Talk:Map (mathematics)

Busted symbols
Some symbols in the article do not appear in the intended way as hooked or two-headed arrows. Bo Jacoby 11:07, 20 October 2005 (UTC)

Content consolidated under function
Around 1/16/06 most meaningful content from this page was moved to function (mathematics), since 'map' and 'function' are largely synonymous. The consolidation will help avoid duplication.

There are several redirects that point to map, that should now point to function, where the content they refer to now lives, e.g. bijective. I've fixed some of these.

Anyone contemplateing adding content here, rather than under function, should consider the consequences: links, duplication with function, etc. LarryLACa 03:00, 20 January 2006 (UTC)

Much of this information that was merged, especially the stuff from Bijection, injection and surjection, has later been removed from those articles. Any merging of these pages should be done via the AFD process, as there is NOT consensus that they should be merged. Perel 03:16, 6 December 2006 (UTC)

As "map" has many varieties of meaning even in the sense of "function", a separate page seems justified to me. Zaslav (talk) 07:56, 26 February 2011 (UTC)

Basics
Every map is a correspondence (mathematics).218.133.184.93 05:37, 5 November 2007 (UTC)
 * So? Why is this not obvious from what's already there.  &mdash; Arthur Rubin |  (talk) 08:22, 5 November 2007 (UTC)
 * And why does it not require a source for it to appear in the article? &mdash; Arthur Rubin |  (talk) 08:23, 5 November 2007 (UTC)

Simply untrue
A function is something whose range is a one dimensional.

For example the object $$ f : \mathbb{R}^m \to \mathbb{R}^n$$ where $$m,n \in \mathbb{N},$$ is called a function if $$n=1$$ and a map (or mapping) if $$n > 1.$$ Dharma6662000 11:21, 1 December 2007 (UTC)


 * I don't think I've ever seen that usage. &mdash; Arthur Rubin |  (talk) 16:51, 1 December 2007 (UTC)


 * Probably someone used it that way but it is not generally known. There is great variety of usage. Zaslav (talk) 08:00, 26 February 2011 (UTC)

umm, what ?, please can you explain
um, an introduction and an example(or two?) applicable to a real world situation may aid understanding here, i'm not particularly uneducated - but understanding this article, i am not. :) —Preceding unsigned comment added by 80.42.179.40 (talk) 11:01, 21 February 2008 (UTC)

Morphism
The given impression is that map and function are effectively synonymous, and this appears true in ordinary contexts; but is it not the case that function has a more strict sense than map? It would presumably not be admissible to call any morphism a function, but would it not be admissible to describe a morphism as a kind of map? I agree, incidentally, that some explanation geared at non-mathematicians would be helpful.

A function is a special case of a map
Formally a map takes an element from a set A and outputs an element of set B. A function although similar has the restriction that every element in set A corresponds to one element in set B. for example the function f(x)=x^2 takes in a value and outputs only one value, for 1 its 1 for 2 its 4 etc. A map ||(x,y)||=1 or alternatively x^2+y^2=1 can output multiple values for an input. For x=1/sqrt(2) for example it outputs 1/sqrt(2) and -1/sqrt(2). Something that cannot happen in a function. —Preceding unsigned comment added by Pirhomaniac (talk • contribs) 17:01, 19 February 2010 (UTC)


 * I believe this not to be standard notation. Have you a reference?  — Arthur Rubin  (talk) 17:52, 19 February 2010 (UTC)


 * I agreed with Pirhomania. Then I checked my book and found I was wrong.

I think the confusion here is that in order to do something like calculus on a map you need to impose other conditions, and so in some lecture courses they talk about functions to mean (those maps which will work) rather than any map. Also (in Warwick, UK anyway) you often have questions along the lines of A map is defined as follows... is this map well defined? When in reality the question is asking "is this a map?" - however the fact that their is no suitable word for what the question is other than map, the brain slowly absorbs the term 'map' for any old thing and 'function' for the maps which really are maps.

As it happens any map is a subset of the Cartesian product AxB, but not every subset (or pairing) is a map. So really the questions from uni could have used the term pairing. I believe they didn't because that would require more knowledge about set theory than they wanted to worry about in the first term of first year, and because it doesn't help to visualise what's going on.

