# Talk:Function (mathematics)

WikiProject Mathematics (Rated B+ class, Top-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B+ Class
 Top Importance
Field: Basics
A vital article.
One of the 500 most frequently viewed mathematics articles.
Wikipedia Version 1.0 Editorial Team / Vital

## Alternate definition???

Calling an author's sloppiness in not mentioning the codomain of a function an alternate definition of the term "function" takes a simple concept and convolutes it to the point where it can and will confuse a novice reader. Most of the section that includes this should be deleted as pure fabrication - an elaboration of a non-existent difference. The definition of a function, a la Bourbaki (1937), has become standard and our article on it should reflect that fact. Bill Cherowitzo (talk) 17:14, 7 April 2013 (UTC)

The alternatives are well cited and quite common and that's all that's necessary in Wikipedia. We document what's out there. Alternatives are used quite frequently nowadays, Bourbaki wasn't the last word on the subject. The alternatives section is well marked as alternative so I don't see the problem with confusion. Dmcq (talk) 20:57, 7 April 2013 (UTC)
I would object to your use of well cited and quite common. I don't have a copy of Apostol to check, but the book is over 45 years old and is a bit dated. The Heins reference does not support the claim. In over 35 years of teaching this material using a large number of common texts, I have never seen this statement; if this was quite common I should have run across it. Of course a couple of good modern citations (say, published after 1990) would show me the error of my ways. I am not a big Bourbaki fan, but I will give them credit when it is due, and they nailed this definition. I am also not clear on why you would want to call this an alternative definition. This is not a different way of looking at an object or a different approach to understanding a concept, it is just being sloppy with a basic definition. Bill Cherowitzo (talk) 03:58, 8 April 2013 (UTC)
Try searching Google books for the phrase "function is a set of ordered pairs". It may make you sad (as it does to me) that they neglect to specify the codomain as part of the function, but it seems to be quite common. —David Eppstein (talk) 05:23, 8 April 2013 (UTC)
Thanks for that sobering suggestion, but I didn't really find anything that I wasn't expecting. The exercise did remind me that the vast majority of texts where this will come up are at the pre-college level. Let me be a little clearer about what is bothering me. The difference between what is found in this level of book and and a true mathematical definition is the level of formality. For quite valid pedagogical reasons, you do not introduce concepts in their most formal guise. Getting across the basic idea that a function is a set of ordered pairs (with a given property) should not be convoluted by introducing unnecessary vocabulary. I would object strongly to seeing the term codomain at an introductory level. It is not even clear to me that it should be mentioned that this set of ordered pairs is a subset of a Cartesian product. But, not saying it does not mean that it is not true! All we are doing, by being mum, is being less formal with our definition. What I am objecting to is calling the same definition stated at different levels of formality "alternative definitions." Under the "alternative" definition given in this article, all injections are bijections because every function is surjective. This nonsense statement comes about only because the level of formality in the definition is too low to support the abstraction of the concepts in the statement. This is not a legitimate alternative, it does not lead to the same conclusions. So, what does this mean for the article? I would strongly prefer to see, instead of these statements about alternative definitions, something more like, "When treated less formally, the codomain of a function may not be mentioned explicitly. This leads to ..." Bill Cherowitzo (talk) 23:37, 8 April 2013 (UTC)
The definition of a function as a set of ordered pairs is completely standard in set theory - all the way to the graduate level. It is the definition used by Halmos Naive Set Theory, Levy Basic Set Theory, Kunen Set Theory, Jech Set Theory - the latter two being the current standard graduate-level books on the subject. Moreover, I thought I don't have them here, there is a longer list of sources at [1] that included books by Apostol, Bartle, Krantz, Kolmogorov and Fomin, Royden, and others. It is completely standard to define a function simply as a set of ordered pairs, and also completely standard to define it as a triple consisting of domain, codomain, and graph. neither definition is more or less formal than the other.
In the setting where a function is just a set of ordered pairs, one does not talk about a function per se being surjective, one only talks about a "surjective function to a set B". Thus surjectivity is not treated as a property of a function, it is a property of a function and a set. — Carl (CBM · talk) 01:13, 9 April 2013 (UTC)
Thanks for the archive link. I clearly walked into this blindfolded and didn't realize it was a hornet's nest. I have no desire to stir up that pot again – far too much verbage has been spilled on what I would call an argument over the number of angels that could fit on a pinhead. Bill Cherowitzo (talk) 04:58, 9 April 2013 (UTC)
One thing that I have always found interesting about editing here is that in a few occasions I have found that the definitions I learned and thought were universal were actually only one of several formally distinct definitions used by different authors in print - and other people learned the other definitions and thought that the ones I learned were crazy. This seems to happen much more often with basic concepts. — Carl (CBM · talk) 12:06, 9 April 2013 (UTC)

## Slowly (!) getting there -- but still full of inconsistencies

The article has much improved since I last saw it several months ago, but remains full of inconsistencies.

A major improvement is agreeing on the definition found in most present-day textbooks: "A function f from X to Y is a subset of the cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset".

Unfortunately, many logical inconsistencies remain, due to using disparate sources, and even internal inconsistencies in many present-day (say, post-2005) textbooks. The issue is muddled on this discussion page by throwing around names of authors without precise references (title, page), and often without having read them. Especially Bourbaki is a victim of such abuse. Hence we start with a few references. Firstly, for historical perspective:

• (A) Tom P. Apostol, Calculus, Vol. I, 2nd ed. (Wiley, 1967), p. 53
• (B0) Nicolas Bourbaki, Theorie des Ensembles (Hermann et. Cie, 1954), p. 76
• (B1) ibid., p. 77
• (Hl0) Paul R. Halmos, Naive Set Theory (Van Nostrand Reinhold 1960), p. 30
• (Hl1) Paul R. Halmos, "Nicolas Bourbaki", in: Morris Kline, Ed., Mathematics in the modern world, pp. 77-81 (Scientific American, 1968), p. 80.

Some useful terminology for later observations: Bourbaki (B0) uses the neutral term functional graph for a set of pairs no two of which have the same first member.

Secondly, a sample of current textbooks, mostly after 2005.

• (DG0) Ulrich Daepp and Pamela Gorkin, Reading, Writing and Proving: A Closer Look at Methematics (Springer, 2003) p. 147
• (DG1) idem, 2nd. edition (Springer, 2011) p. 143
• (K) Steven G. Krantz, Real Analysis and Foundations, 2nd ed. (Chapman et. Hall/CRC, 2005), p. 20
• (Hm) Richard Hammack, Book of Proof, 2nd ed. (CC-BY-ND, 2009), p. 195 [2]
• (R) Charles E. Roberts, Introduction to Mathematical Proofs -- A Transition (CRC Press, 2010), p. 220
• (V) Daniel Velleman, How To Prove It: A Structured Approach, 2nd ed. (Cambridge, 2009), p. 226

All of these definitions (with one exception discussed later) are logically equivalent to the one in the current Wikipedia article (a function being a functional graph). Crucial implications are the following.

