follows from precise definitions. It is important to understand that inconsistency first.  In fact, it appears in more than one recent textbook, and the authors I contacted recognized it easily and immediately.  (b) Since before 1970 (!), I am familiar with the habit of some mathematicians of using "onto" as an adjective.  First, poor language usage by some does not make it right (certainly not for all).  Second, top mathematicians like Halmos, Bartle, Rudin, Royden, Herstein, who were among the originators of the terminology, without exception used "onto" as the preposition it is in natural language and always wrote "onto B" or some other set (please look it up). It is exactly sloppiness like using "onto" as an adjective (or maybe still as a preposition but as a shorthand referring to a set specified in the context) that made alertness drop and caused inconsistencies as described.  Lesson: beware of apprentice sorcerers; let's remain critical and selective. Boute (talk) 19:55, 28 July 2012 (UTC)


 * Surely you are aware the the term "surjective" is due to the Bourbaki group, who were among the most influential writers in the 20th century. It's strange to attribute their terminology to a lack of mathematical acumen on their part. Bourbaki defines a function as a triple $(G,A,B)$, including the graph $G$, domain $A$, and codomain $B$. Now Bourbaki certainly did not use the term "onto", because their original language was French, but particularly in lower-level maths we often use "onto" in English as an adjective synonym for "surjective" (which is also used in English and French). There's nothing mathematical suspect about it, although it is not the convention used in set theory books. &mdash; Carl (CBM · talk) 20:33, 28 July 2012 (UTC)
 * Did you read anything about "surjective" in my comment above? Let's stay with the subject: "onto" (you just repeated what I said, namely that some people use "onto" as an adjective, but it is better to emulate the more numerous people who use it correctly). Even better, let's concentrate on the essence: avoiding inconsistencies in definitions, irrespective of choice of words. About Bourbaki: their heyday was in the time I was a student, so we got plenty of exposure. They are to be admired for what they did well, but were nearly complete theoreticians, and their definition of functions as triples is too narrow to match the wide diversification in applications of mathematics over the past 50 years.  This can be seen even in various fields of engineering based on classical mathematics (e.g., signal processing), but even more in areas that emerged from computing, such as theories of programming languages.  Example: recall the function composition definition from Meyer's book on programming languages, also found in Bartle's book on analysis, and try to do that (or something comparably general, let's be generous) with the triple formulation.  Good luck! Boute (talk) 21:49, 28 July 2012 (UTC)
 * Actually, the typing that is standard in programing language matches much better with the "triple" definition, because a function in that context always comes with a type. But ignoring that, the main point is that both definitions are well established by mathematicians with impeachable credentials, neither definition is superior to the other, and this article covers both equally. It doesn't really matter if you prefer one version over the other; if you want to continue arguing that those who view surjectivity as a property of a function (including Bourbaki) are somehow "incorrect", I don't see much reason to respond further. &mdash; Carl (CBM · talk) 22:21, 28 July 2012 (UTC)


 * You are correct in saying that in progrmming, as in mathematics, functions always come with a type, but the "triple" conclusion is a complete non sequitur. To witness: f : A → B in the context without codomains just restricts f so that its range is included in B (as stated in the ISO standard), in the context with codomains it also makes B a function attribute, in both contexts it makes A the domain.  Also, nothing in the discussion the two of us were having thus far refers to preference for definitions, so please don't muddle issues.  Thus far, we were only talking about (a) (in)consistency (mathematical issue), (b) proper language usage.  (a) the inconsistency mentioned has been recognized. (b) In the context without codomains, proper usage is "onto B" or "surjective on B"; in the context with codomains, "surjective" is the only proper usage. There are so many authors respecting proper usage that there is little excuse for emulating improper usage.  Since now you also brought up (c), preferences regarding codomain-free/codomain-based definitions: no one with minimal mathematical maturity would use the term "incorrect" for a consistent view, and the triple view is certainly consistent (even though the explanation I criticized makes a complete mess of it).  Generality and flexibility, however, are an entirely different matter, and an important citerion in deciding preferences.  I'll come to that after the comment of Dmcq next. Boute (talk) 07:49, 29 July 2012 (UTC)


 * There's no problem doing either form of composition with either definition. The real point is that how often does one want to do either since operations suited for each involve extra work with the other. Dmcq (talk) 00:34, 29 July 2012 (UTC)


 * This brings up a good point: preferences between the two approaches must be decided on a sufficiently thorough comparison of the extra work needed in one formalism to obtain the (real or perceived, important or unimportant) advantages of the other.  This was done years ago (before 1990, in the early phases of the FORFUN project: "Formal description of arbitrary systems by means of functional languages").  I worked a long time with the codomain-based view because it looked more symmetric (it is -- for relations), but the exttra work for defining operators over functions, not just composition, forced me to abandon codomains.  I am very curious whether Dmcq found a simple way; he might still revert my preferences.  The extra work in the other direction, namely when adopting the codomain-free view, turned out to be minimal: writing "onto B" or "surcective on B" rather than just "surjective". Boute (talk) 07:49, 29 July 2012 (UTC)