Anyway I hope that cleared up some confusion. It may be noteworthy to say that colloquially in some universities the term map describes any pairing (subset of cartesian product) when questioning whether or not the map is well defined (i.e. genuinely a map). --92.14.234.142 (talk) 22:33, 14 February 2011 (UTC)


 * The word "function" has a pretty standard meaning throughout math. The term "map" is much more variable and often does not mean a function; it may mean a special kind of function, it may mean "morphism" which is more general than a function.  I adjusted the article in view of this diversity of usage. Zaslav (talk) 07:59, 26 February 2011 (UTC)

Suggestions for improvement
A mapping is simply a tool to 'connect' two abstract (i.e. mathematical) objects. Mappings are usually given by

\begin{align} M : A &\to B \\ a &\mapsto b. \end{align} $$ which expresses how members from two sets of objects interact with one another.

Since functions are also maps—for instance, $$f(x) = x^2$$ can be given as the mapping

\begin{align} f : \mathbb{R} &\to \mathbb{R} \\ x                &\mapsto x^2 \end{align} $$ —the term 'map' and 'function' are sometimes used interchangeably.

p.s. To suggest "map" is a synonym for "function" is like saying "animal" is synonymous with "horse". — Preceding unsigned comment added by 66.11.191.243 (talk) 03:23, 21 September 2012 (UTC)

Suggestion on removal, redirect "map" to "function"
Map is a synonym for function, there is no need to blurry the idea and confuse the readers. How one defines function exactly is still left open for everyone.

The whole thing here to me seems like a setup to change every word usual in set theory into category theory, which built on the perception that a function should have a definite domain and range, hence "map". --14.198.220.253 (talk) 13:35, 7 October 2013 (UTC)
 * Such that other don't get confused too. Map is used by some as a synonym of function, that is true. But this is due to uses in texts written by authors that either were not aware of the difference or didn't want to make explicit the distinction. The key difference between maps and functions, is that map is a function (set of ordered pairs with unique image condition) together with its co-domain. They are the arrows in categories in which morphisms are functions. An elementary book that I remember in which the author was careful enough to perpetuate the misunderstanding is Apostol's Mathematical Analysis. There must be many others. Cactus0192837465 (talk) 02:04, 31 January 2019 (UTC)

Wiktionary or Wikipedia article?
This seems a wiktionary article: it talks about different concepts bound only by the word used to describe them. I think it would be better to create different stubs about each of those concept and link to them from the map (disambiguation) page. What do you think?--Malore (talk) 06:04, 30 September 2018 (UTC)