• Since f is by definition a subset of X × Y where every element of X is represented (exactly once), only X, but not Y, is an attribute of f. This is why the ISO standard (referenced in the Wikipedia article) appropriately states that f : X $\rightarrow$ Y specifies that the domain of f is X and the range of f is a subset of Y.
• Since f is a set by definition, function equality is determined by set equality. Any other characterization of function equality that is not logically equivalent is inconsistent with this definition. For instance, an equivalent (hence consistent) alternative characterization of function equality is found as a theorem in (A) p. 54 and as a "new" definition in (R) p. 223: " f = g iff Dom f = Dom g and f(x) = g(x) for all x in the domain". Adding extras (like Cod f = Cod g) is inconsistent. The reader can verify that similar contradictions are present in nearly all post-2005 references quoted here -- certainly reason for concern.

Clearly, making Y an attribute (called "codomain" or not) is inconsistent with the (well-chosen) definition in the Wikipedia article!

If (!) codomains are wanted, the only way to introduce them consistently is not defining a function as in the Wikipedia article as a functional graph, but as a pair consisting of a functional graph F and a codomain Y, so f = (F, Y). In passing (but not important): some authors use triples (F, X, Y) where X = Dom F, which is clearly redundant.

It is significant that Bourbaki briefly considers the triple-based definition in (B0), but immediately announces (B1) that "function" will often just stand for "functional graph". This illustrates Halmos' observation about the Bourbaki group that "weary of trying to remember their own innovation, the authors slip comfortably into the terminology of the rest of the mathematical world".

Hence the crucial question is: do we want the extra complication of a codomain, in the Wikipedia article or in mathematical usage? In both cases, the answer is a very definite "no".

• For the Wikipedia article, because the definition in the current article is indeed the most frequently used and also the simplest (especially in Apostol's wording).
• In mathematical usage, because
(i) In the entire literature no argument can be found why a codomain might serve a useful purpose at all (except an inconsistent argument in a paper by Shuard).
(ii) Functions defined without codomains lead to a more general and fundamentally as well as practically more useful properties.

Discussing the latter two items is beyond the scope of Wikipedia, but anyone interested in this matter can obtain a note (10 pages) by sending me an e-mail.

Of course, some disagreement about the above arguments is to be expected. However, good decisions require a spirit of logical reasoning rather than near-religious fervor. In particular, given the prevailing definition of "function" as reflected in the current Wikipedia article, codomains at best deserve only very secondary status.

To conclude, logically consistent arguments why codomains might be useful at all are indispensable before giving them more than a brief mention in passing, justified only because the word exists! The fact that Wikipedia is also used extensively by students at all levels imposes a heavy responsibility regarding accuracy, consistency and relevance. Boute (talk) 14:54, 7 June 2013 (UTC)

