Talk:Function (mathematics)/Archive 6

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Definition of function

I read the discussion about definition above with interest. Whatever, the actual definition given at the beginning is hopeless. "The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or output)."

1. It's generally not good practice to begin an article with 'the concept of an X', since this is nugatory. Instead of saying 'the concept of a human is the concept of a rational animal' why not say 'a human is a rational animal' or something like that.
2. 'expresses the intuitive idea' is a category error. A concept can't 'express' anything. Rather, a term expresses something by signifying a concept.
3. 'one quantity ... completely determines another quantity' is better, but why 'quantity'? An argument to a function does not have to be a quantity, if we are talking about logic. I suppose this is about mathematical functions, but then why the stuff lower down about logical functions. A logical function (e.g. 'is running') takes the possible value of a variable as argument (e.g. Socrates himself) and maps it onto a truth-value (true or false).
4. Indeed, a few sentences later it is asserted that sets can be arguments to functions. Sets are not quantities.
5. The notion of 'determine' is a difficult one to explain unless we resort to the idea of a function!

Logicist (talk) 11:06, 23 August 2009 (UTC)

Good points. Why not try a rewrite? Rick Norwood (talk) 15:50, 23 August 2009 (UTC)

I have done so. Hi Rick. Logicist (talk) 15:52, 23 August 2009 (UTC)

Cauchy

According to Grattan-Guinness, the modern idea of a function originates with Cauchy. Is this correct? The article doesn't mention him. The lead is now looking much better, in my view. Thank you Rick, for simplifying the first para. Logicist (talk) 21:16, 23 August 2009 (UTC)

I doubt it, but I'd have to see the quote in context. Granted the following are logicians, but I don't see Cauchy in my Couturat 1914, nor Jevons 1880, nor Boole 1854, nor Davis 2000 Engines of Logic, nor Tarski 1946, nor in my Grattain-Lewis, Bornet (1997) George Boole: Selected Manuscripts on Logic and its Philosophy. (I searched the indices and by pattern-recognition of the .pdf files). In my Eves 1990 Foundations and Fundamental Concepts of Mathematics Cauchy appears twice in the index in reference to resolving a "foundational crisis" quote "by replacing the hazy method of infinitesimals by the precise method of limits" (p. 263), and quote "A great forward stride was made in 1821, when the French mathematician Augustin-Louis Cauchy (1789-1857) successfully executed d'Alembert's suggestion, by developing an acceptable theory of limits and then defining continuity, differentiability, and the definite integral in terms of the limit concept. Essentially these definitions are those found in the more carefully written of today's elementary textbooks on the calculus . . . The demand for an even deeper understanding of the foundations of analysis was strikingly brought out in 1974 [etc]". I highlighted "fundations of analysis" to make the point that this is the context in which the above was written as opposed to deep-foundational attempts to characterize the notion of "function". He is also mentioned once in Kleene 1952 w.r.t. "the theory of real numbers which is currently used as the basis for analysis" (p. 30). He appears twice in van Heijenoort re "eliminating references to infinity".

One person/cource that has not been mentioned is Wittgenstein (1922) Tractatus Logico-Philosophicus *5.54 ff, pp. 19-21. This reference appears in (1927-- 2nd edition) Principia Mathematica (page xiv, i.e. the introduction): "This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values" (page xiv). I haven't had time to explore this or Russell's 1927 emendations and re-takes in his Appendix C. Bill Wvbailey (talk) 22:45, 23 August 2009 (UTC)

GG says in 2.7.2 of Roots that Cauchy insisted that a function must be single-valued, so that even sqrt(x) (x>0) has to be split into its positive and negative parts. GG makes another interesting comparison between the idea of a function and Russell's theory of descriptions, saying that the conditions Russell gives for a definite description denoting something are essentially the same as a function having a value. He also notes that these conditions were given by Peano in a much earlier paper about functions and the eliminability of the word 'the', even suggesting that Russell had copied the idea from Peano without attribution. See Roots 7.3.4 (on Russell) and 5.4.3 (Peano) Logicist (talk) 06:00, 24 August 2009 (UTC)

Wow. Very interesting. This stuff should go in. The origin and evolution of the "single-valued" notion has definitely been a loose end. Perhaps under "Schönfinkel's notion of "function" as a many-one "correspondence" 1924". (Tarski 1946:102 uses the similar example x² + y² = 25 with y=4 yielding x= -3 and +3. Tarski opines that "It seems, however, inexpedient -- at least on an elementary level -- to denote such relations as functions, for this only tends to blot out the essential difference between the notion of a function and the more general one of a relation.") Of course, like Russell, Tarski doesn't provide attribution for this idea; Russell is very bad about precise attribution. (Although in his 1903 he admits he took ideas from Peano.) The business about the descriptive function and the notion of identity and uniqueness is the supposition I had to draw based on the evidence, but it needs a reinforcing quote or at least a footnote to that effect. (This is the main reason why I want to see Schroeder, I suspect "single-valued" came from/through him). But I'll leave it up to you, if you want to do the dirty work; I don't have this book. Or give me the book's full title & publisher etc and I'll come up with it. It sounds like a book I need in my "collection". Bill Wvbailey (talk) 14:34, 24 August 2009 (UTC)

Ivor Grattan-Guiness, The Search for Mathematical Roots, 1870-1940 is already a classic. It's rather short on summarisation and high-level views, and when it does take a view, it comes across as OR (he has some eccentric and amusing views on Frege, and is very funny about Russell) but very long on detailed sourcing down to the very lowest level. It's absolutely essential as a reference work. Buy it now. Logicist (talk) 19:38, 24 August 2009 (UTC)

PS don't be put off by the '1870' part of the title. He actually starts with De Morgan and Boole and works forward from there. Logicist (talk) 19:40, 24 August 2009 (UTC)

Codomain vs Range

I notice that codomain has been changed to range in the first paragraph of the leader. Any consensus which is better in that place or doesn't it really matter? Dmcq (talk) 07:57, 24 August 2009 (UTC)

I think "range" is more common at the elementary level, but in any case, note that the output of the function is in the range, whichever definition of range you use. If D is the domain, R the image, and C the codomain, then f is a subset of DxR (which is in turn a subset of DxC). Rick Norwood (talk) 14:54, 24 August 2009 (UTC)

Although I didn't make these edits, 'range' was changed back to 'codomain'. The problem with set-theoretic diagrams of functions is that they rarely show elements of the domain or elements of the codomain that are undefined by the function. A few of the objections on this Talk page really concern the difference between a set and a minimal set. A codomain is a set (and therefore the universal set is a codomain for any function), while an image is a set contained in the codomain.

As I have mentioned, the lead section is still unnecessarily complicated for nonmathematicians to understand. Therefore, I recommend that the lead section 'wave its hands' just a little for the sake of understandability and define a function as briefly as possible ('a function is a mathematical rule that, for each element in a given set, specifies one and only one element in the same or another set'), along with one or two pictures (the 'transformation machine' or the 'Venn mapping').

Note that this approach doesn't need to mention range, domain, codomain, image, graph, table, dependent variable, independent variable, formula, tuple, or any other unnecessary nomenclature. I think this is the right way to go because it makes the first section brief, yet prepares the reader for deeper understanding in the following sections. David spector (talk) 20:42, 5 December 2009 (UTC)

The concept of the universal set has been shown to cause paradoxes. I see only trouble and confusion coming out of using the word range in the leader. It has two distinct and common uses in maths when related to functions never mind that people have a third which which is more akin to the use in statistics. You seem to think they are looking up function while already having an idea of what range and domain means but not knowing what codomain means and the few cases where that would be true hardly seem worth the confusion to me. Never mind a lot of people reading maths articles seem to get in a right tizzy like Rainman when confronted by a bit of hand waving. Dmcq (talk) 00:17, 6 December 2009 (UTC)

The Principles of Mathematics v.s. Principia Mathematica

This is a minor quibble, but perhaps this article should settle for one of those titles since since using both is bound to confuse some readers. Pcap ping 09:44, 5 September 2009 (UTC)

Apparently, I'm the one confused. Is Bertrand Russell's The Principles of Mathematics: Vol. 1 (1903) not considered part of Principia Mathematica? Was it substantially different than Principia's vol. 1 of 1910? Neither our biography of Russell nor our article on Principia shed much light on this issue... Pcap ping 09:52, 5 September 2009 (UTC)

Principles (1903) predates Principia (1910). — Carl (CBM · talk) 11:45, 5 September 2009 (UTC)

There's also his 1908 in between the two where he introduces his "theory of types"; see van Heijenoort 1967:150ff. During this time he was struggling to figure out how to resolve the paradoxes, in particular his paradox that he found in Frege ( Russell's paradox ), and a long treatment appears in his 1903 Principles. Prinicipia Mathematica would out in two parts 1910 and then 1913 (my cc says First Edition copyrighted 1913) with the 1908 virtually intact (included here and there). A second edition appeared in 1927. By this time Russell had digested Hilbert and Bernays and Weyl and Brouwer, he'd read Sheffer's work (1921) and Wittegenstein's Tractatus Logicus-Philosophicus (1922); in a long introduction in the 1927 2nd edition (46 pages worth) he shows how to incorporate the Sheffer stroke into his 1910-1913. He then substituted a new section *8 for the old (Appendix A) and added a response to Wittgenstein's criticisms and suggestions in an Appendix C; see the references at the end of the introduction to the 1927 edition. (Russell was busily writing philsophy books too, The Problems of Philosphy (1912), Introduction to Mathematical Philosphy (1919-1920), etc.). For a broad brush over the major events see Eves (1990); unfortunately it's spread throughout the latter half of the book (e.g. pages 212 on: 8.3 Sets and the Foundations of Mathematics, p. 234-235, 9.4 Crises in the Foundations of Mathematics, 9.5 Philosophies of Mathematics). Bill Wvbailey (talk) 15:41, 5 September 2009 (UTC)

Source for history section

Was anybody able to open the PDF linked from http://math.coe.uga.edu/TME/Issues/v03n2/v3n2.PonteAbs.html, which is our main historical source here? It appears corrupt, and does not open. It certainly lacks the PDF signature at the start of the file. The URL http://math.coe.uga.edu/TME/Issues/v03n2/v3n2%20pagemaker%20files/ponte seems to indicate it is PageMaker source file not PDF, but I don't have that program. Pcap ping 10:03, 5 September 2009 (UTC)

Google. The PDF is available here and a bibliographic entry is available here. However, I am sure that we could find additional sources for the history of functions. — Carl (CBM · talk) 11:36, 5 September 2009 (UTC)

Thanks for the find. Given that Ponti's paper doesn't cover most of the detailed material in this wiki article, which is taken from the reprints of the original sources, perhaps the wiki article should not mislead the reader that Ponti's paper covers it? I'm not complaining that this is WP:OR since it's fairly straightforward read of the originals; this is math after all, but I have pretty good idea who wrote the history section without even looking at the edit history. :-) Pcap ping 12:59, 5 September 2009 (UTC)

Map vs. Function vs.

