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Towards a coherent article about functions (mathematics)

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 $X$ to $Y$ ; (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)[reply]

Outline for a new version of the article

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 variant

This 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):

"A function $f$ is a set of ordered pairs $(x,y)$ no two of which have the same first member."

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) $G$ is called the domain of $G$ and is written $\mathrm {dom} \,G$ or ${\mathcal {D}}\,G$ . The set of all second members is called the range of $G$ and is written $\mathrm {ran} \,G$ or ${\mathcal {R}}\,G$ . Let $f$ be a function (functional graph). For each $x$ in the domain of $f$ there is exactly one $y$ such that $(x,y)\in f$ . Hence $y$ is uniquely determined by $f$ and $x$ . It is therefore properly called the value of $f$ at $x$ and can be unambiguously denoted by some suitable combination of $f$ and $x$ , the common "default" form being $f\,x$ or $f(x)$ . Other forms may be chosen as convenient by prior agreement, such as $f_{x}$ or $x^{f}$ . A common example of the latter is writing $M^{\mathsf {T}}$ for matrix transposition.

A.3 The function equality theorem (Apostol^[2] p. 54) Functions $f$ and $g$ are equal ( $f=g$ ) if and only if (a) $f$ and $g$ have the same domain and (b) $f(x)=g(x)$ for every $x$ 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 $f$ is fully specified by its domain and the value $f(x)$ for each $x$ in that domain. An illustration follows next.

A.4 Function composition This is the most important operation on functions. For any (plain) functions $f$ and $g$ , the composition $g\circ f$ (also written $f\,;g$ ) is also a function, specified as follows: (a) the domain of $g\circ f$ is the set of all values $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$ and (b) for any such $x$ , the value of $(g\circ f)(x)$ is given by $(g\circ f)(x)=g(f(x))$ or, written with less clutter, $(g\circ f)x=g(f\,x)$ (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 $f$ , $g$ and $h$ , we have $h\circ (g\circ f)=(h\circ g)\circ f$ . This associativity allows making the parentheses optional and writing, for instance, $h\circ g\circ f=f\,;g\,;h$ .

A.5 Conveying domain and range information The literature presents numerous conventions for relating the domain and/or range of a (plain) function $f$ to sets $X$ and $Y$ . A helpful preamble is the following legend.

statement	meaning
$f$ is a partial function on $X$	$\mathrm {dom} \,f\subseteq X$
$f$ is a (total) function on $X$	$\mathrm {dom} \,f=X$

statement	meaning
$f$ is (in)to $Y$	$\mathrm {ran} \,f\subseteq Y$
$f$ is onto $Y$	$\mathrm {ran} \,f=Y$

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):

A (total) function from $X$ (in)to $Y$ is a function with domain $X$ and range included in $Y$ .

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 $X$ to $Y$ is commonly introduced by writing $f:X\rightarrow Y$ , where $X\rightarrow Y$ can be interpreted as the set of all (total) functions from $X$ (in)to $Y$ (Meyer^[23] p. 26, Reynolds^[24] p. 458), in other contexts also written $Y^{X}$ . As a logical consequence, $f:X\rightarrow Y$ stipulates that (a) the domain of $f$ is $X$ and (b) the values $f(x)$ are in $Y$ and can be further specified, for instance, by a formula. This style is very convenient, as illustrated by the following function specifications

${\mathsf {sqra}}:\mathbb {R} \rightarrow \mathbb {R}$ with ${\mathsf {sqra}}\,x=x^{2}$ and ${\mathsf {sqrb}}:\mathbb {R} \rightarrow \mathbb {R} _{\geq 0}$ with ${\mathsf {sqrb}}\,x=x^{2}$ .

By definition, both specify the same function ( ${\mathsf {sqra}}={\mathsf {sqrb}}$ ) which is onto $\mathbb {R} _{\geq 0}$ but not onto $\mathbb {R}$ . Consider also

${\mathsf {posrt}}:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} _{\geq 0}$ with $({\mathsf {posrt}}\,x)^{2}=x$ and ${\mathsf {negrt}}:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} _{\leq 0}$ with $({\mathsf {negrt}}\,x)^{2}=x$ .

Here ${\mathsf {posrt}}$ and ${\mathsf {negrt}}$ are respectively the positive and negative square root function. Both are functions from $\mathbb {R} _{\geq 0}$ to $\mathbb {R}$ but ${\mathsf {posrt}}$ is onto $\mathbb {R} _{\geq 0}$ whereas ${\mathsf {negrt}}$ is onto $\mathbb {R} _{\leq 0}$ .