### Discussion

If you look at the section immediately above this, you will see someone making exactly the opposite argument! There is no benefit in trying to work out which side is "right" -- both sides are very common in practice, and the article somehow has to represent both sides. Whether it does that well is another question, and would be worth discussing, but I see little reason to continue trying to argue that one of the two versions is wrong. I will simply quote from a 2011 Springer textwbook that gives the other side: Ethan D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, Undergraduate Texts in Mathematics, 2nd ed. 2011. From pages 131-132:
It is important to observe that a function consists of three things: a domain, a codomain, and a subset of the product of the domain and the codomain satisfying a certain condition. Indeed, one way of defining a function is as a triple of sets (A,B,F) where F is a subset of A×B that satisfies the conditions given in Definition 4.1.1. However, we avoid this cumbersome triple notation by observing that in Definition 4.1.1 every function is defined as being from a set A to a set B, denoted f:A→B, and therefore the domain and codomain of a fuction are always specified in the definition of a function. Hence, to define a function properly, it is necessary to say "let A and B be sets and let f:A→B be a function." We will sometimes be more concise and just say "let f:A→B be a function," where is is understood from the notation that A and B are sets. It will not suffice, however, to write only "let f be a function" without specifying the domain and codomain, unless the domain and codomain are known from context.
The need to specify the domain and codomain of a function when defining a function is not a mere formality, but a necessity when treating functions rigorously. For example, consider the set F={(n,n2) : n ∈ Z}. The set F is a subset of Z×Z that satisfies the conditions given in Definition 4.4.1, and hence F can be though of a defining a function Z→Z. However, the set F is also a subset of Z×R that satisfies the conditions in the definition of a function, and hence F can be thought of as defining a function Z→R. Such ambiguity is not acceptable when we use functions in rigorous proofs, and so the domain and the codomain of a function must be specified as part of the definition of a function.
The opinion expressed there is very common, and the article needs to take it into account. — Carl (CBM · talk) 15:34, 7 June 2013 (UTC)
Of course, the article has to take into account many views, but the consistency obligation remains, and the decision about relative weights cannot be avoided (in your text, "very common" is still "much less common"!). Also, if you carefully check my observations, the word "wrong" never appears, although in mathematics inconsistency is wrong. Unfortunately, now I must say that the quoted statement "Such ambiguity is not acceptable when we use functions in rigorous proofs" where "such" refers to the explanation that immediately precedes it is entirely wrong. Indeed, the fact that, with the definition in the current Wikipedia article, a function type X → Z (Z for the integers) is a subtype of X → R (R for the reals) in no way hinders rigorous proofs (what an idea!); it does not reflect any ambuguity, but a richer algebra of fuctions and function types. Even more: the note I mentioned is precisely motivated by the concern for rigorous proofs, which is why I found the inconsistencies in the referenced textbooks (about proofs!) so disappointing. If you are interested in the link with formal proofs, feel free to send me an e-mail to receive the note (.pdf file). Boute (talk) 17:09, 7 June 2013 (UTC)
Note that the present Wikipedia article does not define the bare word "function" in the section "Definition". Instead, for all sets X and Y, the article defines "a function from X to Y". Formally, this is an infinite set of definitions, one definition for each pair of sets X and Y (or, it is a relation R(f,X,Y) between the object f being defined, X, and Y. Compare defining "A is a subset of B" versus trying to define "A is a subset" without reference to any other set.) Every "function from X to Y" does have a codomain specified, namely Y. The section "Alternative definition of a function", although it is not very well organized at the moment, talks about the options that are present when one tries to define a "function"; that is the point where different definitions enter the picture. — Carl (CBM · talk) 17:52, 7 June 2013 (UTC)
A small aside, has anyone ever written something good about what saying the set of integers is a subset of the set of reals actually means?Dmcq (talk) 23:26, 7 June 2013 (UTC)
The question of the aside leads to Dedekind cuts, and is similar to the question how integers are a subset of (or embedded in) rational numbers (formally defined as pairs). Dedekind cuts were common fare in high schools 50 years ago, but were at some time eliminated from the curriculum as being "too abstract". Sad but true. Anyway, back to the subject.
Let's assume your interpretation that starting a definition with "a function from X to Y" announces an infinite set of definitions (I rather see it as an attempt to cram 2 definitions into one sentence, but let that be). Even in that interpretation, if a function f from X to Y is defined as a subset of X × Y with the functionality property, f is also a subset of X × Z with the functionality property when Y is a subset of Z, in which case f is also a function from X to Z by an "other" (Z-based) definition in your infinite collection. That's fine by me and by the ISO standard. Hence, even with the infinite collection of definitions interpretation, if you want to associate a codomain consistently, you have to complicate things by defining a function as a pair (F, Y) or, with some redundancy, as a triple (F, X, Y), where F is a functional set of pairs. My main point earlier was how fortunate it is that, in the current literature, for each textbook using such triples (or pairs), there is a vast multiple using the simpler formulation as in the Wikipedia article or Bourbaki on page 77 (op. cit.), defining a function as just a functional set of pairs (the F in the triple). Also, although I have shown how to make the codomain-based definitions consistent, the crucial remark remains: the burden of the proof as to why codomains might be useful is on those who want it. Did you ever see a fitting justification for codomains? Boute (talk) 01:54, 8 June 2013 (UTC)
A commonly mentioned justification is that one cannot say whether a function itself is surjective unless there is a codomain associated with the function. So one is no longer able to ask just "Is that function a surjection?" if the function has no codomain associated with it. But in many fields, particularly algebra, it is completely common to say that a function itself is surjective. And the term "surjective", by the way, is due to Bourbaki, who do define a function as a triple. — Carl (CBM · talk) 02:11, 8 June 2013 (UTC)
(margin reset) The "commonly mentioned" argument in favor of codomains is indeed the only one I ever found in the literature (traceable to Shuard's paper) but its "advantage" (writing just "surjective" instead of "onto T" if the range is T) is rather illusory: arguably, "surjective" is coarser than "onto T"! Recall Halmos's remark (Hl1) that the Bourbaki authors apparently regretted some of their experimental definitions, which explains the decision in Bourbaki (op. cit, page 77) for often using the definition by which a "function" is simply just a functional set of pairs, as in the Wikipedia article. As regards algebra, most "modern algebra" textbook authors in the 1960's were well aware of Bourbaki's work, but (at that time) were careful in writing "onto T" rather than just "onto". Herstein's Topics in Algebra (Xerox Corporation, 1964), arguably the best introduction to algebra even today, is illustrative (definition halfway page 12). Many "completely common" practices might be just sloppiness (sometimes even leading to inconsistencies as mentioned earlier).
More important in this discussion are the genuine advantages of not tying down a function by a codomain. Just two examples (many more available on request).
(a) With codomains, composition g o f for f : XY and g : UV is defined only for the special case when Y = U. This turns out to be very restrictive in actual usage. Without codomains, composition is defined for any pair of functions, the domain of the composition being the set of all x in X such that f(x) is in U. This definition is found in (A, p. 140), and many books not listed above, including most classical and modern calculus books (Axler 2008, Bartle 1964, Larson and Edwards 2009, Stewart 2007). In calculus, the codomain restriction would seem most unwelcome.
(b) With codomains, function typing is very inflexible. Avoiding codomains allows extreme flexibility, such as so-called dependent function types of the form
f : X::xY(x), where Y is a set-valued function allowing to express that the set constraining f(x) may depend on x.
See Halmos (Hl0), p. 36 for this view of the Cartesian product as a function space. Boute (talk) 08:55, 8 June 2013 (UTC)
The definition of a function as having a domain and codomain in mathematics corresponds quite closely with the definition of a function in strongly typed computer languages like Java. So you'd prefer a weakly typed language like LISP where anything goes. Fine for you. Lisp corresponds more closely with the set theory part of maths where functions as you like them are more appropriate and people want to do their own complicated things but don't cooperate with each other much. Python is an intermediate type language which is popular for getting work done quick, it is used in teams but typically where an occasional glitch isn't a problem. A language like Java or C++ is used more where people work in teams with each other and want to be able to depend on what they are using for robust applications that don't fall down. Yes your definition is flexible - but more often extra rigidity is what people really want, they don't want a funtion they want to talk about whole sets of functions of the same type. For everyday maths they want something like Python - tending towards being strongly typed but they fudge around and don't mind winging it a bit.
As to your integers being a subset or embedded in the reals. If you are going for a fully formal set theory approach that simply does not work. The natural numbers would be defined using sets of sets which don't at all match the pairs of a Dedekind cut. Dmcq (talk) 09:29, 8 June 2013 (UTC)
Sorry, Dmcq, but your inferences about what I personally like/dislike are simply wrong and likes/dislikes or programming languages are irrelevant in the effort to build a good Wikipedia article about mathematical functions. Also, linking weak/strong typing in programming languages to a codomain, as well as the idea that, without codomains, "anything goes" or program robustness is compromised goes beyond absurdity. Finally, observe that the aside which you designate as "your integers" was neither introduced by me, nor relevant to this article about the general notion of function. In other words, let's keep focus. Boute (talk) 10:17, 8 June 2013 (UTC)
Speaking of keeping focus: Boute, as far as I can tell you keep trying to argue that the definition without codomains is superior somehow. Such arguments are not really interesting to me, and they are irrelevant for writing this article. Both conventions are common in mathematics, and this article needs to cover both. I looked through the article again, and I didn't see any glaring "inconsistencies". If you focus on the actual content of the article, rather than arguments like "In calculus, the codomain restriction would seem most unwelcome.", it would help focus the discussion and move it forward. It would be a favor to everyone here if you could also keep your posts short and to the point, and avoid repeating arguments that have been covered before. — Carl (CBM · talk) 13:10, 8 June 2013 (UTC)
If you check, note that my observations contain significantly more examples, precise definitions and literature references "per line of text" than those of any other contributor thus far. So picking on the one sentence "In calculus, the codomain restriction would seem most unwelcome." is not serious. If it is a matter of wording, replace it by "The treatment of function composition in most current calculus textbooks indicates that the general form (defining g o f for arbitrary functions) is required, whereas the usual definition for g o f with codomains imposes the restriction that Dom g = Cod f ". In fact, saying "most" is most careful here, since it actually pertains to all current calculus textbooks I checked (about half a dozen, randomly chosen). — Preceding unsigned comment added by Boute (talkcontribs) 10:01, 9 June 2013 (UTC)
On the integers/reals business you said "Indeed, the fact that, with the definition in the current Wikipedia article, a function type X → Z (Z for the integers) is a subtype of X → R (R for the reals) in no way hinders rigorous proofs (what an idea!)" I disagree with saying a function like that is a subtype or subset of the other but if one is speaking loosely I wouldn't object in most circumstances. I brought in the programming languages analogy to try and step back a little and show a similar debate elsewhere and what it means. Your automatically changing reals into integers definitely counts as an operation in most programming languages even if implicit and they write stuff describing it. Dmcq (talk) 13:17, 8 June 2013 (UTC)
I certainly agree with the latter; considerable caution is needed. Note however, that the codomain issue is unrelated to degrees of formality, and "subtypes" are not a matter of "speaking loosely". The degree of formality is orthogonal to the choice of definition, and there are two clearly distinct definitions under consideration:
* (B0) Bourbaki p. 76: A function is a triple (F, A, B), where F is a functional graph and A = pr1 F.
* (B1) Bourbaki p. 77: A function is a functional graph.
Both definitions are equally formal. Since, in Bourbaki (p. 71), a "graph" is defined as a set of pairs, (B1) is equivalent to the definition in the Wikipedia article, the ISO standard, Apostol's book and most current textbooks. Also, in view of (B1), calling f a subfunction of g iff f is a subset of g is formal and unambiguous, and so isis calling XY a subtype of XZ in case XY is a subset of XZ (or, for nonempty X, if Y a subset of Z), where XY is defined as the set of all functions from X to Y (see, e.g., Bertrand Meyer), written by Halmos as YX. Boute (talk) 14:18, 9 June 2013 (UTC)
Overall I agree with CBM. I think the alternatives section definitely could be improved and a contribution there would be most welcome. However arguments about the superiority of a way of doing things would need citations to people comparing the alternatives and would simply be reported in the article rather than mean we should go about proselytizing for their preferred scheme. Dmcq (talk) 15:30, 8 June 2013 (UTC)
This brings up one of the main difficulties, which is that although there are many textbooks that have some description of functions, they don't compare and contrast their definition with other definitions. If there was a source somewhere that actually discussed both definitions, it would be very valuable for the "alternative definitions" section. By the way, I think that the "definition" section does a good job of being written in a way that can be read either way, depending on the taste of the reader. — Carl (CBM · talk) 19:39, 8 June 2013 (UTC)
I wrote about these two definitions here: http://www.abstractmath.org/MM/MMFuncSpec.htm and in the book Handbook of Mathematical Discourse by Charles Wells, Infinity Publishing, 2003. A preliminary version of the latter is online at http://www.abstractmath.org/Books/handbkhyper.pdf.