All this is very well, but we should have these discussions off-line since they delay improving the Wikipedia article. It is evident that, next to the classical function-as-a-set-of-pairs-and-nothing-else view, the codomain-based view and the triples view (they are not the same!) also deserve mention. This brings up two action points The reason for adding the "working mathematician's" qualification is that the triple view is well-explained and prevalent in the category theory literature (I have no "independent source" saying this, but a pile of category theory books proving it to my satisfaction), but reflects a different notion of function than in the rest of mathematics (as remarked by by Pierce). Boute (talk) 07:49, 29 July 2012 (UTC)
 * Finding a reliable source that gives an accurate picture of the relative frequency of the various views in the "working mathematician's" literature. This is one of the Wikepedia criteria for determining relative coverage (although I find the resulting conservatism often revolting);
 * Finding better sources in the "working mathematician's" literature for the triple view than the earlier reference. The concept deserves to be done more justice.

I'd like to say I think the example of apples is terrible. There is no implicit way of distinguishing them and also having both four colours and four apples is asking for trouble - readers have to work out where a four came from. Just saying the number of possible pairs is sixteen implies what? That one can get any subset? If one is going to give a count one should give the number of possible functions too which here is 44 but of course here one gets the problem of where each 4 comes from. Can't we just have something like the matches a team plays {A,B,C,D} and possible results {lose,draw,win}. Explaining that all matches must have a result but not all the results may occur is easy enough. Dmcq (talk) 15:58, 28 July 2012 (UTC)

Also the triple definition of a function seems to have disappeared and the function is a pure set of pairs has been moved to an alternative. I think it is right that the discussion of a function without saying f from A to B be done separately but it shouldn't be shoved to the end like that and the triple definition removed. Dmcq (talk) 16:24, 28 July 2012 (UTC)

I didn't want to be the first to say something, there was an earlier agreement or understanding at least, that major changes to the article were to be the subject of discussion in here first. Not all the changes that have been made recently are major but some of them are not minor and I think ought to be reversed until they have been discussed and agreed to.Selfstudier (talk) 17:02, 28 July 2012 (UTC)


 * I don't think it's awful, although the section on the two definitions is not very clear at the moment. &mdash; Carl (CBM · talk) 17:39, 28 July 2012 (UTC)


 * Well, I don't think it's awful; OTOH, it wasn't awful before, either. Had the changes been limited to presentation and layout, I would have had no complaint.Selfstudier (talk) 18:35, 28 July 2012 (UTC)
 * Three definitions sort of, the main function from A to B definition being a set of pairs can be considered valid whichever way one thinks of an unadorned function. Dmcq (talk) 18:57, 28 July 2012 (UTC)

OK, I see that we are now allowing threaded conversation as opposed to endless tape....:-) Still, I remain with the impression that the net effect of recent changes has not resulted in an improvement of the article. If an editor would like to promote some alternative viewpoint then it is incumbent upon them to find source material in support of their position (and ideally, garner some support (or at least, no objection) for that in here prior to making substantive edits). Selfstudier (talk) 11:31, 29 July 2012 (UTC)

I have no strong objection to parentheses for ordered pairs, but I wonder why are "awful". Rick Norwood (talk) 12:13, 29 July 2012 (UTC)

Back to business
Following my own proposal, I have resolved not to waste time on discussions that delay improving this Wikipedia article, and conduct them off-line to the individual editors' talk pages (insofar as time permits). Also, as a first contribution to the (overdue) action points, I start a list of references, meant as a common working data base, which I hope will be extended by other editors.

Initially, they are put into the following groups:
 * A. The classical formal view: the function as a set of pairs and nothing more
 * B. Other definitions (the function as a rule) without codomains
 * C. Functions with codomains (or equivalent)
 * D. Functions as triples

Rearrangement may be necessary later. Thus far, I listed textbooks only.