Shouldn't the distinction be addressed? I am not sure that a sole definition exists but they all seem to agree that a funciton is a special case of map. For instance, "A map whose codomain is the set of real numbers R or the set ofcomplex numbers C is commonly called a function." found in "Mathematical Physics: A Modern Introduction To Its Foundations" by Sadri Hassani, page 5. --Javalenok (talk) 09:01, 12 October 2009 (UTC)

Function and map are synonyms, except to authors who use them to mean different types of functions. I added this to the article:

A function can also be called a map or a mapping. Some authors, however, use the terms "function" and "map" to refer to different types of functions. Other specific types of functions include functionals and operators.

— Carl (CBM · talk) 10:22, 12 October 2009 (UTC)

Possible flaw

Section "Overview" paragraph 6:

A family of objects indexed by a set is equivalent to a function: for instance, an infinite sequence of real numbers is a family of real numbers indexed by natural numbers; i.e. a function from natural numbers to real numbers.

We all know that a 1-to-1 correspondence of natural numbers with real numbers is impossible. I don't think there exists such sequence. Zhieaanm (talk) 08:45, 18 October 2009 (UTC)

What is said is true but perhaps it could be phrased differently. An example of such a sequence would be: 1, 0.5, 0.333, 1/n and the function here would simply be 1/n where n is a natural number. Also people may not know that the index set need not be countable Dmcq (talk) 12:58, 18 October 2009 (UTC)

The word "to" is not good. "Into" would be better. Rick Norwood (talk) 13:25, 18 October 2009 (UTC)

Simplify lede

I've done some work on the lede. I think, over time, it has become confusing to the lay reader, and needed a few simple examples. Rick Norwood (talk) 13:52, 6 December 2009 (UTC)

caption under the graph

The caption on the first image in the article seemed to me too complicated for an elementary introduction.

Graph of example function,
${\begin{aligned}&\scriptstyle f\colon [-1,1.5]\to [-1,1.5]\\&\textstyle x\mapsto {\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}$

I replaced it with the text you see below the second image, but this was reverted.

Graph of example function,
${\begin{aligned}&\scriptstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}$

I would appreciate other opinions on the subject.

Rick Norwood (talk) 21:44, 6 December 2009 (UTC)

As the one who reverted it, my comment is that it's important for the domain and range and/or codomain of the function to be noted. I have little objection to the "=" vs. $\mapsto$ distinction (the "=" version is technically wrong, but understandable), but the domain and codomain should not be removed from the caption. — Arthur Rubin (talk) 21:55, 6 December 2009 (UTC)

How about this slightly less technical language, then. "In the part pictured, the domain and codomain are both the set of all real numbers between -1 and 1.5." I prefer this based on my experience that even calculus students often do not know the notation for a closed interval. My guess would be that anyone who knows what a closed interval is already knows what a function is. Rick Norwood (talk) 22:24, 6 December 2009 (UTC)

Rick Norwood (talk) 22:24, 6 December 2009 (UTC)

OK. It will look a little clunky, but that seems acceptable. — Arthur Rubin (talk) 22:31, 6 December 2009 (UTC)

Quasipositive function

This article has been proposed for deletion. If you have knowledge of the subject kindly take a look to see if it is worth salvaging. -Arb. (talk) 23:34, 12 February 2010 (UTC)

irrelevant material in history section

The material inserted at the beginning of the history section:

The underlying idea of a function dates back to the Persian mathematician, Sharaf al-Dīn al-Tūsī, in the 12th century. In his analysis of the equation x3 + d = bx2 for example, he begins by changing the equation's form to x2(b − x) = d. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value d. To determine this, he finds a maximum value for the function. Sharaf al-Din then states that if this value is less than d, there are no positive solutions; if it is equal to d, then there is one solution; and if it is greater than d, then there are two solutions, etc

does not seem to have much to do with the function concept. The idea that the work on a simple cubic equation as late as the 12th century, valuable though it may have been, has anything to do with the origin of the function concept is far-fetched. Tkuvho (talk) 16:00, 24 February 2010 (UTC)

Moreover, this material is preceded by a mention of da Ponte. Such a mention is misleading. Indeed, da Ponte did not say a word about Sharaf. Rather, da Ponte writes:

Historically, some mathematicians can be regarded as having foreseen and come close to a

modern formulation of the concept of function. Among them is Oresme (1323-1382),

Thus the current version of the historical section is inaccurate and misleading. Tkuvho (talk) 16:04, 24 February 2010 (UTC)

---

It's not good that the Saraf business has no reference. And I can't locate my cc of Eves to check it for a reference. Can you expand the Oresme quote a bit? It sort of leaves me "unsatisfied"; an example from Oresme would be nice, similar to the Sharaf material. Bill Wvbailey (talk) 16:32, 24 February 2010 (UTC)

It is referenced, actually, to the Katz-Barton article. However, the material is presented in a way that suggests that da Ponte mentions it. I did not look at Katz-Barton yet. Their article talks about algebra, and I expect don't say anything about functions other than the accidental occurrence of the word in describing the cubic term on the right side. That in itself is very questionable, of course, because the original is not in a european language. Tkuvho (talk) 16:38, 24 February 2010 (UTC)

I have now looked up the article by Katz and Barton. They write:

One Islamic mathematician, who was interested in solving cubic equations, began to think in new ways. This was Sharaf al-Din al-Tusi (d. 1213), a mathematician born in Tus, Persia. Let us consider his analysis of the equation x3 +d = bx2. He began by putting the equation into the form x2(b−x) = d. He then noted that the question of whether the equation has a solution depends on whether the “function” on the left side reaches the value d or not. To determine this, he needed to find a maximum value for the function.

Note that the term "function" used by Katz and Barton, is not attributed to Sharaf. It is their own shorthand for the cubic term. It is clear from the full quotation above that they did not mean at all to attribute the function concept to Sharaf. Whoever added Sharaf functions to this article is interested not in history but in promoting an ideological agenda. Tkuvho (talk) 16:47, 24 February 2010 (UTC)

I'd say based on this that you should go ahead and fix the paragraph: purge Sharaf and add Oresme and anything else relevant that you've got a source for; this section leading up to 1850 needs expansion; even the title should be changed to something like "The notion of function prior to 1650", "The notion of function in the Middle Ages", etc. Bill Wvbailey (talk) 17:10, 24 February 2010 (UTC)

Ponte (not de Ponte) does say something more specific about Oresme's "latitudes", but I don't really understand it. Give it a try. Tkuvho (talk) 17:22, 24 February 2010 (UTC)

Cauchy bis

Freudenthal says that the modern concept of continuity was invented by Cauchy. Cauchy was the first (perhaps second, after Bolzano) to realize the significance of general functions, going beyond algebraic expressions and the like. He was the first to realize the need to prove the intermediate value theorem, which he proved (as did Bolzano). I would say both Bolzano and Cauchy should be mentioned here. Tkuvho (talk) 17:25, 24 February 2010 (UTC)

Sounds like a good add to me because it shows the development of the notion of "function". You have the source. Go for it. BillWvbailey (talk) 23:09, 24 February 2010 (UTC)

Revised lead

Over time, the opening paragraph has reverted back to a Bourbaki-like definition, with no hint of the meaning of functions or the scope of their applications. I have restored an older, gentler description of functions in the opening sentence and took an opportunity to clean up the clutter that has accumulated in various places in the lead. By the way, the backwards use of the word "associates" (in the set-theoretic paragraph) bothers me a bit, I prefer "assigns", as in "The function ƒ assigns the value ƒ(x) to each argument x in its domain". Arcfrk (talk) 01:06, 8 April 2010 (UTC)

Codomain and mathematics

The statement "most authors in advanced mathematics outside of set theory prefer the greater power of expression afforded by defining a function as an ordered triple of sets" reflects a personal viewpoint, is misleading in various respects and hence should be replaced by a quite different text. Here are the main flaws.

A. The literature provides no indication that a majority - or even a significant minority - of authors in advanced mathematics outside of set theory prefer the stated definition.

B. It is incorrect that defining a function as an ordered triple of sets provides greater power of expression; below we shall see that quite the opposite is true.

C. The triples mentioned have a useful role in embedding functions in category theory, but such triples represent function arrows and must not be confused with functions in the ordinary mathematical sense.

These points are elaborated below. The conclusion is that selecting ill-advised definitions amounts to painting oneself into a corner.

A. There is no indication that a majority of authors in advanced mathematics outside set theory prefer defining functions as triples of sets; not even a single reference is given. (Aside: why this artificial exclusion of set theory from the purported statistics?).

That some people may have the impression that many definitions of a function include a codomain is is explained by the quotations in earlier discussions on Wikipedia.

The relevant definitions all start with "A function f from a set X to a set Y ...". Yet, it is wrong to conclude from such definitions that Y is part of the definition of f. Indeed, if read correctly, such definitions do not define a function by itself, but a function from X to Y, i.e., they say when f is a member of the set of functions from X to Y. More specifically, judging by the majority of books selected at random: f is a function from X to Y if f assigns to every element x in X a unique element in Y, written f(x). Hence, for every Y' that includes Y, every function from X to Y is by definition also a function from X to Y' (indeed: every f(x) in Y is also in Y').

One can only conclude that a codomain is part of the definition if that is stated explicitly, e.g as codom(f) = Y, or if it becomes evident later on from a definition of function equality, or still later from a definition of function composition. Aside: it is unfortunate that relatively few texts explicitly define function equality - perhaps the most important relation between functions.

In Tom Apostol, Calculus (page 54), functions f and g are defined to be equal iff they have the same domain and f(x) = g(x) for every value in the domain.

A few remarks are in order here (with f, X and Y as introduced earlier).

i) Two references where the above is made clear are Naive Set Theory by Paul Halmos (page 31) and Principles of Mathematical Analysis by Walter Rudin (page 21).

ii) For those who are bashing calculus texts as being "wrong" in saying "range" instead of "image": the referenced pages in the Halmos text (not a calculus text!) and the Rudin text make a clear and useful distinction: if A is a subset of X (the domain) then the image of A under f, written f(A), is the set of y in Y such that f(x) = y. Note the phrasing: "image of A", not: "image of f". The range is then defined as the image of the domain X, i.e., range(f) = f(dom f).

iii) With these conventions, a function cannot be said to be "onto" or "surjective" by itself, but only with respect to a set. Quoting Halmos (or Rudin) again: "if the range of f is equal to Y, we say that f maps X onto Y". Note that Y is mentioned explicitly. This is as it should be for expressiveness.

iv) The domain of a function is just the set of values x for which f(x) is defined. For those who define f as a set of pairs, the domain appears obvious, but if f(x) is defined by an expression, the domain must be specified explicitly. For instance, if f(x) = x/(x-1), in a distant past one would have said that the domain is the set of real numbers except 1, but that is too rigid: the domain is whatever one decides to specify (of course, excluding values for which the expression is undefined).