SixWingedSeraph (talk) 22:28, 8 June 2013 (UTC)

May we assume that in "The value of F at 3, for example, is 2, because the definition says that F(2) = 3." the reversal of 2 and 3 is a typo? Boute (talk) 10:14, 9 June 2013 (UTC)
Infinity Publishing comes under WP:SPS doesn't it? I think Wikipedia will have to do something more about SPS in future because so many people are doing this sort of thing. Dmcq (talk)
How realistic is WP:SPS if any editor with hidden identity can ignore it? Especially given the absence of peer review, perhaps it is rather the hidden identity issue that requires scrutiny.Boute (talk) 13:19, 9 June 2013 (UTC)
The 'self' in SPS refers to the book in that it hasn't passed through editorial control. You're probably thinking of WP:COI. Dmcq (talk) 17:24, 9 June 2013 (UTC)

### Some information about requiring codomains

I collected data from some math books published since 2000 that contain a definition of function; they are listed below. In this list, "typed" means function was defined as going from a set A to a set B, A was called the domain, and B was not given a name. If "typed" is followed by a word (codomain, range or target) that was the name given the codomain. One book defined a function essentially as a partial function. Some that did not name the codomain defined "range" in the sense of image. Some of them emphasized that the range/image need not be the same as the codomain.

As far as I know, none of these books said that if two functions had the same domain and the same graph but different codomains they had to be different functions. But I didn't read any of them extensively.

My impression is that modern mathematical writing at least at college level does distinguish the domain, codomain, and image/range of a function, not always providing a word to refer to the codomain.

If the page number as a question mark after it that means I got the biblio data for the book from Amazon and the page number from Google books, which doesn't give the edition number, so it might be different.

I did not look for books by logicians or computing scientists. My experience is that logicians tend to use partial functions and modern computing scientists generally require the codomain to be specified.

Opinion: If you don't distinguish functions as different if they have different codomains, you lose some basic intuition (a function is a map) and you mess up common terminology. For example the only function from {1} to {1} is the identity function, and is surjective. The function from {1} to the set of real numbers (which is a point on the real line) is not the identity function and is not surjective. Also a lot of modern math research is stated in terms of functors (particularly homotopy functors) which need to make the distinction.

THE LIST

Mathematics for Secondary School Teachers By Elizabeth G. Bremigan, Ralph J. Bremigan, John D. Lorch, MAA 2011 p. 6 (typed)

Oxford Concise Dictionary of Mathematics, ed. Christopher Clapham and James Nicholson, Oxford University Press, 4th ed., 2009. p. 184, (typed, codomain)

Math and Math-in-school: Changes in the Treatment of the Function Concept in ... By Kyle M. Cochran, Proquest, 2011 p74 (partial function)

Discrete Mathematics: An Introduction to Mathematical Reasoning By Susanna S. Epp, 4th edition, Cengage Learning, 2010 p. 294? (typed, co-domain)

Teaching Mathematics in Grades 6 - 12: Developing Research-Based ... By Randall E. Groth, SAGE, 2011 p236 (typed, codomain)

Essentials of Mathematics, by Margie Hale, MAA, 2003. p. 38 (typed, target).

Elements of Advanced Mathematics By Steven G. Krantz, 3rd ed., Chapman and Hall, 2012 p79? (typed, range)

Bridge to Abstract Mathematics By Ralph W. Oberste-Vorth, Aristides Mouzakitis, Bonita A. Lawrence, MAA 2012 p76 (typed, codomain)

The Road to Reality by Roger Penrose, Knopf, 2005. p. 104 (typed, target)

Precalculus: Mathematics for Calculus By James Stewart, Lothar Redlin, Saleem Watson, Cengage, 2011 p. 143. (typed)

The Mathematics that Every Secondary School Math Teacher Needs to Know By Alan Sultan, Alice F. Artzt , Routledge, 2010. p.400 (typed)

SixWingedSeraph (talk) 22:16, 8 June 2013 (UTC)