A. The classical formal view: the function as a set of pairs and nothing else


 * Tom M. Apostol, Calculus, Vol. I, 2nd. ed. John Wiley (1967). Comment: function page 53, equality theorem page 54, composition page 140.
 * Robert G. Bartle, The elements of real analysis. John Wiley (1964). Comment: functions page 13, composition page 16.
 * David Gries and Fred B. Schneider, A Logical Approach to Discrete Math. Springer (2010). Comment: function p. 280.
 * Paul R. Halmos, Naive Set Theory. Van Nostrand Reinhold (1960). Comment: function p. 30.
 * Israel N. Herstein, Topics in Algebra. Xerox College Publishing (1964). Comment: function (called mapping) p. 10.
 * Steven G. Krantz Real Analysis and Foundations. Chapman & Hall/CRC (2005). Comment: function p. 20.
 * Bertrand Meyer, Introduction to the Theory of Programming Languages. Prentice Hall (1991). Comment: function p. 26, composition p. 32.
 * Halsey L. Royden, Real Analysis. Macmillan (1968). Comment: functions p. 8. Identifies functions with their graphs only for functions with sets as domains (e.g. not for the identity function defined by i(x) = x for all x).
 * Daniel J. Velleman, How To Prove It: A Structured Approach (2nd. ed.). Cambridge (5th printing 2009). Comment: function p. 226.

B. Other definitions (the function as a rule) without codomains


 * Sheldon Axler, Precalculus: A Prelude to Calculus, preliminary edition. John Wiley (2008). Comment: functions page 33 (limited to reals, trivially extendable to arbitrary domains), equality page 35.
 * ISO/IEC, Quantities and units --- Part 2: Mathematical signs and symbols to be used in the natural sciences and technology. ISO 80000-2 (2009-12-01). Comment: classification tentative, “function” not defined explicitly.
 * Andrej L. Kolmogorov and Sergey V. Fomin Introductory Real Analysis. Dover (1970). Comment: real functions p. 4, evident generalization p. 5.
 * Leslie Lamport, Specifying Systems: The TLA+ Language and Tools for Hardware and Software Engineers. Pearson Education Inc. (2003). Comment: function p. 48, equality p. 48.
 * Walter Rudin, Principles of Mathematical Analysis. McGraw-Hill (1964). Comment: function p. 21

C. Functions with codomains (or equivalent)


 * Serge Lang, Undergraduate Analysis. Springer-Verlag (1983). Comment: does not use the term “codomain” but the remark on p. 5 shows that functions with respective types A -> B and A -> B' where B /= B' can never be equal.

D. Functions as triples


 * (To be completed)

End of bibliography. Boute (talk) 13:12, 29 July 2012 (UTC)


 * Of course, there is an existing database of sources included already in the article and nothing prevents you from adding these in the appropriate place (if an appropriate place exists)Selfstudier (talk) 13:40, 29 July 2012 (UTC)
 * Isn't Wikipedia great? Still, since I want to be selective (and currently am very busy), I'll do that later.


 * I am a bit baffled by this distinction that is being made; from my own perspective,a function (or map) is a triple (A,B,f) (say), A domain, B codomain (then for all x in A there exists y in B etc etc). For me, f:A arrow B is simply the traditional notation for that triple.Selfstudier (talk) 14:03, 29 July 2012 (UTC)
 * If that is your perspective, you may enjoy or deplore the rich variety of variants that have appeared during the history of the function concept, even if considering only the last 60 years or so (the "formal era"). Enjoy for the curious, deplore for the educator or Wikipedia editor who has to present it systematically.  The best way to appreciate (enjoy) all this is reading the sources.  Still, a brief explanation may suffice here.
 * For group A, a function is nothing else (crucial!) than a set of pairs satisfying the functionality condition, meaning that no two pairs have the same first element. The domain is by definition the set of first elements, the range (some call it image) the set of second elements.  They need no separate definition (no triples).  Writing f : A → B specifies the type of f, and can be seen as a user interface specification.  It says that f is defined for elements of A, the domain (in the input-output view: specifying input values for which the output is guaranteed). Obvious obligation: you may write this only if the set of first elements is indeed A.   It also says that the range is a subset of B (see the ISO standard).  In the I/O view: it guarantees that, if the input is in A, the output will be in B.  You may write this only if all outputs are indeed in B.
 * For group B, a function is defined by some "rule". Group A motivates their view by the fact that "rule" is vague, but that is another matter.  Anyway, let f(x) = x2 be such a rule.  What is the domain? Unlike for case A, where it is the set of first elements for some set of pairs, there is no clue.  It may be prime numbers, negative numbers or complex numbers.  Hence you have to specify the domain explicitly. Writing f : A → B has the same meaning as before, and thereby specifies the domain to be A.  Obligations: ensure that the rule "works" for every element of A and that, for every input in A, the result is in B.  Interesting extra flexibility: assume the rule is (f(x))2 = x.  Then you can use the B in the specification A → B to enforce positive or negative square roots.
 * For group C, writing f : A → B makes B an extra attribute of f. This means that functions with respective types A → B and A → B ' can never be equal unless B = B '.
 * Since you mentioned group D (triples), I asssume it is known to you. Let me mention one subtlety: when writing (A, B, F),  the part F plays the same role as for group A.  However, some authors define A to be the domain, others define it to be a superset of the domain, calling it "source".  The obligations are obvious by now. Boute (talk) 19:16, 29 July 2012 (UTC)