Disambig
As currently it stands, this page isn't really an article but a note on the terminology. (For example, "map" in graph theory has nothing to do a map as a function). Function (mathematics) already discusses the term "map" so this page is redundant and need to be turned into a disambig page. In fact, reading the above, the other editors seem to support my position. -- Taku (talk) 23:02, 30 January 2019 (UTC)
 * Map and function are two different concepts. While it is true that Wikipedia presents the use of them as synonyms, and it should, this is due to the large number of people that use them as such, due to either not being aware, or not being too relevant to what they are saying. But, they are different concepts, and that is mentioned in the article about functions. This page is dedicated to the concept of map, which are functions together with their codomain. Compare with the definition of function as the set of pairs, from which the codomain cannot be obtained. As a subsection this article contains all other possible uses of map in Mathematics. That is why it is an article of its own, because it is a concept of its own. The article is linked to from the actual disambiguation page for maps in all areas, mathematics or not. In any case, if you would like to float the idea of merging you don't have to completely destroy the structure of the article, just add the tag. Cactus0192837465 (talk) 01:54, 31 January 2019 (UTC)
 * I'm not disagreeing that authors make a distinction between map and function. I'm talking about the current structure. This page is currently written as a disambig page; map in graph theory has nothing to do with functions. Whether we need a separate article on map from function is another question. Reading from the above, the answer to that question is no. -- Taku (talk) 02:45, 31 January 2019 (UTC)
 * I have therefore moved some map vs function stuff to function (mathematics) for now. -- Taku (talk) 02:56, 31 January 2019 (UTC)
 * A map in graph theory is a function from the graph, to points and lines in the plane sending vertices to points and edges to lines in such a way that they don't intersect. How is it that has nothing to to with functions? The current structure looks like a disambiguation page because you keep turning it into a bullet list. Cactus0192837465 (talk) 03:01, 31 January 2019 (UTC)
 * I'm not talking about the bullet list and I'm not saying a distinction between "function" and "map" is meaningless. This page, as currently written (bullet list or otherwise), isn't written as one; except in the lead and the first section. We can perfectly discuss for example the importance of keeping track of codomain at function. (Notice I have already moved the relevant stuff to "function" article; so no loss of information.)
 * I said this page is a disambig page because currently the page simply lists various usages of the term "maps" in mathematics, as the other editors have observed. Turning this into a disambig is thus not a unilateral action. For example, a map can mean a morphism in category theory and that, while related, is a different usage. Ditto for ones in graph theory and logic. -- Taku (talk) 03:12, 31 January 2019 (UTC)
 * There is already a disambiguation page for map. The intention of this page it to be the article about the use of map in mathematics. If it doesn't look to you like that, you should try to make it look like it what is intended to be, instead of turning it into another page that has a different use, which already exists. Cactus0192837465 (talk) 03:16, 31 January 2019 (UTC)
 * There are various usages of maps in mathematics, of course. In Wikipedia, we do not have a single article on various distinct usages; we simply have separate articles for different usages and have a disambig page that refers the readers to relevant articles. The question is whether we want the readers to this page instead of function or morphism (category theory). -- Taku (talk) 03:20, 31 January 2019 (UTC)
 * The difference is that all those uses are closely related, all being slightly more general or particular than others. They are all functions in which one adds more information besides the relation itself. They wouldn't warrant independent articles. Even a case could be made that this article be merged with that of function. I personally lean towards not merging, mainly because it would bloat the article about functions even more than it is now. In any case, this one cannot possibly be the disambiguation page, because such a page already exists. Cactus0192837465 (talk) 03:26, 31 January 2019 (UTC)
 * I agree they are not completely "unrelated"; though map in the graph theory, I think, is unrelated; it's something similar to political map. Anyway, the question why a separate article is needed. I don't need to start independent articles since there are already ones; e.g., function (mathematics), morphism (category theory). The main problem is that if this page is not a disambig page, we'll be sending the readers who are interested in functions or morphisms to this article instead of main articles. This is precisely why this page, currently as it stands, is "functioning" as a disambig page. -- Taku (talk) 04:16, 31 January 2019 (UTC)
 * I completely agree it's important to discuss "map as a function together with codomain". But that has to be done at function (mathematics); putting such a key info in a separate article is a disservice to the readers. Since you don't seem to object merging this with map (mathematics), I will try to do that (but not today). -- Taku (talk) 04:19, 31 January 2019 (UTC)
 * This might help: some authors use the term “map” to mean a function together with a domain and codomain. But this usage is not standard; in the formal def, domain and codomain are parts of the def of a function; otherwise a function is not a morphism in the category of sets. (In practice, it’s sometimes useful to be loose about codomain, but that’s still abuse and strictly speaking functions with different codomains are distinguished). So the notion “function with codomain” has to be discussed at function (mathematics). —- Taku (talk) 11:03, 31 January 2019 (UTC)

There's a technical reason why Map (mathematics) should not be a DAB page. (1) All DAB pages should have a redirect to them with the (disambiguation) qualifier. (2) Double parenthetical qualifiers are deprecated verging on forbidden. (3) DAB pages with qualified titles are bot-reported for maintenance attention in Category:Disambiguation pages with (qualified) titles. Narky Blert (talk) 09:57, 1 February 2019 (UTC)


 * Ok. But since the page has been merged with Map (disambiguation), this shouldn’t be an issue. —- Taku (talk) 10:32, 1 February 2019 (UTC)
 * Did you review WP:FIXDABLINKS (or, for that matter, WP:MOSDAB)? --R'n'B (call me Russ) 11:42, 1 February 2019 (UTC)
 * No. But as I said, I don’t think there is any issue now (is there?) I don’t want to sound confrontational but I don’t think we want to be asking the editors to study all the style manuals and all the guidelines before letting them edit. We need to be asking editors to make edits that they think right and fix the issues as they turn up. I didn’t think and I still don’t think it’s wrong to put the disambig page template to a page that was functioning effectively as a disambig page. —- Taku (talk) 12:59, 1 February 2019 (UTC)
 * OK, since the subtle approach didn't work, let me spell it out for you: When creating disambiguation pages, fix all resulting mis-directed links.