Similarly, a partial function from $X$ to $Y$ is a function with domain included in $X$ and range included in $Y$ . For instance, in calculus/analysis most functions are defined on some subset (interval, region, ...) of $\mathbb {R}$ , $\mathbb {R} ^{n}$ , $\mathbb {C}$ , $\mathbb {C} ^{n}$ and so on hence are partial on these sets. For the set of partial functions from $X$ (in)to $Y$ one finds various notations, such as $X\not \rightarrow Y$ (Meyer^[23] p. 26) and $X\;{\stackrel {\longrightarrow }{\scriptscriptstyle \mathrm {PFUN} }}\;Y$ (Reynolds^[24] p. 458).

As a very interesting illustration, the reader can verify that, given $f:X\rightarrow Y$ and $g:U\rightarrow V$ , the composition $g\circ f$ is a partial function from $X$ to $V$ and that $g\circ f$ is a total function from $X$ to $V$ iff $\mathrm {ran} \,f\subseteq \mathrm {dom} \,g$ , which trivially holds in case $Y=U$ .

Important remark: as in natural language, onto is used as a preposition, mentioning $Y$ explicitly (Flett^[6] p. 5, Scheinerman^[22] p. 172; more references follow in the next paragraph). A function that is onto $Y$ is sometimes called surjective on $Y$ or a surjection on $Y$ . Scheinerman^[22] (p, 172) designates omitting $Y$ as "mathspeak", but it is not harmless and may cause misunderstandings.

A.6 A shortcut formulation for a function from $X$ to $Y$ 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 $f$ from $X$ (in)to $Y$ as a subset of $X\times Y$ such that for every $x$ in $X$ there is exactly one $y$ in $Y$ such that $(x,y)\in f$ . 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 $X$ to $Y$ .

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:,

The function equality theorem holds as stated (only mentioned explicitly by Gerstein^[20] p. 113).
Composition $g\circ f$ is defined for any functions $f$ , $g$ , although some authors overlook this and define $g\circ f$ only for $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ , which reduces generality by assuming $\mathrm {ran} \,f\subseteq \mathrm {dom} \,g$ .
Any function $f$ is a function from its domain to any superset of its range. Hence the versatile specification style illustrated by the examples ${\mathsf {sqra}}$ , ${\mathsf {sqrb}}$ , ${\mathsf {posrt}}$ , ${\mathsf {negrt}}$ remains applicable.
As before, onto-ness is specified with respect to a set, using "onto" as a preposition (Bartle^[3] p. 13, Bartle & Sherbert^[4] p.7, Royden^[8] p. 8, Halmos^[14] p. 31, Herstein^[1] p. 12, Gerstein^[20] p. 118, Szekeres^[25] p. 11).

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 $f^{n}$ for $n$ -fold function composition and $S^{n}$ for the $n$ -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 $f$ from $X$ to $Y$ less formally as associating "in some manner" a unique value $f(x)$ in $Y$ with every value $x$ in $X$ , called the domain of $f$ . This can be captured as follows:

A (plain) function is an entity that is fully specified by a domain, which is a collection (set or a class) of values, and by a unique value assigned to each element in this domain.

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 $f$ is then the set of all values $f(x)$ for $x$ in the domain of $f$ . All earlier auxiliary formulations carry through literally as stated, namely, fully general composition (A.4) and conveying domain/range information (A5). The graph of $f$ is then the set of all pairs $(x,fx)$ for $x$ in the domain of $f$ and is denoted by ${\mathcal {G}}\,f$ . Evidently $f=g$ if and only if ${\mathcal {G}}\,f={\mathcal {G}}\,g$ . This may be useful in simplifying certain proofs and definitions (e.g., for inverses).

B. Functions: the labeled variant and the notion of codomain

Recall that, for plain functions, the appearance of $Y$ in $f:X\rightarrow Y$ specifies that $f(x)\in Y$ , without making $Y$ an attribute of $f$ (in contrast $X$ , which is specified to be the domain). How to exploit this flexibility in function specifications was demonstrated by the examples ${\mathsf {sqra}}$ , ${\mathsf {sqrb}}$ , ${\mathsf {posrt}}$ , ${\mathsf {negrt}}$ .