 * I would view f:A arrow B as defining a typed function rather than a function. The type or signature is function from A to B and the value is a set of pairs. The distinction between the definitions comes when one asks what an untyped function is, where one isn't considering them with a whole bunch of other functions of the same type or haven't explicitly specified A and B. With the classical view that is a set of pairs and with the triple view it is a domain, codomain and a set of pairs. There is a difference dor example when one considers composition. If one composes two classical functions without having specified domains and codomains one is talking about finding z so x,y) is in f and (y,z) is in g and different books differ on whether one requires that there always be a (y,z) for every y in an (x,y). With the triple definition one would ask is the codomain of f the same as the domain of g, or one may allow automatic extension and only require the codomain of f be included in the domain of g. So one is very inclusive and the other is very exclusive. However when A arrow B is specified there isn't all that much difference normally in what people do with the functions. Dmcq (talk) 17:54, 29 July 2012 (UTC)
 * The f : A → B is indeed a user specification.  However, note that in all of the views A, B, C, D the function domain is A.  In all views the domain is central, and therefore it must also be defined explicitly for composition (as in Bartle and Meyer, more wishy-washy in Apostol).  Aside: with all views, one can specify A to be a superset by writing f : A /-> B (barred arrow, see Meyer). Boute (talk) 19:16, 29 July 2012 (UTC)


 * OK, I see; well, there is certainly an argument for (more) typing in math, might have to wait a generation, though...:-)Selfstudier (talk) 18:19, 29 July 2012 (UTC)


 * It is already what happens, see above about Boute complaining about Bloch doing it. Dmcq (talk) 18:44, 29 July 2012 (UTC)


 * Yes, I did see that; I don't think that you will find that many mathematicians thinking in terms of typing though, these things change only slowly; the A arrow B is more a thing about custom and practice over many years, look how long it has taken for a category theoretic viewpoint to make inroads... Selfstudier (talk) 18:50, 29 July 2012 (UTC)


 * The interpretation by Dmcq of my earlier comments on Bloch's definition is so far off the mark that I don't know where to start to correct it. My complaint (please read it again) about this definition (please also read it again) is exclusively about the presentation, which is substandard.  Furthermore, I hope that, by my earlier explanation, I have shown that a typing of the form f : A → B plays exactly the same important role in any of the views A, B, C, D, and I even clarified this by an interpretation as a user specification in a common setting to facilitate comparison.  It is important to understand all views and their differences in order to make meaningful choices.  Boute (talk) 19:49, 29 July 2012 (UTC)
 * I have read what you said again and I think you must be having a big problem expressing yourself if you agree that Bloch was using typing and yet wrote what you did. Dmcq (talk) 20:58, 29 July 2012 (UTC)
 * My criticism about the Bloch definition was clearly about its incoherent formulation and the self-contradicting explanation following it. Types were not even mention mentioned, so what I wrote has nothing to do with any agreement about whether or not this definition uses types. So the big problem must in the reading.  Now that you brought up typing: the typing in that definition is very coarse and ad hoc and does not even reflect a more general type system or notation.  In generic functionals you can see how important types are and how flexible and refined function types can be. However, this is beside the work on the Wikipedia article and, as mentioned, I will not go any more into long elaborations on this talk page, but perhaps (time permitting) on your talk page. Boute (talk) 03:44, 30 July 2012 (UTC)
 * No I'd prefer not to have explanations on my talk page thanks. Dmcq (talk) 07:26, 30 July 2012 (UTC)