Before moving an article to a qualified name (in order to create a disambiguation page at the base name, to move an existing disambiguation page to that name, or to redirect that name to a disambiguation page), click on What links here to find all of the incoming links and repair them. You didn't do that. You broke existing links in over 100 other articles. And frankly I think your approach here has been like a bull in a china shop; although well-intentioned, you have simply not listened to anything any other user has said, and just gone forward with the changes that you personally decided you wanted. --R'n'B (call me Russ) 13:49, 1 February 2019 (UTC)


 * ”changes that you personally decided you wanted.” not true. I felt emboldened since the issue has been raisen more than one occasion in the past; see the above threads. I simply implemented the change that multiple editors felt is needed.
 * ”You broke existing links in over 100 other articles.” This is too narrow a view. The links were broken before, effectively speaking as there should have not been incoming links to this page (see what links here). In other words, I didn’t create problems; they existed in the past and the edit (putting the template and then merger) merely brought them up to the surface. I’m interested in problems and fixing them in the real sense; not bot technical sense.
 * Also, are you proposing to undo the changes? Do you think those incoming links to this page are correct? Maybe this is a constructive question: what should an editor do when they find a page that is essentially a disambig page in disguise? I think we should ask them to put the disambig template and fix the broken links as a community. I have fixed so many disambig links before and I’m willing to do that this time too. — Taku (talk) 14:15, 1 February 2019 (UTC)
 * Here is one concession: should I have consulted the disambig page project about how the page like this should be handled? Yes, in hindsight. I admit I have created more works (if the works that are needed). —- Taku (talk) 14:55, 1 February 2019 (UTC)
 * A large number of people simply don't know the difference between the concepts of function and map, and that includes writers of widely used introductory courses. The history went from studying curves given by elementary functions, to the study of arbitrary relations when interesting properties started to be found (like discontinuity everywhere) which were called functions, to then interest began to appear in the study of spaces giving birth to the separate concept of map, to later passing to the study of the maps as just arrows between the spaces giving the notion of morphisms (which can be things that are not functions at all). Functions, maps, and morphisms are three concepts that are different in essential ways, enough that they can hold having each its own independent article. Now, language evolves. The same way that their meaning diverges, it also converges due to use (regardless whether that is in part due to ignorance or economy of words). Maybe in 50 years, the sloppy writers have driven almost everyone to simply identify the three words. Who knows. I don't know how Wikipedia would like to reflect the status of the current literature, if distinguishing the concepts, and commenting on the identification in popular use, or by merging and leaving the distinction as a comment. You have chosen the latter. I personally find it an sloppy exposition of the content. Cactus0192837465 (talk) 14:57, 1 February 2019 (UTC)
 * Again it’s not my choice: this so-called “article” has failed to evolve since the page was essentially functioning as a disambig page (as the others have noted above as well). There are of course distinctions; that’s why we have articles like function, homomorphism, morphism, etc. The term “map” is memerly a shorthand for those. Acknowledging this page was a disambig page is a sharper writing.
 * Note also: you claim a map is a function with a codomain; this is not standard usage. A “function” is a morphism in the category of sets: this is standard in the research level math. The term “map” sometimes is a function in the general sense and the term “function” is something specific; you’re imagining a distinction that is not standard. You make a good point of distinguishing between maps and functions. We do acknowledge that and that’s done at function (mathematics); since there are multiple ways to make distinction, the only sensible approach is to use one article. Picking one way as a standard is original research. -- Taku (talk) 15:26, 1 February 2019 (UTC)
 * No, a function is not a morphism in the category of sets. That's what a map is. In almost every book that you will find a function is defined as a relation, with the unique image property. This is not only what you call 'research level' definition, but also the de facto definition used even by sloppy introductory courses, books in logic, and foundations of mathematics. There is an analogous to map for relations without the unique image property, that is the concept of correspondence. A correspondence is to relation, what map is to function. A correspondence is a triple of two sets and a subset of their Cartesian product. Cactus0192837465 (talk) 16:07, 1 February 2019 (UTC)
 * The article in was not a disambiguation page. It was a broad-concept article, a viable target from many articles. The article should be restored, as should the links that have been removed since, eg  and many others by User:BD2412 (see also: User:Deacon Vorbis's edit at User talk:BD2412. -- Michael Bednarek (talk) 17:23, 1 February 2019 (UTC)
 * As it stands, there is no article, and therefore nothing to which these specific links can point. Per WP:OVERLINK, the fact that a thing exists doesn't mean that every article mentioning the thing needs to have a link to it. bd2412  T 17:30, 1 February 2019 (UTC)