In brief, definition in mathematics is not "right" or "wrong" (unless it is inconsistent), but is to be judged by its usefulness in conceptualization and reasoning (including symbolic calculation). This brings us to the next point.

B. Including a codomain in the definition of a function seriously reduces the power of expression, since the codomain gets "in the way" of combining functions in a reasonably flexible way.

Let us take function composition as just one example. With codomains as part of the definition, composing functions f and g to $f\circ g$ requires that the codomain of g equals the domain of f. Yet, the defining expression $(f\circ g)(x)=f(g(x))$ is also meaningful if the range of g is included in the domain of f, which is a less stringent condition. In that case, the domain of $f\circ g$ still equals the domain of g. However, one can go still further: since f(g(x)) is already meaningful if g(x) is in the domain of f, one can define the domain of $f\circ g$ as the set of those elements x in the domain of g such that g(x) is in the domain of f. Formally, $dom(f\circ g)=\{x:dom\,g\,|\,g(x)\in dom\,f\}$ with, for all x in this domain, $(f\circ g)(x)=f(g(x))$.

Many authors have found this additional power of expression useful, ranging from calculus (see Thomas's Calculus 11th edition, page 40) to programming language semantics (see Bertrand Meyer, Introduction to the Theory of Programming Languages, page 32), all using the last definition given.

Function composition is just one example, but there are many other useful ways of combining functions that all would become considerably less general if they had to be defined for functions having a codomain as part of their definition.

C. Still, triples play a useful role in making functions (in the mathematical sense) correspond to arrows in an appropriate category. Here we refer to Benjamin Pierce Basic Category Theory for Computer Scientists as well as Richard Bird and Oege de Moor Algebra of Programming, slightly merging their formulations in an optimal way for making the explanation uniform. Functions in the mathematical sense (as defined above) give rise to function arrows in appropriate categories, of which we give two examples.

1) In the category of total function arrows, to every fuction f there corresponds a collection of arrows whose source A is the domain of f and whose target B contains the range of f. Pierce (p. 2) denotes this arrow by a pair (f, B), Bird and De Moor (p. 26) as a triple (f, A, B) - slightly redundant since A = dom(f).

2) In the category of partial function arrows, to every fuction f there corresponds a collection of arrows whose source A contains the domain of f and whose target B contains the range of f. Here a triple (f, A, B) is indicated.

Again a few remarks.

(i) Pierce explains the reason for using pairs/triples by an example: the function that takes every real number $r$ to $r^{2}$ can be seen as a function from reals to reals, but also as a function from reals to nonnegative reals (and, of course, similarly for every set that includes the nonnegative reals). The corresponding arrows are considered different, and therefore distinguished by B. Pierce also explicitly recognizes that "functions" in category theory (shorthand for "function arrows") do not have their ordinary mathematical meaning (page 3).

The express inclusion of B in (f, B) by Pierce indicates that B cannot be inferred from the definition of f as a function in the ordinary mathematical sense. If f were defined with a codomain, f would directly correspond to an arrow.

(ii) At a later stage (page 88-89), Bird and De Moor define "simple arrows" and "entire arrows" as special kinds of relational arrows, and introduce the terms "partial function" for a simple arrow and "function" for an arrows that is both simple and entire. It is clear from the context that these are purely category-theoretical terms, not to be confused with the concept of "function" in its normal mathematical meaning.

The rich material in these two references shows that category theory can provide an elegant formalism for capturing many different concepts in a uniform way. However, it can become stifling if taken too rigidly as a pattern for redefining mathematics. Boute (talk) 15:49, 11 June 2010 (UTC)

I'm not sure exactly what you're saying but as to functions having a codomain it seems pretty clear from earlier discussions here and at codomain that what is said here is a reasonable statement about the current state. The statements should have citations though as there have been a couple of disputes. It looks like the main changeover in university textbooks was after about 1970, there's quite a few around that were written before that and have even been reissued with a new date which is confusing. There's a large history at the end of the article, I'll have a look at that - I've sort of stayed out of there before now. Dmcq (talk) 16:36, 11 June 2010 (UTC)

Boute, your points are well taken. We've had many discussions that pitted "formalists" against "intuitivists", and some of the participants had strong opinions on the subject. The passage you have quoted resulted from attempts to reach a compromise, not just within this section, but in the context of the whole article (think of it as editorial horsetrading of sorts). If you have a good way of addressing your points A,B,C that will improve the article without descending to chaos then go for it! Beware of the law of unintended consequences, though. Arcfrk (talk) 01:29, 12 June 2010 (UTC)

Thank you for the feedback. Perhaps some personal experiences in this matter may clarify my reasons for bringing up the codomain topic. I first learned about functions in the "modern" sense (as opposed to the classical real-valued functions in calculus) at the end of the 1960s from C.L. Liu, Introduction to Combinatorial Mathematics (1968), page 130 and Michael Arbib Theories of Abstract Automata (1969), page 24. In both, a function is defined with a codomain (in the cited references called the "range") as part of the definition. Quote (from Arbib): We say two functions, $f:X\rightarrow Y$ and $f':X'\rightarrow Y'$ are equal iff $\,X=X'\;$ , $\;Y=Y'$ and $(\forall x\in X)f(x)=f'(x)$. Of course, shortly thereafter I also encountered definitions without codomain as an essential characteristic (and hence, for equality, $\,f=f'$ iff $dom\,f=dom\,f'$ and $\forall \,x:dom\,f\cap dom\,f'\,.\,f(x)=f'(x)$). At that time, some professors remarked that the variant with codomain was predominant among mathematicians with a European background (Bourbaki?), whereas the codomain-less variant was predominant in the U.S.. Verifying this might be an interesting topic for math historians. Later on, when experimenting with both variants in various areas of applied mathematics (including computer science), I found the codomain a burden rather than an asset; the function composition example above is an illustration.

Perhaps the article should list the advantages and disadvantages of either variant rather than appeal to trends (especially since there seem to be no clear statistics).

I will not attempt to change any article now; some serious thinking about formulation and balance is needed first. Perhaps several people should "compare notes" on both variants.Boute (talk) 09:04, 12 June 2010 (UTC)

I think a citation for that formal definition of equality with domain but not codomain could be useful in that discussion in the article. Do you remember where you saw it? Dmcq (talk) 10:37, 12 June 2010 (UTC)

Arcfrk's statement above mirrors what I would say: there is a lot of discussion about this already in the archives of this talk page, and the language in the article reflects a compromise between various positions there. It is probably not worthwhile to revisit that compromise. The underlying moral is that that language mathematicians use to describe functions is only precise in the ways that we need it to be precise, and is far from precise in many other ways that only seem important when writing articles like this one. — Carl (CBM · talk) 12:10, 12 June 2010 (UTC)

The history section should have a bit about codomain and the way Bourbaki pushed the concept. There is a brief mention in relation to the terms surjection and injection but nothing but the history section ends before then except for brief mention of categories and functors at the end which came after. Was there a connection between the introduction of codomains and the development of topology and category theory for instance or did topology lead to codomains? I for one don't know. Dmcq (talk) 12:27, 12 June 2010 (UTC)

In my researches for the history section I had to stop around 1950 because I couldn't find any good secondary sources RE terminology. This I do know: as an undergrad I took calculus in 1966-67 using Thomas's book, where there is no talk of codomain, only range and domain (cf page 19 in the 1960 edition 3rd printing 1966), and I could never keep them straight. Then in 1984, because I kept reading about some newfangled notion called "set theory" (it wasn't taught to undergrads except the most rarified of math majors), I bought Halmos's 1970 Naive Set Theory. Again there was no talk of codomain, only range and domain. In the intervening years (1970 - 2007) the terminology changed so when I came to wikipedia I was confronted with a whole bunch of new words, like "codomain" and those Latinate confusors like "surjection" and "injection" and "bijection", and notions of "into" and "onto". So something has been happening after about 1970 to change the terminology (See Archive/3 for my canvassing of 18 or so texts to figure out the terminology in Archive/3). If anyone knows a secondary reference/study re the shifting terminology please let me know. BillWvbailey (talk) 16:39, 12 June 2010 (UTC)

Naive Set Theory was originally published in 1960 so sometime after that. Dmcq (talk) 21:39, 12 June 2010 (UTC)

You are correct; I double-checked. 1960. That sort of goes along with Thomas's usage. Here it is, in full:

"Independent and dependent variables. The variable x which yields the first of the two numbers in the ordered pair (x, y) is often called the independent variable, or argument of the function f. The second variable y is called the dependent variable. The set of all values taken on by the independent variable is called the domain of the function. The set of all values taken on by the dependent variable is called the range of the function. If the pair (a, b) belongs to the function f, that is, f(a) = b, we also say that b is the image of a under f. Similarly, the range of a function is the image of its domain.. We also say that the function maps its domain onto its range." (page 19)

Later we also see:

"A function is determined by the domain and by any rule that tells what image in the range is to be associatiate with each element of the domain. For, once the domain and the rule are given, the set of all ordered pairs (x, y) can, at least in theory, be computed by a machine into which the elements of the domain are fed, one after another. The machine [etc, etc]" (page 21-22).

We see here, in the discussion of "the machine" (page 22), a drawing of "function boxes", plus a long discussion of "lookup tables" and "restrictions" about divide by 0. My underlining and (repeated underlining) of the first quote, and then the first part of the second quote (up to but not including the machine-business that I blew off . . . too bad) indicates that I was desperately trying to learn these terms and their import. Now I realize that even Thomas's presentation contains far more nuggets of important stuff than was apparent (e.g. "the function maps its domain onto its range"). Bill Wvbailey (talk) 22:02, 12 June 2010 (UTC)

It appears that the word codomain is shortedn from converse-domain or counter-domain. Some of the following is from my survey to be found in Archive/3. First we see Suppes:

Suppes 1960, Axiomatic Set Theory: "If R is a relation then the domain or R (in symbols: DR) is the set of all things x such that, for some y, <x,y> ε R. ... The range of R ... is the set of all things y such that, for some x, <x,y> ε R. ... The range of a relation is also called the counterdomain or converse domain ... The field of a relation ... is the union of its domain and range." (p. 59).