I just pulled Bartle's book off my shelf and Halmos's just to check them. Perhaps I am missing something, but they both define a function as a subset of the Cartesian product X&times;Y. Isn't that the same thing as a function with domain X and codomain Y? Surely we're not basing this categorization of sources on whether the source uses the exact word codomain for the "Y" set. I must say, I do find it baffling why these would be included in the first group of sources. Presumably there are the same issues with other sources on that list. Sławomir Biały (talk) 20:15, 29 July 2012 (UTC)
 * Bartle allows the domain to be a subset of X, it doesn't have to be the whole of X and uses mapping when the domain is the whole of X and his composition of functions doesn't necessarily have to be valid for every argument that is valid for the first function. Basically they are partial functions and the domain and range are functions of the set of pairs. Dmcq (talk) 20:58, 29 July 2012 (UTC)
 * It seems like that is neither here nor there, then.  Sławomir Biały  (talk) 23:35, 29 July 2012 (UTC)
 * Sławomir Biały 's question is very pertinent and deserves a clear answer. Although Bartle and Halmos wrote in prose, they are very precise. They indeed define a function from X to Y as a subset of X&times;Y (with the usual extra property).  In doing that, they define two different things simultaneously.  First, a function as a set of pairs.  Second, what it means saying that a function is from X to Y (the function's type).  Still, the Bartle and Halmos definitions are not identical.
 * It is easier to start with the common aspect of both definitions. Both define the domain of a function as the set of first elements of the pairs and the range as the set of second elements.  Hence the domain and the range are inherently attributes of the set of pairs, and need no separate specification: knowing the set of pairs, you know the domain and the range. Next, from the function itself, you cannot retrieve the Y mentioned in the definition.  Saying that a function is "(in)to Y " means that its range is a subset of Y (as also stated in the ISO standard).  A function from X to Y is therefore also a function from X to Y ' for any superset Y ' of Y.  Hence Y is not a function attribute, making it meaningless to talk about "a" codomain.  A function is "onto Y " (not just "onto"!) if its range is exactly Y.
 * As for the differences: in Halmos's definition, the domain is defined to be X. In Bartle's definition, it is a subset of X.  In Bartle's definition, even the empty set is a function from X to Y. Dmcq correctly points out that Bartle actually defines "a partial function from X to Y ". Caution: in this setting a function is a set of pairs, nothing else, so with Bartle's definition, you cannot know X from the function's domain: it can be any superset.  Hence the phrase "a partial function from X to Y " cannot be meaningfully understood as saying that the finction itself is partial, only that it is "partial on X ", otherwise one runs into the same problem as with "onto" (pun intended).  Any function is total on its domain, partial on any superset of the domain, and strictly partial on a strict superset of the domain. Boute (talk) 04:49, 30 July 2012 (UTC)
 * Some examples may even be more clarifying. Consider the set of pairs
 * f := {(1, d}, {2, c}, {3, c}}.
 * By the typical definitions ranging from the 1960's (e.g., Halmos, Bartle, Herstein etc.) to post-2000 (e.g., Gries & Schneider, Krantz, Velleman}, all of the following holds.
 * Since no two pairs in f have the same first element, f is a function.
 * The domain (set of first elements ) is D := {1, 2, 3} and the range (set of second elements) is R :={c, d}.
 * Clearly f is a subset of D &times; R and hence is a function from D to R.
 * Given Y := {a, b, c, d}, the function f is also a subset of D &times; Y and hence also a function from D to Y.
 * (Not common to all sources) Given X = {0, 1, 2, 3}, the function f is a subset of X &times; Y and hence, since X is not just D, a partial function from X to Y.
 * Of the sources mentioned in this example, only Gries & Schneider define the term "partial".
 * By Bartle's definition, f is a function from X to Y.
 * The 1960's authors mentioned above call f a function onto R. Liu (1968) and the post-2000 authors mentioned write just onto.
 * Note: although, with the function definition common to all sources cited in this example, writing just "onto" by itself is incomplete, an alleviating factor is that the authors who omit the set w.r.t. which a function is "onto" always seem to implicitly refer to a set defined in the context that they are keeping in mind. Still, from a rigorous viewpoint, in the mind is not necessarily in the function.
 * Note: by coincidence, all sources cited in this example use the term "range" in the sense mentioned. The total lack of any convergence and how to report it in this Wikipedia article is another matter. Boute (talk) 07:10, 31 July 2012 (UTC)
 * It is not true that all the definitions agree that f := {(1, d}, {2, c}, {3, c}} is a function. Some of them say that, the ones where a function is defined as a set of pairs. However others do not and use the 'type' or 'signature' X → Y. f:X → Y is a set of pairs bur a function without a specified domain and codomain is a triple in which the domain and codomain is specified. For those definitions f:X → Y and g:X → Z if Y and Z are different even if the image of both is the same subset of the union of Y and Z. They have for instance a different identity function for every set rather than a single universal identity function. Even with this it is common to consider the set as defining a function but this is just the same sort of business making life easier as considering the integer 3 the same as the real number 3. Dmcq (talk) 10:37, 31 July 2012 (UTC)
 * I often enough said: all sources cited in the example; repeat: Halmos, Bartle, Herstein, Gries & Schneider, Krantz, Velleman. They all belong to group A. That there are other definitions is exactly why I introduced groups B, C and D. These are irrelevant to answering Sławomir Biały 's question, which was about Bartle and Halmos. Elementary courtesy to people asking specific questions is not muddling the issue. Boute (talk) 11:16, 31 July 2012 (UTC)
 * So why does Halmos for instance always refer to f:X → Y when talking about functions and why does he have the identity function for a set rather than just the identity function? You are simply reading what you want into the sources. In Halmos Naive Set Theory a relation is not just a set of pairs, it is a set of pairs from a specified Cartesian product. Dmcq (talk) 13:25, 31 July 2012 (UTC)
 * First, note that Halmos's definition fully justifies my claim that any function from X to Y is, by his definition also a function from X to any superset Y' of Y. If you disagree, give a counterexample. To answer your questions:
 * (a) f:X → Y means that f is a function from X to Y which, by his definitions, means that dom f = X and ran f is a subset of Y.
 * (b) Why he has an identity function for a set is simply because every function must have a domain, and "just the identity function" would require a domain that is the set containing "everything". In most variants of set theory (except those with a universal set, which are not common fare) there is no such set.
