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Rationale The following outline is a suggestion for prospective editors. For various reasons (stability of edits and links to related articles) I prefer not doing any editing myself, but present these notes as a resource for others. The basic rationale is that the article should center on helping the readers rather than on the personal preferences of the writers/editors. Therefore I will not start with my own preferred definition but with a reader-oriented one with the following characteristics: (a) utmost simplicity; (b) enhancing clarity by adhering to the principle of separation of concerns, in this case separating the concept of function pure and simple from characterizing a function as being from to ; (c) the most general one in view of its algebraic properties, especially around composition; (d) prevalent in basic university/college textbooks in mathematics; (e) a convenient logical basis for explaining/understanding/comparing other variants. It is fortunate that all these properties happen to coincide. Also fortunate is that in the current literature there are essentially only two variants, simply distinguished by whether or not the notion of a codomain plays any role, so covering both remains very manageable. Also clarifying for the readers are brief justifications of the design decisions behind the definitions, without turning the article into a fully-fledged tutorial that is too long for Wikipedia. In view of the many misconceptions observed in the printed literature and on the web (including Wikipedia), a substantial package of references is indispensable. The text follows next. Boute (talk) 13:18, 15 February 2022 (UTC)
Outline for a new version of the article | ||||||||||||
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Outline for the article (text starts here)The concept of a function or a mapping has been described (Herstein[1], page 9) as "probably the single most important and universal notion that runs through all of mathematics". Evidently this also pertains to all other branches of science (physics, engineering etc.) where mathematics is used. In present-day mathematics, there are essentially two major variants of the function concept, and in a balanced account both must be addressed. For this purpose, we designate them as (A) the plain variant and (B) the labeled variant, which has a codomain. The subject matter also requires ample references, also because different formulations often define the same variant, thereby clarifying each other. About a dozen paragraphs suffice for giving the reader a structured guide through the rather varied literature. A. Functions: the plain variantThis variant is the simplest and also the most widespread throughout the sciences, including (but not limited to) calculus/analysis[2][3][4][5][6][7][8][9][10], set theory[11][12][13][14][15][16][17], logic[18][19], algebra[1], discrete mathematics[20][21][22], computer science[23][24], and mathematical physics[25]. Authors and specific page numbers will be mentioned later. A.1 Basic definition One of the simplest formulations is provided by Apostol[2] (p.53):
In general, a collection of ordered pairs is called a graph or a relation and is called functional[12][23] or determinate[21] if no two pairs have the same first member (or component). Thus the preceding definition can be rephrased by saying that a function is a functional graph ((Bourbaki[12] p. 77). Formulations that are equivalent in content and style appear in calculus/analysis (Apostol[2] p. 53, Flett[6], p. 4), set theory (Bourbaki[12] p. 77, Dasgupta[13] p. 8, Quine[15] p. 21, Suppes[16] p. 57, Tarski & Givant[17] p. 3), logic (Mendelson[18] p. 6, Tarski[19] p. 98), discrete mathematics (Scheinerman[22] p. 73), computer science ((Meyer[23] p. 25, Reynolds[24] p. 452). The wordings differ but all define the same concept, apart from the fact that some authors[11][15][17] apply them to classes instead of mere sets. A.2 Conventions The set of all first members of the ordered pairs in a graph (or relation) is called the domain of and is written or . The set of all second members is called the range of and is written or . Let be a function (functional graph). For each in the domain of there is exactly one such that . Hence is uniquely determined by and . It is therefore properly called the value of at and can be unambiguously denoted by some suitable combination of and , the common "default" form being or . Other forms may be chosen as convenient by prior agreement, such as or . A common example of the latter is writing for matrix transposition. A.3 The function equality theorem (Apostol[2] p. 54) Functions and are equal () if and only if (a) and have the same domain and (b) for every in this domain. This theorem follows directly from set equality and holds for all formulations (preceding and following) of the definition of plain functions. It implies that a (plain) function is fully specified by its domain and the value for each in that domain. An illustration follows next. A.4 Function composition This is the most important operation on functions. For any (plain) functions and , the composition (also written ) is also a function, specified as follows: (a) the domain of is the set of all values in the domain of such that is in the domain of and (b) for any such , the value of is given by or, written with less clutter, (see Apostol[2] p. 140, Flett[6] p. 11, Suppes[16] p. 87, Tarski & Givant[17] p. 3, Mendelson[18] p. 7, Meyer[23] p. 32, Reynolds[24] p. 450,452). Composition has the interesting property that, for all functions , and , we have . This associativity allows making the parentheses optional and writing, for instance, . A.5 Conveying domain and range information The literature presents numerous conventions for relating the domain and/or range of a (plain) function to sets and . A helpful preamble is the following legend.