We go back in time to Tarski (Dover 1995, unabridged 9th edition (1961) of 1946 2nd edition, first edition 1941),Introduction to Logic. He uses "counter-domain (converse-domain) of a relation, 87 f" (cf INDEX on page 232). He does not use "range" (!) Moreover, if I understand him, he defines functions backwards from typical analysis and set-theoretic conventions , as in "x R y" being equivalent to "x = R(y)" (cf pp. 99-100). updated wvbaileyWvbailey 19:37, 10 September 2007 (UTC), with emendations here and there on 06-13-2010.

Indeed, Tarski's reversal of modern usage comes from Principia Mathematica, in particular Russel's "descriptive function". So I go back to my pdf of Russell's 1903 Principles of Mathematics and voila -- Russell defines "converse-domain" page 97:

"These two classes I shall call respectively the domain and converse domain of the relation; the logical sum of the two I shall call the field of the relation" (p. 97)

At least now we can see that usage was available from the published literature as far back as 1903. I dunno if it goes back even further; Russell was lousy at foot-notes and attribution. Perhaps the mystery should be why "range" came into usage as opposed to "codomain". Bill Wvbailey (talk) 15:56, 13 June 2010 (UTC)

Somewhat earlier Dmcq correctly observed that "a citation for that formal definition of equality with domain but not codomain could be useful in that discussion". The citation I gave is from Tom Apostol Calculus, page 54, but this is rather old. Looking in recent literature was very instructive. The problem is that definitions of function equality in the literature seem rare somehow (not found in about a dozen of recent math and CS books where such a definition could be expected). Googling on "function equality" or "equal functions" was equally ineffective, but "functions f and g are equal" yields many results (mostly lecture notes). Some definitions of function equality include codomain equality, some do not. Here are a few that do not include codomain equality, as requested:

http://solitaryroad.com/c303.html
http://www.stanford.edu/class/cs103/handouts/25%20Functions.pdf
http://www.bus.ucf.edu/sgerking/ECO%203401/Chapter%202.ppt
http://www-math.cudenver.edu/~wcherowi/courses/m3000/lecture8a.pdf
http://www.cse.unl.edu/~cse235/files/Functions.ppt

An interesting subtlety: many of the latter group define a codomain, but do not include it in the definition of function equality. As Leibniz would say, this means that they do not really consider the codomain as part of the definition of a function. More subtle: some talk about "the domain of f" but about "the codomain" (without "of f").

A pragmatic observation: authors including a codomain (say Y) in the definition of a function seem to do very little with it, except (i) defining when a function is "onto" instead of "onto Y", (ii) letting it become a hindrance for the ways in which functions can be combined, for instance, by composition. In other words, the issue is not just important for survey articles such as this one, but has practical consequences. Therefore I think it is more important to weigh technical advantages and disadvantages than to make frequency statistics. Boute (talk) 11:41, 13 June 2010 (UTC)

Functions in analysis

I'm surprised that we don't appear to have any proper treatment on Wikipedia of the notions of a function in analysis. For instance, in complex analysis "function" often refers to a holomorphic function. In real analysis, elements of the L^p spaces are called "functions", but in actuality are only equivalence classes of functions. The term "function" applies respectively to a specialization and a generalization of the usual mathematical concept of function in these two areas of analysis. Sławomir Biały (talk) 11:35, 4 September 2010 (UTC)

3.2: Functions with multiple outputs is ambiguous as to the notation of mapping two inputs to a single output

The general example of the two part notation, in 3.1("Notation"), doesn't intuitively extend to functions with multiple outputs. I think an example of the notation in 3.1, of a function with multiple inputs and outputs would make that section(3.2) of the article more accessible. —Preceding unsigned comment added by 96.235.67.230 (talk) 07:31, 26 January 2011 (UTC)

In mathematics, the definition of a function is that it has a single output for a given input. If in a particular context multiple outputs are desired, then the codomain of the function is usually defined as the power set of some universal set, so that the output is a single set rather than a single number. If something has multiple outputs (such as "less than") it is called a relation rather than a function. On the other hand, a function may have different inputs with the same output. Such functions are not one-to-one, but are still functions. Thus f(x) = x^2 is a function, f(x) = plus or minus the square root of x is not a function. Rick Norwood (talk) 13:07, 26 January 2011 (UTC)

I though the section "Functions with multiple inputs and outputs" was pretty clear. Multiple inputs and outputs are simply treated as functions with a single input and a single output which are a cartesian product of the various inputs or outputs. There is a simplification for multiple inputs of removing a set of brackets but In what way is that ambiguous? There is a simplification for multiple inputs of removing a set of brackets for a tuple representing a cartesian product but they are both tuples. Dmcq (talk) 15:26, 26 January 2011 (UTC)

You are correct, the section was clear on the use of f(x,y)=[mapping here] style definitions. I'm interested in how the f:D->C, where f is a function with D domain and C codomain, style definition, and in particular the second half, the half that maps the relation, would be used to define a multi-variable function. It was unclear, to me, how the example of the two part notation exemplified in 3.1 would extend to multi-variable functions.

Well using the example of swap there which has domain R×R and codomain R×R you'd get using the notation in the section above

${\begin{aligned}swap\colon \mathbb {R} \times \mathbb {R} &\to \mathbb {R} \times \mathbb {R} \\\left(x,y\right)&\mapsto \left(y,x\right)\end{aligned}}$

though of course with other mappings one might have to use something more complicate instead of that second line. I suppose one could if one were being unhelpful even use one symbol for the pair as in swap(p) where p is (x,y), in fact I think most people would forget about the brackets round p unless it was an expression, and write the mapping as for instance p → (tail(p), head(p)). Of course many mappings can't be written straightforwardly in a line like that anyhow. Dmcq (talk) 12:32, 27 January 2011 (UTC)

As a more interesting example you could consider the integer divide function, where the range is the (quotient, remainder) pair. Formally this gives

${\begin{aligned}intdiv\colon \mathbb {Z} \times \mathbb {N} &\to \mathbb {Z} \times \mathbb {N} \\\forall (x,d,q,r),d>0\implies {\mathit {intdiv}}\left(x,d\right)=\left(q,r\right)\\{\text{where}}~x=q\times d+r\land 0\leq r<d\end{aligned}}$

with apologies for the bad layout which I haven't yet learnt to handle. SamuelTheGhost (talk) 16:32, 27 January 2011 (UTC)

That clears up any question I had. Is that definition worth putting in the article, or should we leave it at this? —Preceding unsigned comment added by 108.10.194.223 (talk) 23:09, 27 January 2011 (UTC)

I don't think there's anything new there but if somebody thinks it worth sticking in some more explicit example of what it means, well I haven't any strong feeling about that which means I'm not doing anything but I wouldn't object if somebody else did. Dmcq (talk) 00:14, 28 January 2011 (UTC)

Current state of the lead paragraph:

The lead paragraph ought be briefer and more concise; together with the rest of the lead, it needs to beg the reader to hang around to learn more about the notion of a 'function'.

The lead paragraph could be a simple (incomplete) definition. Some texts I have reviewed have a very simple statement that defines a function, such as, "a function is a mathematical rule between two sets which assigns each member of the first, exactly one member of the second" [McGraw-Hill Dictionary of Scientific and Technical Terms: Fifth Edition, 1993, McGraw-Hill, page 816], or "a function is a rule that assigns to each element from one set exactly one element from another set" [Lial, Greenwell, and Ritchey (2002) Calculus With Applications: Brief Version: Seventh Edition, Pearson Education, page 50], or "a function is an association between two or more variables, in which to every value of each of the independent variables, or arguments, corresponds exactly one value of the dependent variable in a specified set" [Gullberg (1996) Mathematics From the Birth of Numbers, W.W. Norton & Company, NY, page 336]

The second and third paragraphs can carry the details that make the definition come alive and inviting to the reader, while serving to make the definition more formally correct.

The notion of 'function' is so fundamental to mathematics that it screams out for perfection in its explanation. That perfection, however, needn't be completely captured in the first paragraph.

The typical person that I imagine so enjoys Wikipedia that she might type "function" into the search bar is looking first for a simple, comprehensible, first-order definition with which she might gain a little-toehold in her quest for true enlightenment, and should not be scared away by a profusion of confusing adjectival modifiers that overwhelms the notion being explained. Also, a reader needs to be encouraged (invited by dint of the way the notion is expressed in language) to read the entire article.

That wasn't my experience when I typed in 'function', and I have a reasonable, yet clearly dated, background in mathematics. When I began reading that first paragraph something smacked me in the face. I had always thought of a function as having a domain and range; I found the term "codomain" stunning! After urgently reading the commentary on this discussion page and realizing what was meant by the term 'codomain', indeed, it seems to have a place in the definition of a function, but I shouldn't have had to go through all of that "what the heck is going on; am I crazy?" reading adventure just because the term 'range' is so familiar to me.

In my opinion, that first paragraph simply tries to do to much work.

Without a mention of the term 'range' in the text of the lead, notice that just to the right is an illustration with the caption:

Graph of example function, \begin{align}&\scriptstyle \\ &\textstyle f(x) = \frac{(4x^3-6x^2+1)\sqrt{x+1}}{3-x}\end{align} Both the domain and the range in the picture are the set of real numbers between −1 and 1.5.

So, my beloved 'range' was there all along!

In his book, "The Road To Reality: A Complete Guide to the Laws of the Universe," [Vintage Books, Copyright by Roger Penrose, 2004] Roger Penrose introduced the notion of 'function' in the following way; I found it suitably engaging, and quoted him:

"To Euler, and the other mathematicians of the 17th and 18th centuries, a 'function' would have meant something that one could write down explicitly, like x^2 or sin x or log(3-x+e^x), or perhaps something defined by some formula involving an integration or maybe by an explicitly given power series."

"Nowadays, one prefers to think in terms of 'mappings', whereby some array A of numbers (or of more general entities) called the domain of the function is 'mapped' to some other array B, called the target of the function. The essential point of this is that the function would assign a member of the target B to each member of the domain A. (Think of the function as 'examining' a number that belongs to A and then, depending solely upon which number it finds, it would produce a definite number belonging to B) This kind of function can be just a 'look-up table'. There would be no requirement that there be a reasonable-looking 'formula' which expresses the action of the function in a manifestly explicit way."