 * (c) A relation in Halmos is defined in the sentence starting at the bottom of p. 26 and ending at the top of p. 27 just as a set of pairs, no reference to a Cartesian product (CP). More: even in definitions by other authors in group A who do go via a CP, the "specified" CP is not visible in the set of pairs.  In other words, from the function as a set of pairs you cannot infer a unique Y such that f : X → Y.  Any superset of the range goes.  As I said, often some Y is defined in the context and "kept in mind", but it is not in the function.  So why insist on seeing things that are not there?
 * So, if you dissent with my earlier reasoning, that is your right, but it carries the obligation of giving a counterexample. BTW, if you got my e-mail with the attachments, I'd appreciate in return one of your papers that gives me a deeper insight in your views than is possible with these Wiki discussions. Boute (talk) 18:46, 31 July 2012 (UTC)
 * And where does Halmos ever justify saying that a function is the same whatever Y you use provided it includes the range? He talks about the inclusion function and the identity function, they would be the same function if that were true. I already said to you that I do not wish to discuss things with you elsewhere. Also it does not really matter all that much about f:X → Y as the results come out the same whichever way people go about it provided people do suitable amounts of language all around. However I see nothing in Halmos to indicate he thought a function was the same if a superset was substituted for Y. Dmcq (talk) 22:51, 31 July 2012 (UTC)
 * Halmos does not have to say everything that logically follows from his definition; the universe would be too small to encompass that.
 * As for the inclusion map, quoting Halmos: "If X is a subset of a set Y, the function f defined by f(x) = X for each x in X is called the inclusion map (or the embedding, or the injection) of X into Y." First note Halmos's definition: "the function f defined by f(x) = X for each x in X "  defines f without mentioning of Y, which indicates that the domain and the mapping suffice!  We all would call f correctly the "identity function on X ". However, if X happens to be a subset of Y, that same f is also correctly called an inclusion map from X into Y.
 * Maybe you do not want to discuss things with me elsewhere, but I feel increasingly embarrassed having to explain obvious things here, thereby taking up other people's space and time. So any remarks by you that do not bring new elements will have to be discussed elsewhere.
 * As for f : X → Y, this widely used notation (not just by Halmos) is best of all explained in Meyer (op. cit.) page 28: it specfifies the type of f. It does make a huge diference: specifying f : {0, 1, 2} → {0, 1, 2} by f(x) = x or, more illustratively, by f = {(0,0}, {1, 1), (2, 2)} is type correct, specifying g : {0, 1, 2} → {0, 1} by g(x) = x or by g = {(0,0}, {1, 1), (2, 2)} (same set of pairs) is type incorrect.
 * Given his own definition, logically Halmos could not have thought anything else than that a function was the same if a superset was substituted for Y.
 * Anyway, I think I found a good way to clear up such misunderstandings. Consider the following definitions:
 * (1) Let S be a set. A subset of S is a set such that all its elements also belong to S.
 * (2) Let X and Y be sets. The Cartesian product (CP) of X and Y is the set of all pairs whose first member is in X and whose second member is in Y. The CP of X and Y is written X &times; Y.
 * (3) Let X and Y be sets. A relation between X and Y is a subset of X &times; Y.
 * In Halmos pp. 26-27: just a set of ordered pairs. Important: this is the same definition (see below)!
 * (4) Let X and Y be sets. A function from X to Y is a subset of the X and Y such that (etc.).
 * In Halmos p. 30: a function from X to Y is a relation f such that dom f = X and such that for each x in X there is a unique element y in Y with (x, y) isamemberof f.
 * Consider (1): although the definition starts with "let S be a set", you do not know from a subset T of S what S is (except that it is any superset of T). Yet, you can say that T is a subset of S.
 * Consider (3): although the definition starts with "let X and Y be sets", you do not know from a relation R from X to Y what X and Y are, since R is just a subset of X &times; Y. Still you can see that X is a superset of dom R (set of first elements of R) and Y a superset of ran R (set of second elements of R). Most clarifying: you can see from R that it is a subset of X &times; Y, and hence a relation from X to Y.  It is automatically also a subset of any CP that contains X &times; Y, with the consequences I stated many times.
 * Similar observations hold for (4). Boute (talk) 04:50, 1 August 2012 (UTC)
 * As said, I feel embarrassed taking up space unnecessarily. Do other people feel that my explanations are helful, or just belaboring the obvious?  Or does anyone else feel that I am misrepresenting Bartle, Halmos, Herstein, Gries & Schneider, Krantz or Velleman?  In that case, please say so with concrete and specific counterarguments, avoiding repetition. I will act accordingly, taking anyone's arguments seriously. Boute (talk) 04:50, 1 August 2012 (UTC)
 * I think the effort should be directed towards improving the article, not towards discussions about the subject matter on the talk page. The article needs more references and inline citations, so why not be bold and start working on the definition in the article? Isheden (talk) 07:25, 1 August 2012 (UTC)
 * Agree with that. My objection to Boute is their explaining things in their terms and trying to make out another widely accepted way of going round things is wrong. We should have both ways in the article, and I think we really need the usual f:X → Y as a common form with two ways of talking about it. I'd leave Bartle's use for what is a partial function as an alternative where it is at the end but the pairs definition and the triple should both be after the f:X → Y. And as to Boute can I say I feel the same frustration as trying to explain eπ is multiple valued if e is a complex number if somebody says but it is just a real but I really don't think going on about it will help this article. Dmcq (talk) 09:30, 1 August 2012 (UTC)
 * I resent Dmcq's implying that I am distorting things. Moreover, as I said before, no mature mathematician would use the term "wrong" (at worst, "inconsistent").  I also said that f : X → Y was of huge importance, so what's the complaint about that? I gave an informal proof of the statement that (with the definition from all authors cited in my answer to  Sławomir Biały 's question - group A) a function is the same if a superset was substituted for Y.  I can prove it formally (someone interested?), but there is no need: any mature mathematician who finds the informal proof erroneous would have provided a counterexample right away, instead of acting anti-Wikipedian and become personally insulting. Boute (talk) 10:26, 1 August 2012 (UTC)
 * I have an equally low opinion of your ability to understand explanations. Dmcq (talk) 11:27, 1 August 2012 (UTC)
 * Which explanations? You gave not a single explanation, only made statements that are non-sequiturs with respect to the definitions considered (group A), which I logically refuted. The only explanation supporting your viewpoint on those definitions would be a counterexample. This is my last reply to remarks accompanied by personal insults. Boute (talk) 12:06, 1 August 2012 (UTC)
 * Fine by me. Now perhaps the article can be developed on the basis of what authors actually say rather than your extrapolations and proofs of what they would say according to you if they agreed with you. Wikipedia is supposed to summarize stuff not be our ideas. Dmcq (talk) 17:10, 1 August 2012 (UTC)