For instance (Apostol[2] p. 578, Flett[6] p. 5, Dasgupta [13] p. 10, Scheinerman[22] p. 169, Meyer[23] p. 26, Reynolds[24] p. 458):
Flett[6] (p. 5) warns that such phrases only conveys information about the domain and the range but does not define a new kind of function. A function from to is commonly introduced by writing , where can be interpreted as the set of all (total) functions from (in)to (Meyer[23] p. 26, Reynolds[24] p. 458), in other contexts also written . As a logical consequence, stipulates that (a) the domain of is and (b) the values are in and can be further specified, for instance, by a formula. This style is very convenient, as illustrated by the following function specifications
By definition, both specify the same function () which is onto but not onto . Consider also
Here and are respectively the positive and negative square root function. Both are functions from to but is onto whereas is onto . Similarly, a partial function from to is a function with domain included in and range included in . For instance, in calculus/analysis most functions are defined on some subset (interval, region, ...) of , , , and so on hence are partial on these sets. For the set of partial functions from (in)to one finds various notations, such as (Meyer[23] p. 26) and (Reynolds[24] p. 458). As a very interesting illustration, the reader can verify that, given and , the composition is a partial function from to and that is a total function from to iff , which trivially holds in case . Important remark: as in natural language, onto is used as a preposition, mentioning explicitly (Flett[6] p. 5, Scheinerman[22] p. 172; more references follow in the next paragraph). A function that is onto is sometimes called surjective on or a surjection on . Scheinerman[22] (p, 172) designates omitting as "mathspeak", but it is not harmless and may cause misunderstandings. A.6 A shortcut formulation for a function from to Quite a few authors (Bartle & Sherbert[4] p. 5, Royden[8] p. 8, Halmos[14] p. 30, Herstein[1] p. 10, Gerstein[20] p. 110, [25] Gries & Schneider[21] p. 280, Szekeres[25] p. 10) do not start from the basic definition given earlier but directly define a function from (in)to as a subset of such that for every in there is exactly one in such that . Less often, some authors (Bartle[3] p. 13, Gries and Schneider[21] p. 280) use a formulation that amounts to replacing "exactly one" by "at most one", which effectively defines a partial function from to . Important remark: appearances notwithstanding, this shortcut formulation logically defines exactly the same kind of function as the basic definition with exactly the same properties and conventions. In particular:,
A.7 Separating the plain function concept from its graph Whereas defining a function as a graph is very precise and rigorous, it creates some ambiguities for certain common conventions. Just two examples: (i) writing for -fold function composition and for the -fold Cartesian product, and (ii) defining sequences (in particular pairs) as functions on some subset of the natural numbers. Some definitions (Carlson[5] p. 182, Kolmogorov & Fomin[7] p. 5, Rudin[9] p. 21) avoid this by defining a function from to less formally as associating "in some manner" a unique value in with every value in , called the domain of . This can be captured as follows:
As noted by Royden[8] (footnote p. 8) this formulation can be made precise by taking the statement of the function equality theorem (A.3) as an axiom. The range of is then the set of all values for in the domain of . All earlier auxiliary formulations carry through literally as stated, namely, fully general composition (A.4) and conveying domain/range information (A5). The graph of is then the set of all pairs for in the domain of and is denoted by . Evidently if and only if . This may be useful in simplifying certain proofs and definitions (e.g., for inverses). B. Functions: the labeled variant and the notion of codomainRecall that, for plain functions, the appearance of in specifies that , without making an attribute of (in contrast , which is specified to be the domain). How to exploit this flexibility in function specifications was demonstrated by the examples , , , . Dasgupta[13] (p. 10) points out that making an attribute of in a proper fashion requires explicitly attaching to to form a triplet . Mac Lane[26] (p. 27) calls this modification labelling. In general,
The set is called the source of and is called the target of or the codomain of . The domain and the range of are those of . Similar formulations, sometimes identifying domain and source, are given by Bourbaki[12] p. 76, Adámek & al.[27] footnote p. 14, Bird & De Moor[28] p. 26, Pierce[29] p. 2. Some of the major differences with the plain varianr are:
References
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Any hidden (collapsed} text needs a visible summary or abstract. Summary This proposal is a resource for editors to improve the article. It starts with the simplest and most widely used definition (in essentially two formulations), which also serves as the basis for explaining more complicated ones. Function composition receives special attention because it is the most important operation on functions. Various misconceptions are resolved, especially regarding codomains. About 30 literature references support the information given. These sources are selected for reliability but, whenever possible, also for being easy to find. Boute (talk) 05:35, 17 August 2022 (UTC)
I have collapsed the proposal for distinguishing easily this long proposal from the discussion about it.