While we cannot steal Penrose's work, we must have, amongst the mathematicians who contributed to this discussion page, at least one who, like Penrose, can write better than the way the lead paragraphs of this topic are currently written.Langing (talk) 00:59, 18 May 2011 (UTC)

I've rewritten the lede with the above ideas in mind. Comments? Rick Norwood (talk) 13:02, 19 May 2011 (UTC)

Well I'd have much preferred you'd not stuck in all that stuff about range and accepted that it is an ambiguous and confused word. The article itself points this out, I don't know why you had to then just one meaning in the lead. There's other sources mentioned here besides a dictionary and a popular science work. Dmcq (talk) 13:43, 19 May 2011 (UTC)

I have no problems with what Langing said. However I hope you can see that 'range' was assumed by Langing to mean something because of previous experience so there is no real reason for preferring it to codomain if you are talking about someone new looking up this article, and there is a lot against sticking it in so often in the lead as a main thing when it has no agreed definition. Dmcq (talk) 13:53, 19 May 2011 (UTC)

The word "range", like the word "ring", has different definitions in different, equally authoritative, books. We need to mention both definitions of "range" or neither. My reason for mentioning the definition that defines range as image is that it avoids the "ordered triple" definition. If a function is defined by a formula, there is no way to know what the codomain is unless it is stated explicitly. Rick Norwood (talk) 16:02, 19 May 2011 (UTC)

Well image is the word if you really mean just the values that actually occur or codomain if you are talking about functions in general and the possible target values for instance real valued functions. The ordered triple is the usual definition nowadays as it allows people to deal with the concept of for instance real valued functions properly as it allows dealing with things like composition easily, the other way of going about it is only useful for particular special functions or in things like set theory when you want everything pared down to the minimum rather than being useful in itself. Dmcq (talk) 16:12, 19 May 2011 (UTC)

The point is that all mathematicians already know what a function is, and that anyone who turns to this article for information is almost certainly trying to understand the meaning of function when it is understood that the function is a real valued function of a real variable. Therefore, function as ordered triple belongs further down in the article. A Wikipedia article must not say anything that is wrong, but should not try to say everything that is right. The rule is: address the lede to the layperson. Rick Norwood (talk) 17:59, 19 May 2011 (UTC)

I think the current third paragraph needs to go. The lead should summarize the article and provide an accessible introduction. The lead is not the place to discuss fine terminological distinctions such as the one between range, image, and codomain. Nor does it seem helpful to discuss how these are often "understood" in context at this point. These are relatively minor considerations that should be addressed elsewhere. I do agree that referring to the range rather than the codomain improves the overall accessibility, though. Sławomir Biały (talk) 19:23, 19 May 2011 (UTC)

I propose the following as a replacement for the first several paragraphs:

A function, in mathematics, takes as argument a set of quantities, and assigns to each and every quantity one value. The set of all quantities input to a function is called its domain; the set of all quantities output by a function is called its range.

A particular function's argument, also called input, and its value, also called output, could both be the set of real numbers. But a great many functions exist in mathematics, so a function's argument and value can be elements from any possible sets of mathematical entities.

A simple example of a function is f(x) = 2x, where x is any real number. This function associates every real number with a real number twice as large. So, for example, 5 is associated with 10, written f(5) = 10. Notice that for this function the domain is the set of real numbers, and the range is also a set of real numbers; the two sets are not identical. Langing (talk) 19:41, 19 May 2011 (UTC)

Agree with Bialy--- Current third paragraph, that starts describing the codomain, must go.Langing (talk) 19:55, 19 May 2011 (UTC)

I have no objection to moving codomain further down in the article. On the other hand, I see serious problems with Langing's proposed lede. I do not think it is standard usage to have "argument" and "domain" be synonyms, but Langing defines both as the set of inputs. I think standard usage is for the argument to be an element of the domain, not the domain itself. Langing's first paragraph says functions take as argument a set of "quantities", the second paragraph says they may take anything as an input, not necessarily only quantities. And I'm not sure what the reader is supposed to understand by the assertion that the set of real numbers is not identical to the set of doubles of real numbers. Two sets are in the interest identical if they have the same elements, and domain and range of f(x) = 2x have the same elements. Rick Norwood (talk) 20:19, 19 May 2011 (UTC)

Referring to range the way the lead does now is a really bad idea. Range c an mean either image or codomain in mathematics and in statistics it can mean an interval. It may be a more common name for a layman but that doesn';t mean it is a better word to use because it doesn't necessarily mean what they think it means. You're far better off with words like image and codomain which only have one well defined meaning in maths. Cutting the codomain out of the lead entirely is to destroy the article as any sort of description of the concept in modern maths. Could you please not try turning this back to the time of Euler. There is no point in describing something that bears little relation to modern maths just because you wanted something more accessible. Dmcq (talk) 23:10, 19 May 2011 (UTC)

Have a look at Range (mathematics) and seer the problems. It has got to be mentioned because so many older texts mention it and computer science uses it in the sense of codomain but it just causes confusion. Dmcq (talk) 23:16, 19 May 2011 (UTC)

The exact same problem occurs with "ring". Are the even integers a ring or not? But mathematicians all understand the ambiguity and have no choice but to live with it. The lay reader needs to be told that "range" has two meanings, or else the lede should not use the word at all. Many people, not mathematicians, remember the word "range". They have a right to be told what it used to mean, and what it means today. What you have written is fine, except for a minor typo which I fixed. Rick Norwood (talk) 00:17, 20 May 2011 (UTC)

Personally I can't think of any ring without unity that can't be treated as an ideal just as well or better but yes it's good example - but at least people are very aware there is that problem. The range business is just chaos though with people assuming what it means. In computing too they have the same confusion, normally it means the codomain but it also means an interval so they refer to an argument being out of range and mean the image when they ask what values the output ranges over. Dmcq (talk) 09:56, 20 May 2011 (UTC)

Note to TheDinParis

I'll let you finish your edit, and then continue mine, but please note that I had not finished removing the repeated definitions, so there is still a lot of repetition. Rick Norwood (talk) 13:39, 21 May 2011 (UTC)

Ordered triple of sets?

There is something that doesn't make sense to me on the definition of function as an ordered triple of sets (domain, codomain, graph), because a triple is an Tuple of 3 elements, and citing the wiki of Tuple: An n-tuple can also be regarded as a function whose domain is the tuple's set of element indices, and whose codomain is the tuple's set of elements.

So a function is a tuple, and a tuple is a function. Isn't this a circular definition? Wich of the definitions should be dropped in that case? — Preceding unsigned comment added by 186.58.68.193 (talk) 19:09, 27 May 2011 (UTC)

"So a function is a tuple, and a tuple is a function." A tuple is regarded as a function and a tuple is not defined as a function. In set theory a tuple is a set and in Martin-Lof's type theory a tuple is primitive. Actually I am unaware of a theory which defines a tuple as a function. --Beroal (talk) 11:46, 18 July 2011 (UTC)

Didn't notice this query, but the definition using a triple is no more strange than the alternative definition of a function in terms of just the graph which is also a set. That doesn't mean that sets are defined as functions. Dmcq (talk) 12:05, 18 July 2011 (UTC)

Image of a set (notation)

I admit I'd never seen f`x and f``A; however, Set Theory for the Mathematician by Jean E. Rubin uses f "A. (at least, I think that's the character used; it could be f’’A or $f''A$.) — Arthur Rubin (talk) 09:34, 2 August 2011 (UTC)

I just noticed in the Image (mathematics) article a mention of f "A as a notation used in set theory, and it didn't have a citation. So I'll copy your citation over to that article. It also featured more prominently which is why I missed that before some other notations so I'm not sure about its due weight here. Dmcq (talk) 17:09, 2 August 2011 (UTC)

Edits, 4th February 2012

When we define a function as a subset of D × C, the codomain is C. This definition is standard.

Please edit the article to improve it, rather than reverting everything.

Rick Norwood (talk) 01:09, 4 February 2012 (UTC)

You gave me a total of 19 seconds to add my comment here. I submit that is not enough time for me to reasonably respond. I was the person who added the "ordered triple" definition in the first place. But, challenged by the request for a reference, I read a number of standard sources, and none of them used the "ordered triple" definition. Convinced that it is non-standard, I attempted to say what my sources (Holstein, Manin, Halmos, Rudin) say. You've restored it, and claim you will find a reference. I hope you do. I certainly remember it from grad school. But it does not seem to be standard today. Rick Norwood (talk) 01:13, 4 February 2012 (UTC)

You seem to be doing a lot of editing, and I think we are both serious about wanting to improve the article, so I'm going to call it a night, and see what you've accomplished in the morning. Rick Norwood (talk) 01:37, 4 February 2012 (UTC)

I have finished, there is very little there except putting in a citation and moving the things around a bit plus I removed the big triple that I though added nothing.

The comment should have been added before saying go to the talk page to discuss.

I'll try and have a look at the books you've mentioned to see what they do about the domain and codomain. Dmcq (talk) 01:56, 4 February 2012 (UTC)

I've put in a bit about defining a 'function from X to Y' as a set of pairs. The set of pairs isn't a 'function' with this, it is a 'function from X to Y'. Dmcq (talk) 02:39, 4 February 2012 (UTC)

---

So far I like Dmcq's edit. It's nice and succinct. . . excepting this, a sentence that was there before Dmcq's edit:

"The domain X may be void, but if X = ∅ then F = ∅. The codomain Y may be also void, but if Y = ∅ then X = ∅ and F = ∅. Such void functions are not usual, but the theory assures their existence."

It's entirely possible for an algorithm instantiated in a computational mechanism (the whole assemblage a "function box") to have no input at all (i.e. input is void) but have output (cf Knuth: "An algorithm has zero or more inputs" . . . these inputs are taken from specified sets of objects", and "an algorithm has one or more outputs" Knuth 1973:5). An example is the busy beaver function with this function-triple: ({∅}, {|, blank}, F: busy beaver algorithm instantiated a Post-Turing machine).

This "triple" definition helps me think about a two-column table-as-function. There's an input alphabet of symbols, an output alphabet of symbols perhaps the same, perhaps not, and the ordered pairs that define each row in the table, the symbols of which are drawn from the appropriate alphabet, i.e. <input-symbol, output-symbol>. Applying my question to this tabular function, it's entirely possible to have a table that has this row <Ø,☹> perhaps written as < ,☹> i.e. with null input the table outputs a frowny-face. What am I missing here? Thanks. BillWvbailey (talk) 15:48, 4 February 2012 (UTC)

An algorithm may have no inputs and still have outputs, but this is not true of a function, at least not as the word is now understood in mathematics. If the domain or codomain are empty, the cross product is empty, and the only subset of the empty set is the empty set. Rick Norwood (talk) 16:20, 4 February 2012 (UTC)

I think the problem here is an algorithm has a tuple of inputs, for functions this tuple is the input. It is perfectly possible for an algorithm to output a constant every time it is called with a tuple of no inputs. This would correspond to a domain consisting of a single element the empty set rather than the domain itself being empty. Basically the empty set is not the set consisting of just the empty set. Dmcq (talk) 18:49, 4 February 2012 (UTC)

There is a slight difference between the function viewpoint and the mathematical viewpoint. An algorithm that runs with no input, when interpreted as a function from $\mathbb{N}$ to $\mathbb{N}$, should be interpreted as a constant function rather than as the empty function. For example, the algorithm "return 3;" computes the constant function f(x) = 3. The issue is just in picking the right function to correspond to the algorithm. — Carl (CBM · talk) 00:35, 5 February 2012 (UTC)

This is the most convoluted, abstruse definition anybody ever conceived.