Logic corroborated by sources
Apparently the conclusions of correct proofs are not equally palatable to everyone. Here are some literal quotations from sources that use the same definition of a function as Halmos, and on that basis define the following terminology.
 * Mendelson, Introduction to Mathematical Logic (3rd ed., 1987), p. 6:
 * A function f with domain X and range Y is said to be a function from X onto Y.
 * If f is a function from X onto Y and Y is a subset of Z, then f is called a function from X into Z.
 * Tarski & Givant, A formalization of set theory without variables (1987), p. 3:
 * We say that a function F maps A into B if Do F = A and Rn F is a subset of B.
 * We say that a function F maps A onto B if Do F = A and Rn F = B.
 * Consistent with this, the ISO standard defines the notation f : A → B to mean f maps A into B, as usual. Boute (talk) 20:21, 2 August 2012 (UTC)
 * I don't see those as showing anything at all one way or the other. Perhaps best to leave this till a particular problem needs to be resolved in the article. Dmcq (talk) 22:07, 2 August 2012 (UTC)
 * It clearly shows that, for group A (all Halmos-like defs: a function as a set of pairs), if f is a function of type A → B and B is a subset of C, then f is also a function of type A → C. This is not a problem (to the contrary, it yields a richer function algebra) but people drawing wrong conclusions hit inconsistencies. Boute (talk) 10:39, 3 August 2012 (UTC)
 * Those weren't clear but I have had a look at a bit of the rest of the stuff that they have written and I believe you are correct about them. However I do not believe you have established any sort of credible case about Halmos nor do I see his Naive Set Theory as backing you up. An interesting addition for the article I think is that Tarski & Givant say a relation is a class of pairs rather than a set which means they do accept the power function as a proper function by their definition. Dmcq (talk) 13:20, 3 August 2012 (UTC)

