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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)."
Logicist ( talk) 11:06, 23 August 2009 (UTC)
Good points. Why not try a rewrite? Rick Norwood ( talk) 15:50, 23 August 2009 (UTC)
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 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)
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)
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)
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)
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)
Section "Overview" paragraph 6:
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)
The word "to" is not good. "Into" would be better. Rick Norwood ( talk) 13:25, 18 October 2009 (UTC)
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)
The caption on the first image in the article seemed to me too complicated for an elementary introduction.
I replaced it with the text you see below the second image, but this was reverted.
I would appreciate other opinions on the subject.
Rick Norwood ( talk) 21:44, 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)
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)
The material inserted at the beginning of the history section:
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)
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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)
I have now looked up the article by Katz and Barton. They write:
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)
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)
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)
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 requires that the codomain of g equals the domain of f. Yet, the defining expression 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 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 as the set of those elements x in the domain of g such that g(x) is in the domain of f. Formally, with, for all x in this domain, .
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 to 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)
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, and are equal iff , and . Of course, shortly thereafter I also encountered definitions without codomain as an essential characteristic (and hence, for equality, iff and ). 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)
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)
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)
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 Lp 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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 .) — Arthur Rubin (talk) 09:34, 2 August 2011 (UTC)
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)
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So far I like Dmcq's edit. It's nice and succinct. . . excepting this, a sentence that was there before Dmcq's edit:
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. Bill Wvbailey ( talk) 15:48, 4 February 2012 (UTC)
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 that" seems more readable than "function associates, etc." but my change was reverted. Tkuvho ( talk) 19:38, 9 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)
"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)
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)
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)
...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.
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:
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)
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.
Tkuvho ( talk) 09:13, 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)
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."
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:
Here's an interesting "object" created by the above:
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. Bill Wvbailey ( talk) 16:42, 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)
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)
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)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 4 | Archive 5 | Archive 6 | Archive 7 | Archive 8 | → | Archive 10 |
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)."
Logicist ( talk) 11:06, 23 August 2009 (UTC)
Good points. Why not try a rewrite? Rick Norwood ( talk) 15:50, 23 August 2009 (UTC)
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 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)
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)
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)
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)
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)
Section "Overview" paragraph 6:
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)
The word "to" is not good. "Into" would be better. Rick Norwood ( talk) 13:25, 18 October 2009 (UTC)
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)
The caption on the first image in the article seemed to me too complicated for an elementary introduction.
I replaced it with the text you see below the second image, but this was reverted.
I would appreciate other opinions on the subject.
Rick Norwood ( talk) 21:44, 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)
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)
The material inserted at the beginning of the history section:
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)
---
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)
I have now looked up the article by Katz and Barton. They write:
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)
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)
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)
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 requires that the codomain of g equals the domain of f. Yet, the defining expression 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 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 as the set of those elements x in the domain of g such that g(x) is in the domain of f. Formally, with, for all x in this domain, .
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 to 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)
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, and are equal iff , and . Of course, shortly thereafter I also encountered definitions without codomain as an essential characteristic (and hence, for equality, iff and ). 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)
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)
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)
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 Lp 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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 .) — Arthur Rubin (talk) 09:34, 2 August 2011 (UTC)
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)
---
So far I like Dmcq's edit. It's nice and succinct. . . excepting this, a sentence that was there before Dmcq's edit:
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. Bill Wvbailey ( talk) 15:48, 4 February 2012 (UTC)
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 that" seems more readable than "function associates, etc." but my change was reverted. Tkuvho ( talk) 19:38, 9 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)
"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)
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)
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)
...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.
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:
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)
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.
Tkuvho ( talk) 09:13, 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)
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."
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:
Here's an interesting "object" created by the above:
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. Bill Wvbailey ( talk) 16:42, 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)
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)
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)
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)