Thanks for that. Our article Binary relation has exactly the same problem as this article. There some people use the word 'correspondence' instead of relation for the triple and relation for the set of ordered pairs. And they have the exact same problem about saying if a relation is surjective etc. Personally I can't see why we should treat a set of ordered pairs as a function when we already have the term functional relation - but of course then the problem has been regressed to what definition of a relation one is using so one would have to have a relation not being a correspondence. Unfortunately that makes how all the terms apply different. Anyway I'm happy this article mainly talks about functions from X to Y and deals with 'function' on its own as the problem case. Dmcq (talk) 11:16, 9 June 2013 (UTC)
The addition by Charles Wells to previous lists is much appreciated. Perhaps all those lists should be collated somehow.
One potential subtlety requires closer scrutiny. Let's consider all definitions where a function is defined as a functional set of pairs whether ot not the sentence starts with the preamble "from X to Y " . and compare the two groups.
(PRE) With preamble (or "typed" in the terminology proposed by Wells), as in the Wikipedia article and most current textbooks: "A function f from X to Y is a subset of X × Y such that every element of X is the first component of one and only one pair in f.".
(NOPRE) Without preamble, as in Bourbaki (B1) and Apostol (A): "A function is a functional set of pairs, i.e., a set of pairs no two of which have the same first member."
Now, let f be a function from X to Y in the sense of (PRE). Clearly, f is a functional set of pairs, and hence also a function in the sense of (NOPRE). Conversely, let f be a function in the sense of (NOPRE), and define, as in Apostol p. 53, Dom f to be the set of first components and Ran f to be the set of second components. Let X = Dom f and Y a set such that Ran f is a subset of Y. Clearly f is a subset of X × Y, and every element of X is the first component of one and only one pair in f. Hence f is a function from X to Y in the sense of (PRE).
The resaoning is rigorous and can be fully formalized.
Finally, add to (NOPRE) the definition "A function f is said to be from X to Y iff Dom f = X and Ran f is a subset of Y (expressed by f : XY, as in the ISO standard). The bottom line is that both definitions are equivalent. If there is any problem, it is with (PRE), which tries to cram two definitions into one: what a "function" is, and when a function is from X to Y.
The psychological problem is that people who have not not fully grasped the cartesian product think that a function defined as a subset of X × Y somehow inherits X and Y in its DNA. For X this causes no problem, due to the extra stipulation in (PRE) that every element of X is represented (exactly once) as the first component. For Y, this does not hold.
Conclusion: rather than improving the definition, the preamble does not help understanding and makes (PRE) easy to misinterpret. A two-step approach, first (NOPRE) from Apostol, then the addition introducing XY, eliminates the problem. Boute (talk) 15:43, 9 June 2013 (UTC)
The proper function of an encyclopedia article is, among other things, to describe the usage of technical language in the subject of the article. It is clear to me the following usages for "function" are common:
• A function $f:A\to B$ has a graph, which is a relation on $A\times B$ with the functional property. Thus it is "typed" in the sense of my previous post on this Talk Page.
• Texts may or may not regard two functions with the same graph but different codomains as the same or different. Thus there are two different meanings of equality for functions and the particular text may not tell you which is being used, and (it appears to me) in many cases it does not even matter, and in other cases it is an important distinction.
• Many texts define a function as a functional relation, period.
• Some texts assume that "function" means what the rest of us call a "partial function".
The main Wikipedia article on functions should describe all these usages in understandable terms and using easy examples. (For example, is an identity function different from an inclusion function?) Mathematicians will continue to use common terms with varying definitions and there is no Terminology Academy to stop them.
I recommend the article use "function" for the typed case -- so there are two meanings of equality -- and "functional relation" for the other case. Some authors have recommended that the word "map" be used for the typed case with the stricter meaning of equality (so codomains matter). I think that it is an excellent idea, but it is not common practice.
Arguments for or against particular usages do not belong in Wikipedia. Of course, you can reference arguments in other texts.
Note: I wrote this post without reading the post by Boute
SixWingedSeraph (talk) 16:31, 9 June 2013 (UTC)

Here are several books that explicitly state the condition that the codomains must be equal when defining equality of functions. Although there are more, I tried to pick a sample to show that this usage goes beyond just abstract algebra.

• Jacobson, Basic Algebra I, second edition, page 5 - a graduate level abstract algebra monograph
• Devlin, Sets, Functions, and Logic: An Introduction to Abstract Mathematics, 3rd edition, page 92
• MacLane and Birhoff, Algebra, 3rd edition, page 5
• Eccles, An Introduction to Mathematical Reasoning, page 93
• Joshi, Introduction to General Topology, page 33
• Haddad, VijaySekhar Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach, page 33

— Carl (CBM · talk) 14:16, 10 June 2013 (UTC)

## A Modest Proposal

It is clear that after much good work by many people there is no longer a bull goose mathematician capable of settling the question of codomains -- the age when Hilbert said it, I believe it, and that settles it is long past. (The same is true of rings.)

I suggest: the article begin with a definiton for the non-specialist. A function assigns to any given input a unique output. We then need some examples. Later in the article, below the ToC, we need a statement that different authors use different definitions, and a clear statement of the three primary points of view: 1: a function is a set of ordered pairs, 2: a function is an ordered triple: domain, codomain, set of ordered pairs, 3: (the naive definition) a function is a rule.

I really do think that we can discuss this until the cows come home and not arrive at a single definition of "function".

Rick Norwood (talk) 17:29, 9 June 2013 (UTC)