Beginning of article
The article needs to begin with a statement that is accurate but is something a non-mathematician can understand. All that is necessary at that level is an imput and an output and some examples. Where the input and output come from, and how one gets from one to the other, are questions that are too technical for the non-mathematician and already understood by the mathematician. They can be covered in the second paragraph. Rick Norwood (talk) 17:26, 29 July 2012 (UTC)


 * Unless my memory is playing tricks or I am misreading the edits, the introductory sentence has not been altered at all from what it was prior to the raft of recent amendments; are you proposing that we change it?Selfstudier (talk) 17:39, 29 July 2012 (UTC)


 * Can you just leave the intro alone till the main part of the article settles down again please. The lead should follow the article. Dmcq (talk) 18:46, 29 July 2012 (UTC)

Sorry. No, I was not suggesting changing in the lead. And I still don't know why are awful. Rick Norwood (talk) 18:58, 29 July 2012 (UTC)
 * The less-than and greater-than signs should be used to denote inequality, and not as brackets. These were used as brackets in the early days of ascii, but now there are better alternatives (e.g., &lang; and &rang; are the typographically correct bracket symbols, but these don't work in some browsers.)  Sławomir Biały  (talk) 20:02, 29 July 2012 (UTC)
 * A normal bracket is fine I think. I've used square brackets from Python sometimes but I'd leave the angle brackets for bra ket vectors. Dmcq (talk) 21:04, 29 July 2012 (UTC)
 * As David Turner demonstrated with SASL a lifetime ago, the comma suffices to recognize a pair, so you can write a, b as well as (a, b), just like, in arithmetic, one can write a + b as well as (a + b). With this rationale, parentheses are used only for emphasis or for overriding operator precedence, as in (a + b) &times; c.  Braces and brackets are "scarce resources", not to be wasted unnecessarily. Boute (talk) 05:04, 30 July 2012 (UTC)
 * Well, if you insisted on such conventions, you'd be going against basically every established convention in mathematics. You probably don't want to go there.   Sławomir Biały  (talk) 15:22, 30 July 2012 (UTC)
 * Which "established" conventions? Most mathematicians don't use brackets, just parentheses.  The SASL view is a refinement of the "established" conventions, showing that you can see a comma as an infix operator (like + in arithmetic) to build a pair and repeated comma's as its variadic application (like a + b + c + d) to build a list (and sublists for trees). If you don't use the extra flexibility and algebraic properties, no mathematician will even notice the difference!  Anyway, it is not constructive to let established conventions (or someone's perception thereof) stand in the way of a good idea. Boute (talk) 07:59, 31 July 2012 (UTC)
 * I wasn't proposing we follow anything like Python's rules with commas for tuples. This is about mathematics. Dmcq (talk) 18:01, 30 July 2012 (UTC)
 * Yet, one should not be too shy about importing good ideas from other disciplines: it is a matter of cross-fertilization versus inbreeding, and we know where the latter leads. Boute (talk) 07:59, 31 July 2012 (UTC)
 * I say again: you don't want to go there. We'll just use the notation that the vast majority of mathematics sources use.   Sławomir Biały  (talk) 19:40, 31 July 2012 (UTC)
 * I replaced them with parens for the reason Sławomir said. I also tried to ensure that the minus signs were right by using entities for them. There should have been no content changes unless I made a mistake. &mdash; Carl (CBM · talk) 02:22, 30 July 2012 (UTC)

We have a fairly common usage: a, b > 0, meaning a and b are both positive. Unless this usage can be done away with, some symbol of grouping is necessary for ordered pairs. Rick Norwood (talk) 15:15, 30 July 2012 (UTC)

I assume that the consensus is that all the recent amendments be allowed to stand? that is, we may take the article as it is now as a basis for any amendment going forward...?Selfstudier (talk) 12:27, 31 July 2012 (UTC)

Towards equivalent word definitions and symbolic definitions in the article
I think the following is important in implementing Rick Norwood's suggestions. Subject to the following conditions: I have some ideas how to do that, but we should first agree on the principle. Boute (talk) 20:55, 2 August 2012 (UTC)
 * The definitions in the first part of the article should be word definitions from a reliable source, quoted literally.
 * The definition in the Formal Definition part of the article is a symbolic definition from a reliable source, also quoted literally. A good choice seems the following up-to-date text, available online [Functions] as well as from Amazon [] proposed earlier by Selfstudier.
 * The crucial definitions should be provably equivalent (not necessarily demonstrated in the article).
 * For secondary definitions, internal inconsistencies in [Functions] are tolerated, since correcting them would again be a matter of endless debate, and I have not seen better sources reflecting the various views as Selfstudier remarked. (I hope everyone appreciates what a sacrifice tolerating inconsistencies will be to me :-)

Assessment comment
Substituted at 20:35, 2 May 2016 (UTC)