The person who recently added this comment to the article put it in the wrong place, but I hope he or she will help us to understand the source of their confusion. Please explain here just what it is about the definition you find hard to understand, and we will try to improve it. Rick Norwood (talk) 13:33, 9 February 2012 (UTC)

Function is a rule

"Function is a rule that" seems more readable than "function associates, etc." but my change was reverted. Tkuvho (talk) 19:38, 9 February 2012 (UTC)

In the 18th Century, most mathematicians would have agreed that a function is a rule. In the 19th century it became apparent that there were serious problems with the idea that a function is a rule, and by the 20th Century, with Cantor and set theory, it was possible to prove that identifying a function with a rule led to contradictions. We don't need in this article to go into the technicalities of countable and uncountable sets and, yes, the idea that a function is a rule is more readable. But we should not say something that is false just to be more readable. Rick Norwood (talk) 19:44, 9 February 2012 (UTC)

Saying that a function is a rule only makes sense if we redefine "rule" to mean "function", at which point it becomes tautologous. The usual meaning of rule, which has to do with a finite expression that defines the function, goes horribly wrong in general, because there are too many functions for us to possibly describe each one with a finite expression. The current language - "a functions associates ... " - is meant precisely to avoid all these issues about how the association is performed. We don't want to say in the first sentence "a function is a set of ordered pairs", either. — Carl (CBM · talk) 21:52, 9 February 2012 (UTC)

This is the usual confusion between pre-mathematical intuitions and their mathematical implementations. The point is that the word "rule" has an intuitive meaning for a general public, whereas "function" does not. Imposing a straitjacket of a rigid definition of "rule" in the context of complexity theory upon the common term "rule" is one of the misconceptions I would have hoped to clarify by now. Tkuvho (talk) 08:24, 10 February 2012 (UTC)

The intuitionists would never have his problem in the first place and could quite happily write rule! :) I had a quick look at if rule could be put in quietly with function or algorithm but I couldn't see a decent way of phrasing it. Dmcq (talk) 09:23, 10 February 2012 (UTC)

The substitution of rules for functions in some intuitionistic settings is one reason those settings use intensional equality. But ordinary mathematics never uses intentional equality for functions (and sets), only extensional equality. — Carl (CBM · talk) 12:11, 10 February 2012 (UTC)

"Rule" certainly does have an intuitive meaning, but that meaning is the one that cannot be used to define the concept of a function, because most functions do not have a rule in that sense. For example, there are uncountably many functions from N to N but there are not uncountably many rules in the intuitive sense. Moreover, although 2x and x+x are different rules, they define the same function on N. The concept of a function is just a completely different concept than the concept of a rule; the relationship is that a rule can be used to define a function, but not that a function is a rule. — Carl (CBM · talk) 11:53, 10 February 2012 (UTC)

That's an important point that's probably worth spelling out in the article I think, that two algorithms or formulae may be completely different and yet correspond to the same function. Dmcq (talk) 14:07, 10 February 2012 (UTC)

This point can be found in Rogers 1987:1-2. The following appears in Algorithm: " Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms". " Bill Wvbailey (talk) 15:06, 10 February 2012 (UTC)

Thanks I've stuck in that cite. Dmcq (talk) 01:58, 11 February 2012 (UTC)

At the level of a reader who does not know yet what a function is, the notion of a function is closely related to that of a rule, just as it was historically. The tried and true method of introducing the notion of a function to a novice is to describe it as a "rule that does such and such". I am not sure what high-minded aspects of intensional equality have to do with this. Tkuvho (talk) 19:51, 11 February 2012 (UTC)

I undid another edit to the first sentence which said, correctly, that a function was historically defined as a rule. The first sentence is not the place to go into history, I think; compare all of our other articles. Really we don't want to encourage the reader to think of a function as a rule, we want them to think of it as an arbitrary association between elements of the domain and codomain, which might "not have a rule" in the informal sense of "rule". — Carl (CBM · talk) 12:52, 12 February 2012 (UTC)

I undid one more and I'm done for the day. I believe it is a mistake to use the word "rule" in the intro in that way. A function is not a rule in the informal sense, and clearing up the confusion between the two is vital for understanding what a function is. The relationship is that a function can be defined by a rule, and I suppose I would not mind saying that in the intro. But I object to any sentence which tries to claim, even with some hedging, that a "function" is or should be thought of as a "rule". — Carl (CBM · talk)

I've chopped out rule completely in the first paragraph and also made the initial sentence more nouny rather than verby or whatever the word is for that. Dmcq (talk) 13:25, 12 February 2012 (UTC)

• teh word for that is probably objectificationing, or management speak, or materialism, or grounding, at a guess, though good copy-edit in this instance. NewbyG ( talk) 14:46, 13 February 2012 (UTC)

No problem, although it does remove the "defined by" point. I was typing the following and got an edit conflict. This is a quote from Devlin's book Sets, functions, and logic page 92:

"This definition of function equality means that we should not really speak of a function as being a rule that takes arguments from the domain and produces values in the codomain. Rather a function is determined by such a rule. It is not the rule itself that is the function, even assuming that we are careful to specify the domain and codomain (as we should). It is the argument-to-value association the rule determines that is "the function".

The italics are in the original. — Carl (CBM · talk) 13:26, 12 February 2012 (UTC)

I was thinking about moving the 'Different algorithms may implement the same function, only the outputs need be the same' at the end of the third paragraph a bit up but didn't see how to do it cleanly before mentioning a function could have many different implementations. Dmcq (talk) 13:32, 12 February 2012 (UTC)

I was looking for a place lower in the article to expand on this issue, but didn't really see a good one there, either. — Carl (CBM · talk) 13:37, 12 February 2012 (UTC)

I've tried putting it in a version at the end of the first paragraph. There's also a citation in the overview section just after the Square definition. Dmcq (talk) 13:53, 12 February 2012 (UTC)

Discussion break

"a function is an association" is much worse than the "a function is a rule". That's not a meaning of the word "association"; not in English, nor in math. For the former just go take a look at wiktionary, where the closest match would be saying the a function is an "act", which it is not; the latter is clear. The fact of the matter is that every function is a rule, i.e. the rule x maps to f(x). While Carl derides this as "tautological", I'd call it "circular"; what's wrong with saying something in the first sentence that is tautologically a synonym anyway. And who cares if we're circular in the first sentence? The point of the first sentence is to be accessible; mathematicians have this thing they "call" a function, but really everyone else would call this a "rule". I think the "anti-rule" people are conflating two issues: (1) that some people think that all functions are given by explicit rules, (2) that a function itself can be thought of as a rule. Worse comes to worst, how about "a function is a way to associate..."? RobHar (talk) 17:47, 12 February 2012 (UTC)

The "association" thing dates from... today and hopefully will not remain until tomorrow. What you are proposing is similar to what the page had yesterday: "a function associates, etc." Your proposed "a way to associate" is better. I still think "a rule that assigns" is even better. The concern that not every function is a "rule" understood in a technical sense can be addressed by writing that "a function is roughly a rule, etc" or something of that sort, which was exactly my edit that was reverted most recently. The current version is WP:PEDANTRY at its best. Tkuvho (talk) 17:51, 12 February 2012 (UTC)

• wp:notpedantry,anyone ?!! NewbyG ( talk) 15:00, 13 February 2012 (UTC)

I found a nifty description in a book with the title "How to prepare for the Virginia SOL":

"A function is a special way of matching the members of one set, called the domain, with the members of a second set, called the range. You can think of a function as a machine that takes the domain elements as inputs, and transforms them into range elements as outputs. But a function must also follow this important rule: Each input must be paired with exactly one output."

Perhaps we could write something similar in the lead of this article? Isheden (talk) 18:15, 12 February 2012 (UTC)

"Special way" is certainly better than "association". Then again, the pedants will say that there are only countably many "special ways". After all, how special can each and every member of a continuum be? Tkuvho (talk) 18:29, 12 February 2012 (UTC)

Just take out the 'special way' and say it matches instead of associates. I can see people having problems with that though as matching sounds like regex to me. Dmcq (talk) 19:35, 12 February 2012 (UTC)

"Matches" sounds both too anthropomorphized and too informal to me. I don't see anything wrong with "association", it seems like a perfectly reasonable way of describing the "functionality" of a function. But I'm not tied to "association" being in the lede as much as I am tied to "rule" not being there. — Carl (CBM · talk) 22:28, 12 February 2012 (UTC)

Re Robhar: I was using "tautological" in the sense of tautology (rhetoric) which basically means "circular". But I think we can do better than give a circular (i.e. meaningless) statement in the first sentence of the article). I also don't like "a function is a way to associate" because a function is not a "way", it is a mathematical object, which is an association between the input set and the output set. — Carl (CBM · talk) 22:26, 12 February 2012 (UTC)

That is not an appropriate use of the word "association", as I stated above. I agree that a function is not a "way", which is why I prefer "rule". As for being a mathematical object: as a mathematical object a function is its graph; so then what is the graph of a function? A function? My point is that while we have a specific standard way of defining formally what a function is, that does not mean that that is what a function is. A natural number is not "an element of every inductive set", it's something we use to count. Similarly, a function is something that models the idea of transforming an input into an output. Like a rule (or a transformation for that matter). And sure it also associates the output to the input, and I'm not against using a word that has as its root "association", but the current use is not an accepted use of the word "association" by anyone. RobHar (talk) 00:15, 13 February 2012 (UTC)

I included a quote above that uses "association" in that way. I see from Google that Serge Lang also used it that way in his Basic Mathematics, first sentence of ch. 14: "We note that a function is an association.". But I'm not really tied to the word "association", which was not in the article until today and is not in the article as I write this sentence.

The main issue I have is that a function is certainly not a rule in the informal sense. It is a "rule" only when we have redefined "rule" to mean "function", which is how authors manage to sometimes say "a function is a rule" with a straight face. We can do better than that circular definition, I believe. Moreover, if someone reads "a function is a rule" without realizing that "rule" has been redefined, they are likely to get exactly the wrong idea, namely that the expression 2x is a function. — Carl (CBM · talk) 00:30, 13 February 2012 (UTC)

I forget which critical originally quipped this sentence should be taken outside and shot, but it could well apply to the current opening sentence In mathematics, a function is a correspondence that associates each input with exactly one output. Using the word "rule" is by far the lesser evil. But whether we eventually settle on "rule" or "association" or something else, surely we can do better than this.