I think the article does start off in a reasonably non-specialist way and it most deals with the case where the differences don't matter much, i.e. where X and Y are specified. I'm not too keen on confusing people by having say exp(x) be the same function whether we're talking about reals or complex numbers and I'm pretty certain students in the future will often have some exposure to computer languages which not only distinguish the two but have explicitly given possible inputs and outputs. The article used to be more up front about the two main variants where one said function without specifying X and Y and discussed them in parallel, but it was changed to the current form with a main and alternate view after having people come along arguing that one side or the other was the greatest most wonderful way and one side basically ran over the other - in fact I believe that argument was initiated by people arguing that not having a codomain was the main way so it kind of backfired on them. Dmcq (talk) 18:14, 9 June 2013 (UTC)
I agree with the proposal. Also, any explanation saying that exp(x) is the same function whether we're talking about reals or complex numbers would indeed be very confusing, but (fortunately) thus far no one has proposed such a thing. However, continuous personal remarks intending wanton misrepresentation are harmful to the final quality of the article. Indeed, in the preceding observations, I often unambiguously endorsed representing both views. Far from "running over" something, I even showed how to make the codomain-based view consistent (using pairs or triples), altbeit properly balanced with factual mathematical properties (no hyperboles!) indicating that choices do matter.
Unfortunately, the current Wikipedia article is very problematic. On one hand, the stated definition is indeed the simplest and most frequently used in modern texts, namely as a functional set of pairs, as in Bourbaki (B1) (no "extra's", as in the triple of B0). On the other hand, the same article without blinking talks about codomains as an attribute of such a function. This is inconsistent and, if diehards persist in denial, will backfire on the entire article.
Since apparently symbolic arguments are basically run over or simply not understood, I make a final attempt by providing an example and then stop wasting my time (ironic cheers predictable). The example comes literally from Hammack, chosen because it is freely available to everyone [Hammack, Book Of Proof], Example 12.2 page 196.
(Begin quote) Let $A = \{p, q, r, s\}$ and $B = \{0, 1, 2\}$ and
$f = \{(p, 0), (q, 1), (r,2), (s,2)\} \subseteq A \times B$
This is a function $f : A \rightarrow B$ because each element of $A$ occurs exactly once as the first coordinate of an ordered pair in $f$ (End quote)
By definition 12.2 that precedes this example on the same page, $B$ is the codomain. However, letting $C = \{0, 1, 2, 3\}$, we also have $f = \{(p, 0), (q, 1), (r,2), (s,2)\} \subseteq A \times C$ and hence $f : A \rightarrow C$, so $C$ is the codomain. Since $C \neq B$, this is a contradiction.
Comparing the definitions shows that this is exactly the same situation as in the Wikipedia article. I rest my case. Boute (talk) 20:39, 9 June 2013 (UTC)
You stuck in f both times but that doesn't mean the two f's are the same. One f is a function from A to B, the other is a function from A to C.It is like in the real or complex number case saying that 1.0 is a real or 1.0 is a complex number. They may look the same but they're different. Of course that's working from the opposite viewpoint to you, you are right from your own viewpoint. Dmcq (talk) 21:33, 9 June 2013 (UTC)
In both cases, the function is simply the set $\{(p, 0), (q, 1), (r,2), (s,2)\}$, irrespective of the name. You can even rephrase the argument without the name, just in terms of this set. If you want to enforce a difference under Hammock's Definition 12.1, you have to use tagged sets or a similar device (as in the disjount union of a set with itself - see Meyer, Introduction to the Theory of Programmig Languages, 1990). In most of mathematics and engineering, enforcing a difference is restrictive. In programming, you can do it in the manner just explained when you need it. This is the advantage of not irrevocably building the difference into the definition from the start. Boute (talk) 05:26, 10 June 2013 (UTC)
You are still in the mode of trying to persuade us that one definition is better than the other. This is the wrong approach. You should be trying to persuade us that one definition is *more accepted* than the other. Or at least, that the level of coverage of one approach in our article is out of proportion to its level of acceptance in the literature. —David Eppstein (talk) 05:56, 10 June 2013 (UTC)
Don't patronize. I'm not trying to persuade anyone; people can only persuade themselves by careful thought about arguments given. I don't even believe that "the level of coverage of one approach in our article is out of proportion to its level of acceptance in the literature". To the contrary, I repeatedly emphasized that the current definition of a function in the article is, and even by far, the most accepted one, as I carefully checked. At worst, the coverage of codomains might be out of proportion, but judging that requires statistics that are not available.
So please stop wasting time and concentrate on the essence of the arguments I gave, which concerns the remaining inconsistencies in the current article. Try to get it right; it's easy for any mathematician used to formal reasoning. Boute (talk) 07:18, 10 June 2013 (UTC)
Afterthought: I consider it only polite to keep responding to remarks addressed directly or indirectly at me, but would prefer devoting my time to other things and see how the article stands in, say, a few months. Boute (talk) 07:29, 10 June 2013 (UTC)
The two functions are only the same because of your definition of a function. Most of the article talks about a function from X to Y rather than a function all on its own and does not try comparing it to some other function from A to B. It is only in the alternatives section it talks about functions without such a qualification. If a function from X to Y is considered as typed then its value can be given by a set of ordered pairs, that does not mean it is the same as some other type of function. It is only if you start with the assumption that it is completely defined by the set and the qualification is irrelevant that of course the qualification is irrelevant because we're assuming functions are untyped. To go back to your considering they are the same from the value without considering the type consider
2 in a representation of numbers using set theory may have the value 2={{}, {{}}}
When talking about ordered pairs a way of representing ordered pairs a short way is (a,b)={a,{a,b}}, this means for instance that (0,0)={{},{{}}}. And in a representation of the integers by the equivalence class of a-b that is 0.
However we do not conclude that 2=(0,0)=0 because the set represents different things each time. It is typed.
To fully specify a typed function complete with its type so we can say function on its own meaning some type of function then we need to specify a triple. But if one's not interested in specifying the domain and codomain and just want untyped functions then of course your way of doing it works. However outside of logic and set theory most people are more interested in sets of functions with a given source and destination set whatever about the flexibility of being completely free. Dmcq (talk) 09:04, 10 June 2013 (UTC)
(MARGIN RESET) Referring to the definition in the current Wikipedia article, also found in most present textbooks: the interpretation just described can be found nowhere in the literature. To the contrary, the notions of cartesian product, subsets and set equality are defined everywhere (i.e. in all references that came up in these discussions thus far) in a clear and unambiguous way that logically excludes this interpretation. Also, consider the analogy with the (PRE) style mentioned above: assume someone states: "DEFINITION: Let T be a set. A subset S of T is a set whose elements are all in T ". Now someone else might start arguing that T is an attribute of S, and that a set defined as a subset of T is somehow different from the same set defined as a subset of U in case U is different from T. However, equality between sets is defined as having the same elements, irrespective of the (many!) sets of which the considered sets might be a subset. Hence, if a function is defined as a subset of X × Y (with the extra property as in the article), it is a set and does not "change" if one defines it as a subset of X × Z -- assuming, of course, that the subset relation indeed holds. Note that using the same X is imperative, and that the function type that logically follows from the Wikipedia definition is that of the ISO standard, stating that the type definition f : XY specifies that Dom f = X and Ran f is a subset of Y. Such functions are not, in any meaningful sense, "less typed" than in the alternative definitions (e.g., as triples), only typed differently. In the example definition "sqrt : R>= 0 → R>= 0 defined by (sqrt x)^2 = x" the type specifies the nonnegative root. Obviously, in this setting, the "distinguishing effect" you are trying to obtain by a nonstandard view on set equality can be obtained simply by using tagged sets for Y and Z, as is common practice in programming language semantics (recall Meyer). Boute (talk) 13:42, 10 June 2013 (UTC)
As far as I can see, you are still talking about the definition of a "function" rather than "a function from X to Y". Instead of continuing that analysis, I would like to ask you to focus on describing any inconsistencies you see in the actual text of the article. You have claimed there are some, but I did not see them when I looked through the text. — Carl (CBM · talk) 14:10, 10 June 2013 (UTC)

(MARGIN RESET - long text by necessity!) I apologize for the length of this text, but please don't read it diagonally, look at all references provided (just 1 click away every time to make it easy) and, in case there is still disagreement, point specifically to the particular step in the arguments and state why you don't consider it logically valid.

I was definitely talking about "a function from X to Y" (or "A to B") as in Hammack's Definition 12.1, p. 195), in the current Wikipedia article and nearly all present-day textbooks. Let's call this definition (X2Y).

(I) I'd also prefer skipping the analysis, but some preliminary arguments in small steps are indispensable to see the inconsistencies in the rest of the article. For distinguishing definition variants, rigorous logic avoids (a) reading into a definition things that are not there, (b) overlooking things that are (already) there.

An illustration: by Example 12.2, p. 196, quoted literally earlier, Hammack demonstrates that he really means what he says in Definition 12.1 -- as befits a book about proofs. He concludes that the example function (a set of pairs) is really a function from A to B from two facts:

(i) the function is a subset of A × B;
(ii) each element of A occurs exactly once as a first component.

This is exactly what his (and the Wikipedia) definition says, nothing more, nothing less, leaving no room for other interpretations. Replacing B by any superset C of Ran f (or Im f) throughout Example 12.2 yields a proof that the same function (set of pairs!) is also a function from A to C under definition (X2Y).

Conclusion: all that (X2Y) says about the relationship between f : XY and the sets X and Y is that Dom f =X and Ran f (or Im f) is a subset of Y. This logically obvious point is made explicit in Bartle, p. 5, in Zakon, pp. 10--11 and by Labl, p. 14, referenced in the Wikipedia article on real analysis, and what the ISO standard says about the notation f : AB. It is consistent with all modern texts using the (X2Y) definition, up to the point where some texts themselves contain internal inconsistencies (the latter's existence is quite scary).