 * I have restored this article to its state prior to the merge. There was clearly no consensus for that action, and per Cactus and others this is a valid topic in its own right. &mdash; Amakuru (talk) 17:45, 1 February 2019 (UTC)

Ok. It is clear by now that there was no consensus; I have thus asked more editors to weight in the matter at the math project. —- Taku (talk) 18:24, 1 February 2019 (UTC)
 * I have restored the incoming links, but if a consensus to disambiguate this term does arise, please fix the incoming links before changing the nature of the article. bd2412  T 18:57, 1 February 2019 (UTC)
 * But why? Wouldn’t it be more efficient if we turn the page into a disambig page and then fix broken links instead of making preliminary changes and then turn the page into a disambig page?. Remember Wiki works by breaking things first and fix them, as opposed to going through preliminary procedures to ensure there will be no breakage. Presumably the former scenario is more efficient since we can ask the communities of the editors, including anonymous users to change the links. —— Taku (talk) 21:13, 1 February 2019 (UTC)

Oppose in the strongest possible terms. When I saw the template at the top of the page I thought the suggested target must have been an inadvertent error. This page is not a disambiguation page, should not be turned into one, and doesn't even remotely look like one. If Mr TakuyaMurata thinks it does I would suggest he needs to buy a new pair of spectacles. David Wilson (talk · cont) 22:13, 1 February 2019 (UTC)
 * "I don’t think we want to be asking the editors to study all the style manuals ..." – not all, just the relevant ones; it's a matter of WP:COMPETENCE and WP:Collaboration.
 * "The links were broken before, ..." – links to broad-concept articles are not broken; links to DAB pages are, which is what TakuyaMurata created.
 * "Wiki works by breaking things first ..." – eh?
 * I suggest that editors familiar with the subject visit the incoming links and modify them there to point to a more specific/precise article. Once there are no more incoming links, the content/form/purpose of this page can be discussed again. -- Michael Bednarek (talk) 08:00, 2 February 2019 (UTC)
 * You haven’t responded to my central question: quoting myself, “But why? Wouldn’t it be more efficient if we turn the page into a disambig page and then fix broken links instead of making preliminary changes and then turn the page into a disambig page?. ... [this] is more efficient since we can ask the communities of the editors, including anonymous users to change the links” . As I wrote before, I view the links to this page are already broken in the practical sense if not in the technical sense. —- Taku (talk) 21:54, 2 February 2019 (UTC)
 * That approach has been rejected before here by User:R'n'B, quoting WP:FIXDABLINKS. If you consider incoming links broken, visit those articles and fix it. -- Michael Bednarek (talk) 05:13, 3 February 2019 (UTC)
 * Again you haven't responded to my question. Why is that approach less preferable? -- Taku (talk) 01:46, 4 February 2019 (UTC)

Concept article
I think that maybe an approach that can serve to make this article a clear separate entity from others, is to dedicate it to the use of map as a concept in mathematics, specially when it is not used as a synonym of function. It seems to me that explicitly saying that this is the purpose of the article, and making it so, would serve to both organize and give stability to its content. Otherwise, people convinced that being a synonym of function is the only use would be constantly leaning the content in that direction, people that are used to other uses going in the other direction.
 * Clearly the first very thing that the article should say is that very widely map/mapping and functions are used as synonyms. This first sentence should have, and it should be easy to provide it with, many references. There would necessarily be a link there to the article about functions. That article is already quite extensive, and contains all (if something is missing it can always be added) the information that map would have when they are synonyms.
 * But then say, as a second sentence, that the rest of the article will be dedicated to the use of map when this is different from the notion of function. This would include, both the cases in which the different notion is a homonym, like that use in graph theory mentioned in the article, but also for what distinctions are made by authors that purposely define functions and map/mapping as different concepts.
 * Links from other articles pointing here, would read soon the second sentence and be able to decide if they want to use the link in the first sentence to jump to the general article about functions, or stay in this one to see the particular material that the second sentence is telling them that this article is about.