To my mind, the statement "a function is a rule..." is only problematic if you redefine "rule" to mean "finite composition of elementary functions". The usage of the word "rule" in contemporary conversation is not the same as the usage in 19th century mathematics. I think it's reasonable for the lead to describe something informally, noting that rigorous definitions appear further down the page. Jowa fan (talk) 00:52, 13 February 2012 (UTC)

The difficulty is that "a function is a rule" would only be helpful to a reader who doesn't already know what a function is if that person already knows what a "rule" is. But the meaning of "rule" that they will have in mind is something like "a finite expression that defines the function", which is exactly the problematic definition from the 19th century. So by saying "a function is a rule" we will make them think of the 19th century definition. What we want is for them to think that a function is some fixed but arbitrary correspondence or association between elements of the domain and elements of the codomain, which may not be given by a "rule" in the sense they are familiar with. — Carl (CBM · talk) 01:03, 13 February 2012 (UTC)

What you think people think "rule" means and what I think people think "rule" means are quite different. But here is what Michael Spivak thinks (from Calculus, introduction to Chapter 3, "Functions"):

...We will not even begin with a proper definition. For the moment a provisional definition will enable us to discuss functions at length, and will illustrate the intuitive notion of functions, as understood by mathematicians. Later, we will consider and discuss the advantages of the modern mathematical definition. Let us therefore begin with the following: Provisional definition. A function is a rule which assigns, to each of certain real numbers, some other real number.

I think this approach, beginning with ideas that are easily grasped intuitively, and leaving until later the examples that are classically considered "pathological", is a good model for us. Jowa fan (talk) 01:18, 13 February 2012 (UTC)

That is fine for a textbook, but our aim is to be an encyclopedia, which is somewhat different. Unlike a textbook, we can't use a "provisional definition"; our lede should say more directly what's going on with the actual matheamtical definition, because that's the topic of the article. At the very least, when read literally our lede should not be vacuous and should not be wrong. I have a suggestion below. — Carl (CBM · talk) 01:25, 13 February 2012 (UTC)

Bravo Spivak and User:Jowa fan. Carl's concern seems to be mainly that the same function may be defined by different rules. I would suggest modifying a "rule"-based opening to clarify this point. Thus, we could write "a function is given by a rule, etc". We could add a sentence later making the point even clearer, e.g. adding the italicized comment in the previous line. Tkuvho (talk) 08:22, 13 February 2012 (UTC)

break 1

One reason the first sentence is somewhat odd is that, in the past, editors agreed not to put the words "domain", "codomain", "range", or "set" there, to try to keep it simple, but at the same time there is a goal to keep it from being vacuous. If we use these words, we can say:

In mathematics, a function gives a correspondence between two sets, the domain (set of inputs) and the range (set of outputs), so that each element of the domain corresponds to exactly one element of the range. A function can be viewed as a tranformation that takes an input and returns the corresponding output. For example, the function f defined by f(x) = 2x, when given a number x as input, returns the number 2x.

The reason to avoid "codomain" is that "range" is more familiar, and the sentence above is literally correct as written, even though the function might also specify a codomain in addition to giving a correspondence between the domain and range. Note that the sugggestion does not say what a function is, it says what a function does, avoiding the identity issue. — Carl (CBM · talk) 01:16, 13 February 2012 (UTC)

If you think that a distinction between "is" and "does" allows us to avoid certain technical issues, then why not "A function can be viewed as a rule...". I'm looking at Wikipedia:Manual_of_Style/Mathematics#Article_introduction, which says rather clearly that the lead should be informal and not rigorous. I'd prefer not to mention sets in the very first sentence, for the sake of accessibility.

I notice that the MOS gives topology as an exemplar. That article begins with the informal notion of "continuous deformations", which isn't an accurate description of the whole of topology. We don't kick off with "a topology on a space is a collection of subsets satisfying the following axioms..."; the correct definition appears half way down the article, but not in the lead. Likewise, I think the lead here should convey an intuitive (if not completely accurate) idea of what a function usually is, but should not contain a complete and correct definition.

In fact, I wouldn't object too strongly to the article starting "In mathematics, a function can be viewed as a tranformation that takes an input and returns the corresponding output. For example..." I think transformation is actually more misleading than rule (it implies preserving some sort of geometric structure), but at least the prose style isn't so forbidding. Jowa fan (talk) 05:08, 13 February 2012 (UTC)

Reading as far as here Furthermore, functions need not be described by any expression, rule or algorithm: indeed, in some cases it may be impossible to define such a rule. For example... in the fifth para of the Function (mathematics)#Overview does not seem too difficult, I think. Up to there, the lede that is, seems well-worked. Um, this is from the perspective of someone who did first year math many years ago, so I ought to have said IMHO. NewbyG ( talk) 06:41, 13 February 2012 (UTC)

I am not sure I agree with the claim that a function defined, say, by an invocation of the axiom of choice cannot be described as being defined by a rule. In a way it is indeed defined by a rule. The rule is, apply the axiom of choice, then apply Hilbert's epsilon, and you got your function. One only runs into a problem with "rule" if one agrees to wear the straitjacket of a computational-theoretic definition of "rule", which we needn't feel duty-bound to do. Tkuvho (talk) 08:27, 13 February 2012 (UTC)

I think transformation is a bad name for it. A function whose output is 5 when the input is 2 does not transform the 2 in any way. The standard meaning of transform is you keep the original but bend or mould it in some way like producing car bodies out of sheet steel. Dmcq (talk) 08:39, 13 February 2012 (UTC)

Yes, the word transformation has been transformed to the word correspondence, <23:14, 12 February 2012‎ CBM .. (90,015 bytes) (→Lede: Is "Correspondence" better?)> which seems more suitable, IMHO NewbyG ( talk) 09:27, 13 February 2012 (UTC)

Re Tkuvho: if you interpret "rule" in that way, it simply means "function", at which point trying to explain what a function is by referring to a "rule" is circular. But most readers will not think of "rule" as a synonym for "function", so by using "rule" we make most readers think the wrong thing". Both "Association" and "Correspondence" are better in that they do not connote a reason for the association, just that it exists. On the other hand "rule" connotes... a rule. — Carl (CBM · talk) 12:09, 13 February 2012 (UTC)

(ec) And the current footnote 10 has if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x.... NewbyG ( talk) 12:14, 13 February 2012 (UTC)

proposed first paragraph of lede

I propose the following:

In mathematics, a function is a rule that assigns exactly one output to each input. The output of a function f with input x is denoted f(x) (read "f of x"). For example, f(x) = 2x defines a function f that assigns to any input number, the number twice as large. If x = 5 then f(x) = 10. Two different rules may define the same function if they make the same assignments, for example f(x) = 3x−x defines the same function as f(x) = 2x. Leibniz originally introduced the notion of function in the context of the study of curves. A planar curve can often be viewed as a rule (function) assigning the y-coordinate to the x-coordinate of a point on the curve.

Proposed

Proposed

Tkuvho (talk) 09:13, 13 February 2012 (UTC)

Views

• Oppose. The first paragraph cannot be contradictory/circular: A function is defined as a rule, and then two different rules may define a function? Also, "assign" is used in the article in another meaning: Assign a symbol or value to a variable. Isheden (talk) 10:23, 13 February 2012 (UTC)

Here is the text of our current footnote 10 using the word "assign" in the usual sense: Eves asserts that Dirichlet "arrived at the following formulation: "[The notion of] a variable is a symbol that represents any one of a set of numbers; if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x. The variable x . . . is called the independent variable and the variable y is called the dependent variable. The permissible values that x may assume constitute the domain of definition of the function, and the values taken on by y constitute the range of values of the function . . . it stresses the basic idea of a relationship between two sets of numbers" Eves 1990:235. Your point concerning "rule" is well-taken. I therefore propose the following:

In mathematics, a function is given by a rule that assigns exactly one output to each input. The output of a function f with input x is denoted f(x) (read "f of x"). For example, the rule f(x) = 2x defines a function f that assigns to any input number, the number twice as large. If x = 5 then f(x) = 10. Two different rules may define the same function if they make the same assignments, for example f(x) = 3x−x defines the same function as f(x) = 2x. Leibniz originally introduced the notion of function in the context of the study of curves. A planar curve can often be viewed as a rule (function) assigning the y-coordinate to the x-coordinate of a point on the curve. Tkuvho (talk) 11:49, 13 February 2012 (UTC)

• This is self contradictory: "a function is a rule" ... "two different rules may define the same function". The word "is" is a powerful word. But if you change that to "correspondence" or "association", it would be OK with me. — Carl (CBM · talk) 12:07, 13 February 2012 (UTC)

You seem to have overlooked the fact that my second version says "a function is given by a rule". My impression is that most editors here don't like 11-letter words and prefer a 1-syllable word. Tkuvho (talk) 12:15, 13 February 2012 (UTC)

I see. The issue there is that not every function is given by a rule in the sense of "rule" that naive readers will have in mind. It's much more direct to say "a function is a correspondence" or "a function gives a correspondence" than to say "a function is given by a rule". — Carl (CBM · talk) 12:20, 13 February 2012 (UTC)

I'd say a function is a relation between the domain and the range, which associates every element in the domain with exactly one element in the range. Put simpler, a function is a relation between (a set of) inputs and (a set of) outputs, which associates every input with exactly one output. Isheden (talk) 13:01, 13 February 2012 (UTC)

Yes, me too, this is the mathematical definition. But I think it is too hard for the first paragraph. In the past we made a particular effort not to have the set-theoretic definition up front. — Carl (CBM · talk) 13:22, 13 February 2012 (UTC)

• Oppose rule as well. A function just isn't a rule or algorithm. There's no point saying it is one and then that it isn't. And like Isheden assigns sounds wrong to me too. Dmcq (talk) 12:45, 13 February 2012 (UTC)

A rule is not an algorithm. Applying the axiom of choice and then Hilbert's epsilon so as to produce a function is also a kind of a rule. The advantage of a definition in terms of a "rule" is that it is both immediately accessible to a novice, and also technically correct with a sufficiently broad interpretation of "rule". Tkuvho (talk) 13:07, 13 February 2012 (UTC)

I believe the lede as is is reasonably accessible to a novice reader, presuming that reader is interested enough about functions to want to read a few hundred or what ever words. It would be more confusing by starting out as a rule, and then admitting that rule is imprecise, and not in line with current terminology, with good reason. NewbyG ( talk) 13:13, 13 February 2012 (UTC)

Wouldn't "relation" be as accessible as "rule" or "correspondence", with the important difference that relation is also what a function is to a mathematician (with the additional requirement of one output for each input)? Isheden (talk) 13:22, 13 February 2012 (UTC)

I think the only advantage of "correspondence" (or "association", although other people seem to dislike that) is that it is more in line with the regular English meaning, while "relation" is a more specialized term. — Carl (CBM · talk) 13:28, 13 February 2012 (UTC)

(indenting reply) Well, relation has many specialized usages, otherwise it is rather a common-place word, intuitively understood, however there is the slight verbal clash with the even commoner *relationship* in the marital sense!?! NewbyG ( talk) 13:55, 13 February 2012 (UTC)

OTOH correspondence ‘’could’’ give rise to thoughts of Ye olde letter-writing, whilst ‘’association’’ gives us, far-fetchedly, football, or trade-unionism. Interesting?!? Correspondence works fine, association is fine, relation would be fine, I don't favour it though. NewbyG ( talk) 14:05, 13 February 2012 (UTC)

The function of the lede is not to give a complete and accurate definition, but to introduce the topic in an accessible way (see MOS for mathematics as mentioned above). I suggest this:

In mathematics, a function can be thought of as a rule assigning to each possible input exactly one output <footnote: quotation from Spivak as above> ...