This proposal contains interesting ideas. However, the present article result from a consensus involving many editors. It is written for beginners in mathematics, who must not be confused by technical considerations that are outside their knowledge. At a first glance this has not being considered by the author of the proposal. So, IMO, the proposal can be useful for improving some points of the article and sourcing, but not is not worth to be expanded into a new version of the article. D.Lazard (talk) 15:20, 15 February 2022 (UTC)
Given two sets X and Y, a function f from X to Y is an assignment of an element of Y to each element of X. X is called the domain of f and Y is called the codomain of f.This would fix Boute's above argument ("ad D"). I'd also apply the same change to the lead, but this can be discussed separately. (By the way: In the "Definitions" section, I'd like to see a formal version using ∀∃, or at least an English sentence without any ambiguity w.r.t the quantification order; cf. also User_talk:Jochen_Burghardt#Function_(mathematics),_"one"_or_"an".)
"put some string in, get some integer out", no need to know Cartesian products), without needing an understanding of the set implementation. However, Boute's outline (necessarily) starts with implementation details almost from the beginning.
"If and are sets, then a function from (or on) to (or into) is a relation such that and such that, for each in there is a unique element in with ."
Alas, (today) the definition stated in the article is still self-contradicting. Moreover, the citations do not match the statement. Halmos says something quite different (not bogged down by codomains). The cited reference in the encyclopedia also says something quite different (introduces codomains without contradiction) Boute (talk) 03:47, 16 July 2022 (UTC)
- If X and Y are sets, a function from (or on) X to (or into) Y is a relation f such that dom f = X and such that for each x in X there is a unique element y in Y with (x, y) f
174.112.98.128 (talk) 17:42, 10 May 2023 (UTC)
Addendum Taking a brief look at how this article fared in the past two years, I still see the same defects. In particular, let f be a function from X to Y according to the current Wikipedia definition. Now let U be any subset of X and V be any superset of Y. Then f maps every element of U to an element of V, so its domain is U and its codomain is V with the current Wikipedia definition. So both domain and codomain are in fact ill-defined. Boute (talk) 11:49, 12 February 2024 (UTC)
set theory definition is older– the set theory definition dates more or less from the late 19th century, but didn't become widely adopted until the 20th century sometime. Our page History of the function concept does a decent job of covering this the history through the mid 20th century (the parts about recent history could use further elaboration). –jacobolus (t) 15:57, 14 February 2024 (UTC)
Note that both definitions are in use, and it is not that one definition is right and the other is wrongI agree with you (although I used to think differently) - and this is reflected in the change I made - as I added a section "Reduced formal definition" in which there is an introduction where I wrote that both definitions are commonly used. Kamil Kielczewski (talk) 15:25, 12 March 2024 (UTC)
The current definition is overcomplicated, and at the same time jagged and incomplete, and seems not to meet the requirements of modern mathematics.
This is because
Let us introduce the following, more direct, simple, complet and uniform "new definition" (it is actually not new concept):
Solution - new formal definition
A function is a Set consisting of following elements:
where !} means "there is exactly one"
Notation
In traditional notation we usually separate domain and codomain definitions from graph e.g.: "The function is given by the formula " (formula represents graph W). By new definition this notation just describes following set: . Of course, using new definition we not need to change traditional notation.
If you omit to provide the domain and codomain for a given function in traditional notation, it means that this information must be derived from the graph or context - which is often the case (and justifies separating the definition of the graph in the notation).
Examples
Lets look on following four examples with similar graphs:
distinction:
properties that can be easily derived from the new definition:
Conclusion
New definition:
— Preceding unsigned comment added by Kamil Kielczewski (talk • contribs) 13:14, 23 February 2024 (UTC)
A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Nevertheless, a sentence near the end of the lede was ambiguous, because the lack of "Given its domain and its codomain" before "a function is uniquely represented ...". I have fixed this.
I suggest to cease discussing, or to base future contributions on reliable sources (WP:RS). This is what David Eppstein proposed on 23 Feb already. - Jochen Burghardt (talk) 17:21, 27 February 2024 (UTC)
The above lengthy discussions revealed me that, in many contexts, "function" is used in place of "partial function". This may confuse readers. This is my reason for adding a subsection § Partial functions of the section § Definition. I have tried to explain this confusing terminology with examples. D.Lazard (talk) 15:25, 5 March 2024 (UTC)
The section Partial Functions which has no source, contains this paragraph:
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