Consequence A: making a codomain a function attribute in the sense that, given f : XY, one can write Cod f = Y is inconsistent with (X2Y), since it would allow Cod f = Y and Cod f = Z with Y /= Z. Obviously, this does not prevent calling Y the "codomain" of the area of discourse.

Consequence B: with (X2Y), "onto" is a preposition, as in "onto Y" (Halmos, Bartle, Kolmogorov etc.), and using "onto" as an adjective assumes Y from the context. Example: in the April 2013 Bulletin of the AMS, Kontorovich, pp. 199 and 210 uses both forms whereas Fuchs, pp. 241, 242, 244--246 uses "onto" as a preposition only.

Occasionally the remark surfaces that one cannot talk just about a "function" an sich but only about "a function from X to Y for given sets X and Y". Its validity critically depends on the definition variant. A complete analysis is skipped here, but consider:

(a) The definition in Apostol (A) and Bourbaki (B1) where a "function" an sich is simply a functional set of pairs is in no way problematic and is used formally in Suppes, Axiomatic Set Theory (Dover 1972), Definition 39, page 86 -- so let's call this definition (ABS).
(b) It has been shown that any function from X to Y as defined by (X2Y) is a function by (ABS), and that any function f as defined by (ABS) is a function from Dom f to any superset of Ran f by (X2Y). This follows from the definitions if one puts aside all dogmata other than rigorous logic. Conclusion: the remark that the phrase "from X to Y" is essential in the definition of a "function" does not pertain to the Wikipedia definition (which can be rephrased as ABS), but only to some other definitions that one might discuss in an "alternative definitions" section.

(II) This careful preparation allows to be more brief in discussing the residual inconsistencies in the Wikipedia article, and how to correct them with very minimal changes. Aside: keeping the "codomain" issue in a separate Wikipedia article is inviting inconsistencies due to incompatible editings.

(IIa) Inconsistencies in the "function" article In brief: residual inconsistencies in the article arise whenever it is said or implied that a function according to (X2Y) has a unique codomain. The text of the Definition section with (X2Y) is itself consistent, but the diagram is inconsistent with (X2Y), since the set of ordered pairs shown is the function by definition (X2Y), not just a component of it.

Similarly, saying that "A function is properly defined only when the domain and codomain are specified" is inconsistent with (X2Y), according to which a function is fully defined as a set of pairs.

(IIb) Correction with minimal changes It suffices to make clear that "codomain" in the context of (X2Y) is not an attribute of the function itself but a set assumed known in, or implied by, the context.

(i) In the introductory text. For instance, changing the text "called the domain, a set containing the outputs, called its codomain" into "called its domain, a set containing the outputs, called codomain" and adding "In some definitions, the codomain is part of the context only, in others it is also part of the function itself.
Definitely misleading is the suggestion that function spaces in real analysis involve codomains. I checked all references and links in that article where the definition of "function" was visible (Google Books is more irritating than Amazon): nearly all of them use definition (X2Y) but none of the (X2Y)-users mention the term "codomain", and the function space XY, also written YX, is simply as defined in the ISO standard. Note that this function space is isomorphic to the XY variant with codomain.
(i) In all the rest of the article: similar diligent care.

The codomain article, whether or not integrated in this "function" article, needs more references to compensate for its strong bias. I'll add them when time permits. Boute (talk) 06:49, 15 June 2013 (UTC)

The problem is encapsulated in the example of the problem you are having about whether you aredealing with a typed functionor just functions where the type is not specified. Now how about applying your own logic to
Natural number 2 ={{}{{{}}}
Ordered pair (0,0)={{{].{{}}}
Integer 0=0-0={{}.{{}}}
Therefore 0=2.
Two sets that look the same can mean two entirely different things if they are typed. If the domain and codomain are given then the function can be specified by a set of ordered pairs. That does not mean it is the same as a function to some other codomain. This applies to both the source and target for relations from one set to another. The big difference comes when one talks about a relation without assuming one is talking about a relation between one set and another. Some people say you map the relation on one pair of sets to the on one the other pair whereas your way is to say it is exactly the same relation. Even though that first reference you gave talked about the graph of a function you say they showed their true colours because they wrote f=set instead of graph(f)=set but you ignored the inclusion in the set product at the end of the line so in fact they were saying the f was a function from A to B. Don't you think it would be better to accept people might actually mean what they say they mean rater than try and find something to show they mean something other than what they say?
It is perfectly okay your path and for the 1={{},{{}}} I guess one aught to put extra brackets and operators round it like saying Graph(f) instead of f above if one were working in set theory and yet wanted to deal with functions from A to B. However trying to show what you want is better and interpreting other people's work in your way is not leading anywhere useful. Why can't you just try and improve the alternatives section rather than trying to prove a point of view? 12:50, 15 June 2013 (UTC)
I am quite aware of the problem you mentioned with natural numbers, ordered pairs etc. (just google "representational and denotational semantics"), but that is an entirely different issue. Indeed, that problem stems from representing quite different kinds of objects by sets. Unavoidably this causes artefacts, as also extensively discussed by Halmos in Naive Set Theory in various places, e.g., p. 25. Here we are dealing with function objects only.
The sentence in my analysis,
"(i) the function is a subset of A × B"
highlights rather than ignores (quote) "inclusion in the set product at the end of the line".
Moreover, this inclusion is the crux of the matter. If B is a subset of C then A × B is a subset of A × C. Hence, if f is a subset of A × B it is, by transitivity, a subset of A × C. Note that, in all the (X2Y) definitions in the ever-growing reference list, it is the function itself that is defined as a subset of A × B (not its graph, which is not even mentioned).
It may interest you learning that my own preferred definition of "function" (which I carefully avoided discussing at all on these pages) is quite more abstract than (X2Y) or (ABS), essentially to avoid artefacts as discussed by Halmos. So my arguments are not motivated in any way by "what I want" for definition, but by the importance of getting the Wikipedia article right. Boute (talk) 18:31, 15 June 2013 (UTC)
A function from A to B is completely represented by its graph but according to the first alternative in the article is typed a function from A to B. It is quite reasonable to represent it with a set of pairs. The product specified the A and B and the author went and specified them. It really makes very little sense that you go around saying an author really meant something else when they talk about domains codomains and graphs. There are other works that say differently and agree with the alternative view. As to being correct, we are not some sort of Bourbaki group. As WP:POV says we should present the significant views fairly and in proportion. Dmcq (talk) 20:10, 15 June 2013 (UTC)
I'm not talking about alternate views or which of them is "correct", I am talking only and exclusively about the definition of a function from X to Y as in the Definition section of the Wikipedia article, and in all the other texts that happen to use exactly the same definition. In each of my comments, I've added new references pertaining to this definition -- when will you give some as well? Please read Bartle, p. 5, Zakon, pp. 10--11 and Labl, p. 14. It's free! Anyway, the time I allotted for this is up. I'll see in a few months how it goes. Boute (talk) 21:10, 15 June 2013 (UTC)
I haven't Bartle but I looked at the other two and basically we just don't see eye to eye and we've gone over it all before. I believe the f:A->B description in the main part of the article covers both viewpoints quite adequately, and I really don't see why you keep on trying to say as far as I can see that one of the alternatives is wrong and the other is the only one that is truly consistent with the main part of the article. Dmcq (talk) 21:54, 15 June 2013 (UTC)