Since Spivak asserts that people do frequently think about functions this way (and the discussion so far suggests that he is not entirely alone in this), we have an accurate and sourced statement, and people can easily scroll down to the section headed "Definition" if they want the formal version. Jowa fan (talk) 13:35, 13 February 2012 (UTC)

I quoted above the following from Keith Devlin:

This definition of function equality means that we should not really speak of a function as being a rule that takes arguments from the domain and produces values in the codomain. Rather a function is determined by such a rule. It is not the rule itself that is the function, even assuming that we are careful to specify the domain and codomain (as we should). It is the argument-to-value association the rule determines that is "the function."

So we also have a sourced and accurate statement that a function should not be spoken of as being a rule...

The lede is meant to be an introduction to the article, but it should also be accurate the to extent we can manage. We may choose not to cover fine details in the lede (such as the two definitions of "domain") but the difference between a "rule" and a "function" is not a minor detail, it's a fundamental thing that a reader has to grasp to understand the concept of a function. This does not mean we should have a formal definition in the lede, but the definition should not be misleading. — Carl (CBM · talk) 13:57, 13 February 2012 (UTC)

• Oppose. I'm with Carl on this issue. The notion of "function" is not trivial and "rule" is simplistic, as my example below shows. However, there does seem to be a "meta"-issue here having to do with the notion/generalization/rule/process/method by which one creates a "function". I noodled on it last night (you folks have been busy!) then cut it so I could sleep on it: This is what I wrote (more or less):

"I like associates. A nice strong active verb. It means: "to bring together or into relationship". [ And what happened to relationship? ] But corresponds works too. “To be” is the most powerful, as in “exists, is”: "The collections C of c-things exists, and the collection D of d-things exists and to every d-thing exactly one c-thing is assigned [placed in correspondence, or associated]". This is the core of a *potential* modus ponens, *potential* because it requires an actual D (aka universe of discourse) from which the d are drawn, before their associated c can be produced.

∃C & ∃D & ∀d ∈ D: d → E! c ∈ C. "

From my dictionary: pp of L. fungi to perform: "a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set." (Webster's 9th Collegiate). Except for the definition itself, nowhere in this is the notion of a "rule". To demonstrate the point: here's a listing of ordered pairs generated with random assignment from two collections:

=CONCATENATE("<", INT(9*RAND()), ",", INT(3*RAND()), ">")

Here's an interesting "object" created by the above:

{ <3,0> <6,1> <0,2> <5,2> <1,2> <0,1> <4,2> } : not a function because of <0,2>, <0,1>, but . . .

{ <3,0> <6,1> <0,2> <5,2> <1,2> <4,2> } : is a function because I manually pulled out <0,1>.

This specific object { <3,0> <6,1> <0,2> <5,2> <1,2> <4,2> } embeds no specific rule for the individual assignments inside the ordered pairs, guaranteed by the rand() functions. But it is an object created by a generalized-to-all-functions rule/process/method for the formation of any function; we can see this in the order of the symbol-assignment inside the CONCATENATE instruction, plus the (random) extraction of symbols from two collections { 0-9 }, {0-3}. The concatenation-process itself failed to be a function; there was still the matter of me checking by hand to see to be sure the assignment not one-many. BillWvbailey (talk) 16:42, 13 February 2012 (UTC)

Certainly if it's not single-valued it's not a function. It is hard to see what you have added to the discussion. Tkuvho (talk) 16:47, 13 February 2012 (UTC)

Devlin quote

The Devlin quote is the best proof that we should use "rule" in the lede. The reason Devlin elaborates on this is because "rule" is the best intuitive way of introducing the notion of function, and he warns against interpreting it in certain ways that are not correct. Clearly describing a function as being given by a rule is enough to respond to Devlin's concern (note that Devlin himself wrote "Rather a function is determined by such a rule", which presumably means that he thinks this statement is correct). Similarly, formalist pedantry was recently overcome at the lede of Dirac delta function where common sense prevailed in the end, and the reader has the advantage of learning the necessary intuitions before proceeding to the more technical part of the page. Tkuvho (talk) 14:02, 13 February 2012 (UTC)

This isn't "pedantry" or formalism, it's entirely about how to help a naive reader. Describing a function as a "rule" is a pedagogical disservice to them. Readers should not build the intuition that there is any sort of "reason" or "rule" behind a function, because that intuition is wrong and has to be overcome to learn what a function actually is: an arbitrary association of inputs to outputs. The word "rule" has a false connotation that makes it harder to learn what a function is. — Carl (CBM · talk) 14:10, 13 February 2012 (UTC)

I understand that this seems to be your position. However, note that the quote from Devlin does not support this at all. Tkuvho (talk) 14:19, 13 February 2012 (UTC)

My argument is exactly that "we should not really speak of a function as being a rule". It's fine if we say that a function can be defined using a rule, that is perfectly correct. In fact I put that in the lede [1] but someone else removed it. I am just saying that any implicit or explicit claim that a function is the same as a rule should be avoided. — Carl (CBM · talk) 14:24, 13 February 2012 (UTC)

There could be this, err, compromise. We should not really speak of a function as being a rule though we often used to  think of a function as being a rule ??? No NewbyG ( talk) 14:34, 13 February 2012 (UTC)

This is close to Isheden's proposal below which I support. Tkuvho (talk) 14:37, 13 February 2012 (UTC)

Regarding the lead: According to MOS:MATH, the lead section should contain an informal introduction to the topic. According to WP:LEADSENTENCE, the first sentence should give a concise definition: where possible, one that puts the article in context for the nonspecialist. I'd suggest mentioning that a function can be thought of as a rule in the lead section, but not in the first sentence. Isheden (talk)

Sounds like a good idea. Regardless of the "rule" issue, I think it is useful to have some mention of history in the lede, as in the suggested paragraph above. Tkuvho (talk) 14:17, 13 February 2012 (UTC)

History, yes. A sub=section on historical usage of terminology, or development of terminology or... NewbyG ( talk) 15:06, 13 February 2012 (UTC)

Discussion, per

The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.[1][2][3] The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge.[4] It was introduced by theoretical physicist Paul Dirac.

Looks like the lede from Dirac delta function. Alles in ordnung. smileyface NewbyG ( talk) 14:15, 13 February 2012 (UTC)

The discussion there was whether to say that the function is zero everywhere except at the origin, or whether not to say that. The difficulty is that there are really two different things being discussed in that paragraph: the δ function and the δ distribution. I argued that the lede was fine because the δ function is defined to have to properties in the first sentence (except that, alas, no such function exists). On the other hand, this article is just about regular mathematical functions, so we don't have quite the same issue, because a function is not defined to be a rule. — Carl (CBM · talk) 14:18, 13 February 2012 (UTC)

Devlin and Spivak say the same thing

English prose can be subtle. When Devlin says "we should not really speak of a function as being a rule..." the word really flags the fact that it often happens even though it's not technically correct, which is the same point made by Spivak. (Notice that "we should not really..." has quite a different meaning from "we really should not..."!) Then Devlin goes on to say "...a function is determined by such a rule." So he's cautioning us regarding the word "rule", but not trying to ban it entirely.

By now it's clear that having the word "rule" in the first sentence is not going to be supported by a consensus any time soon. But there are several of us who think it should be mentioned somewhere near the top of the article. What about this:

In mathematics, a function is a correspondence <footnote 1> that associates each input with exactly one output. The output of a function f with input x is denoted f(x) (read "f of x"). For example, the rule <footnote 2> f(x) = 2x defines a function f that associates any input number with the number twice as large: if x = 5 then f(x) = 10. Two different rules define the same function if they make the same associations; for example f(x) = 3x−x defines the same function as f(x) = 2x.

where footnote 1 is the Halmos quotation that was recently added, and footnote 2 mentions both the Spivak and Devlin quotations given above? Jowa fan (talk) 23:42, 13 February 2012 (UTC)

A function can certainly be determined by a rule as he says, but when he says a function is not really a rule that is also true. What's the point of dragging in another word here though? It doesn't seem to aid in anything that I can see. Dmcq (talk) 23:54, 13 February 2012 (UTC)

The point is to help mathematically unsophisticated readers to gain some intuition as to what functions are and how they're commonly used in practice. Or, to put it another way, the point is to make the lead appear less technical and more accessible. Jowa fan (talk) 05:02, 14 February 2012 (UTC)

I added some comments as you suggested. If you think it is appropriate, I will include the sentence about Leibniz as above. Tkuvho (talk) 15:23, 14 February 2012 (UTC)

I think the discussion about function as a rule would fit much better in the third paragraph, which starts out with "There are many ways to describe or represent a function." Isheden (talk) 08:35, 14 February 2012 (UTC)

I restored CBM's comments about rules. Tkuvho (talk) 15:52, 14 February 2012 (UTC)

Quick Google Books survey

• "a function is a rule" -- about 28,000 books
• "a function is a correspondence" -- about 8,940 books

I would support the wording "a function is a rule" as the most common and the most comprehensible. The phrase "a function is a correspondence" is taken from Bourbaki, who do not give a terribly clear definition of a function (not a good basis for Wikipedia). In any case, the word "correspondence" is a poor translation of what Bourbaki said: they actually define a function to be a particular kind of binary relation. But that is really defining a model of a function, not an explanation of the concept. The phrase "a function is a rule" defines the concept more clearly. -- 202.124.75.226 (talk) 05:17, 16 February 2012 (UTC)

Please register with a username rather than using an IP if you get a chance. Tkuvho (talk) 07:35, 16 February 2012 (UTC)