Thoughts?

Would the title require to reflect such intention of the article too? I don't know if there are rules in Wikipedia about that. Cactus0192837465 (talk) 01:56, 2 February 2019 (UTC)


 * I think we need a tie-breaker. To reiterate my position, I still don't think how this "page" can be turned into a genuine Wikipedia article. There are "concepts of maps", obviously; but, no single "concept" of a map. In my view (and in fact also of the others; e.g., ), these concepts of maps are already covered elsewhere; function (mathematics), morphism (category theory), homomorphism, etc. In short, in the practical sense, this page functions as a disambig page. I don't know how to change that, without duplicating existing articles. -- Taku (talk) 23:53, 2 February 2019 (UTC)
 * I think it would make a perfectly good set index article. —David Eppstein (talk) 04:41, 3 February 2019 (UTC)
 * Agree; either that or leave it as a broad-concept article. -- Michael Bednarek (talk) 05:13, 3 February 2019 (UTC)

I can has cheez "Map (mathematics, non-function)"? Functionally, at least. Purgy (talk) 07:15, 3 February 2019 (UTC) Only partially obsolete for what I missed as of writing that. Purgy (talk) 07:56, 3 February 2019 (UTC)

Not insisting on something but to explain my view: I suppose for me "map" is simply an ambiguous term: "Springfield" perfectly can refer to a specific location, as long as the context in which the name is used is clear. In the same way, in practice, the meaning of "map" can be inferred since the term is almost always used in some specific context; one never talks about a map between some arbitrary mathematical objects. (If those objects are objects in the sense in category theory, a map will simply mean a morphism; see, the context makes the term "map" unambiguous). Anyway, again, this is why I thought (and still do) this page can only be a disambig page; but I will not insist on this view. -- Taku (talk) 01:39, 4 February 2019 (UTC)

Requested move 29 April 2020

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 

The result of the move request was: not moved. (closed by non-admin page mover) Jerm (talk) 19:06, 6 May 2020 (UTC)

Map (mathematics) → Mapping (mathematics) – As indicated in the introduction of the article, "map" is the short name and is listed after the full name "mapping". Maggyero (talk) 18:04, 29 April 2020 (UTC)
 * This is a contested technical request. GeoffreyT2000 (talk) 18:40, 29 April 2020 (UTC)


 * I think that this move needs discussion. It seems that we need to consider WP:COMMONNAME here. GeoffreyT2000 (talk) 18:40, 29 April 2020 (UTC)
 * Not move per WP:COMMONNAME: "map" is used 35 times in the article, while "mapping is used 12 times. Moreover, my feeling is that the meanings are slightly different: "mapping" is used for the action of applying the function on its domain when the domain is a set, while "a map" denotes the function (or morphism) itself. For example, one says "a mapping of $E$ on $F$" and "a map from $E$ to $F$". D.Lazard (talk) 19:42, 29 April 2020 (UTC)
 * Keep in place. To me, "mapping" is the process of applying a map to a mathematical object, while the thing itself is a map. See also short map, a specific kind of map, for which the gerund version is not even used. —David Eppstein (talk) 17:00, 30 April 2020 (UTC)
 * Comment: I made this point before but I still think the article is redundant given that there is function, which is the precise mathematical term for mapping. I suppose the question if there is a mapping that is not really a function (in a broad sense, including multi-valued or partially defined one). —- Taku (talk) 22:53, 30 April 2020 (UTC)
 * I agree that this article may be a redirect to the first note of function (mathematics), but we must wait the closure of this request before discussing this. D.Lazard (talk) 10:12, 1 May 2020 (UTC)


 * The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.