In view of the discussion above, wasn't the old version of the section Formal definition [3] actually better than the present one, which only mentions the definition of a function as a set of ordered pairs? Isheden (talk) 15:58, 30 June 2013 (UTC)

I put some work into that old section but it seemed people here wanted things more black and white confuse people with different possibilities. The article now has a main usage and an alternatives section. I believe the alternatives deserves a higher profile but the point of view warring like above has rather polarized the whole business. Dmcq (talk) 20:17, 30 June 2013 (UTC)

## Function (set theory)

If we remove a few lines in bottom, then this page is perfect for Function (set theory) instead of Function (Mathematics). The bottom section where people hardly notice is (imho) an accuracy dispute, since when do we agree that function has to has something to do with set and set theory? And the informal concept of function becomes history? --14.198.220.253 (talk) 10:56, 9 November 2013 (UTC)

I think the situation is practically completely opposite to what you say. It is mainly in the foundations like set theory where people use functions without a given domain and codomain. Dmcq (talk) 12:23, 9 November 2013 (UTC)
I say the situation is independent of what you think. It is mainly in the foundations like set theory where people use functions with a given set of domain and codomain. --14.198.220.253 (talk) 12:52, 9 November 2013 (UTC)
That is simply asserting your opinion again. Either provide evidence that shows the article should change or wait for more opinions on this talk page to back your opinion. Dmcq (talk) 14:22, 9 November 2013 (UTC)
The same goes for you, that is why i see it as an uncited opinion and turn the "joke" back on you, here is your comment, you can verify it. --14.198.220.253 (talk) 14:25, 9 November 2013 (UTC)
You're the one wanting to change things. Dmcq (talk) 14:29, 9 November 2013 (UTC)
I don't know how one can explain the previous edit then, everything is an edit on Wikipedia. --14.198.220.253 (talk) 14:35, 9 November 2013 (UTC)
Many edits are not objected to and consensus can be gained at the talk page for some which are objected to. Dmcq (talk) 14:42, 9 November 2013 (UTC)

## Discuss reverted change casting doubt

Please do not just try forcing things into an article when another editor objects and gives reasons. Please read WP:BRD. Sources say that is how it is said. Provide a source saying it is said some other way. Dmcq (talk) 14:33, 9 November 2013 (UTC)

First, BRD is not a policy. Second, is previous edit a reliable source? If not, then an edit has to has a source as long as it is reverted? Where is bold? You see.
Third, you think it is "casting doubt", I don't think it is. Let me explain further,
\begin{align} f\colon \mathbb{N} &\to \mathbb{Z} \\ x &\mapsto 4-x. \end{align}
The first part can be read as:
• "f is a function from $\mathbb{N}$ (the set of natural numbers) to $\mathbb{Z}$ (the set of integers)" or
• "f is an $\mathbb{Z}$-valued function of an $\mathbb{N}$-valued variable".
Your edit changes from "The first part can be read as:" to "The first part is read:".
When you say "is read", it is not clear that whether it has to be read that way or not. For example, "a function f from N to Z.", "a function f maps from domain N to codomain Z", "f is a function has domain N and codomain Z.", "f colon N arrow Z is a function f from N to Z", Are they all wrong?
Is there a transformation rule to those two forms? What makes them so special? Why should it be in those two forms and keep all the words and structure in check?
That's why I think "is read" is in fact "casting doubt", or in good faith, unclear.
I think you can say $f\colon \mathbb{N} \to \mathbb{Z}$ means "f is a function from $\mathbb{N}$ to $\mathbb{Z}$", you can at best suggest people to read it in certain way, you can say it in different way in English even though it makes use of the word "from" "to" "function"..etc. but you don't say "it is read" because it sounds like it has to be read in the following way. --14.198.220.253 (talk) 15:19, 9 November 2013 (UTC)
You changed the 'is read as' to 'can be read as'. The 'can be' means there are other forms also. Have you found other forms? Dmcq (talk) 15:33, 9 November 2013 (UTC)
Have you read my post? --14.198.220.253 (talk) 15:35, 9 November 2013 (UTC)
Yes. Have you found a citation for a new version? Dmcq (talk) 15:48, 9 November 2013 (UTC)
Have you read the article? Have you found a citation for the old version? Yes?--14.198.220.253 (talk) 15:51, 9 November 2013 (UTC)
If you are doubting those forms then put in {{citation needed}} for places where you require a citation and fill in the reason. Dmcq (talk) 15:59, 9 November 2013 (UTC)
Thanks, it is a useful advice, but I am not doubting those forms.
Thought you said something about consensus above. You didn't respond to my examples yet, "a function f from N to Z.", "a function f maps from domain N to codomain Z", "f is a function has domain N and codomain Z.", "f colon N arrow Z is a function f from N to Z", Are they all wrong? --14.198.220.253 (talk) 16:14, 9 November 2013 (UTC)
They're descriptions of a function f okay, but are they readings of what's written or descriptions of f which just mean the same? 1+1 is the same as 2 but it is read differently. Saying f is a function from domain N to codomain Z for instance is saying N is a domain whereas it is a function from N to Z which has domain N and codomain Z. The alternative definitions section goes into the difference between a function and a function from N to Z and the various ideas of what a function might be. It is certainly okay mostly to just say function here but reading it that way sounds problematic to me. At best it sounds like loose language to me and there's enough loose language already around functions. So yes I rally would like to see someone saying it is read one of those ways rather than you making them up. Dmcq (talk) 16:39, 9 November 2013 (UTC)
I didn't say "f is a function from domain N to codomain Z", but even if your interpretation is right, we can just take that one out. So, why can't I read it as "a function f from N to Z"? Is it wrong?
It is certainly okay mostly to just say function here but reading it that way sounds problematic to me. At best it sounds like loose language to me and there's enough loose language already around functions.
You see, you just said there are "good read" and "bad read", so there is no "must read" or "is read". If you have some kind of transformation rule, for example, to read $f\colon \mathbb{N} \to \mathbb{Z}$ literally, you map each symbol to an English word, "f colon N arrow Z". If you can do that for its semantics too, then we can discuss.
"is read" is inaccurate for now, we should change it to "can be read", "it is recommended that" "it means"..etc. --14.198.220.253 (talk) 17:33, 9 November 2013 (UTC)
Please provide a citation saying is is read otherwise. We are supposed to summarize what is out there, not have people arguing about their own ideas and then stick them in. Dmcq (talk) 19:01, 9 November 2013 (UTC)
I have now had a look around at some places where they say how it is read and there are some variations that justify not being dogmatic about it so 'can' can be put in so I'll self revert. Dmcq (talk) 20:00, 9 November 2013 (UTC)