Dasgupta^[13] (p. 10) points out that making $Y$ an attribute of $f$ in a proper fashion requires explicitly attaching $Y$ to $f$ to form a triplet $\langle f,X,Y\rangle$ . Mac Lane^[26] (p. 27) calls this modification labelling. In general,

A (labeled) function is a triplet $F:=\langle f,X,Y\rangle$ where $f$ is a (plain) partial function from $X$ to $Y$ .

The set $X$ is called the source of $F$ and $Y$ is called the target of $F$ or the codomain of $F$ . The domain and the range of $F$ are those of $f$ . 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:

Equality of labeled functions requires equality for source, domain, codomain and images.
For a labeled function $F:=\langle f,X,Y\rangle$ , the following terminology holds: (i) $F$ is partial means that $\mathrm {dom} \,f\subseteq X$ , (ii) $F$ is total means that $\mathrm {dom} \,f=X$ , and (iii) or $F$ is surjective means that $\mathrm {ran} \,f=Y$ .
Composition $G\circ F$ is defined only in case $\mathrm {cod} \,F=\mathrm {src} \,G$ . In case $G$ is total this means that $\mathrm {cod} \,F=\mathrm {dom} \,G$ , which is quite restrictive when compared to the plain variant.

References

^
ISBN 0-536-00258-4
.

^
ISBN 9971-51-396-X
.

^ ^a ^b ^c Bartle, Robert G. (1964). The Elements of Real Analysis. New York: Wiley.
^
ISBN 9780471433316
.

^
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.

^ ^a ^b ^c ^d ^e ^f Flett, Thomas M. (1966). Mathematical Analysis. New York: McGraw-Hill.
^
ISBN 0-486-61226-0
.

^ ^a ^b ^c ^d Royden, Halsey L. (1968). Real Analysis. New York: Macmillan.
^ ^a ^b Rudin, Walter (1964). Principles of Mathematical Analysis (Second ed.). New York: McGraw-Hill.
ISBN 0-321-24335-8
.

^
ISBN 0-486-66637-9
.

^ ^a ^b ^c ^d ^e Bourbaki, Nicolas (1954). Theorie des ensembles. Paris: Hermann & Cie.

^
ISBN 978-1-4614-8853-8
.

^ ^a ^b ^c Halmos, Paul (1960). Naive Set Theory. New York: Van Nostrand Reinhold.

^
ISBN 0674802071
.

^
ISBN 0-486-61630-4
.

^
ISBN 0-8218-1041-3
.

^
ISBN 0-534-06624-0
.

^
ISBN 0-486-28462-X
.

^
ISBN 978-1-4614-4264-6
.

^
ISBN 1441928359
.

^
ISBN 0-8400-4942-0
.

^
ISBN 0-13-498502-8
.

^
ISBN 978-0-521-10697-9
.

^
ISBN 0-521-82960-7
.

ISBN 0-387-90036-5
.

^ Adámek, Jiří; Herrlich, Horst; Strecker, George E. (2004). Abstract and Concrete Categories - The Joy of Cats. open source: GNU Free Documentation Licence.

ISBN 0-13-507245-X
.

ISBN 0-262-66071-7
.

Last updated: Boute (talk) 13:18, 15 February 2022 (UTC)[reply]

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)[reply]

Discussion

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)[reply]

With very few exceptions, all material in the proposal was written to be understandable by beginners. A good indication of the level is that all material comes from properly referenced introductory textbooks. Moreover, the material is presented in such a sequence that the interested reader can stop at any time, even (in the extreme) after the first paragraph with the initial definition. I understand the complaint about length of the proposal, but in reality the proposal is quite short in comparison with the reality it must properly reflect. Don't forget that the article about functions is considered "vital". The definition in the current Wikipedia has been shown to be logically defective and definitely does not reflect the view on functions prevalent in the open literature, as demonstrated by the references. Talking about codomains before the simpler variant has been presented is about the most confusing thing one can do to beginners. Unfortunately, in half a dozen randomly selected introductory textbooks that attempt to introduce the concept of codomain, this causes a logical error (contact me by email for the references). Boute (talk) 08:30, 16 February 2022 (UTC)[reply]

I disagree definitively with your proposal:

Your "plain" definition implies that the range of a function is a set. So, it is impossible to specify a function without specifying first a set that contains the range. This is this set that is called the codomain. So, in the practice of most users, only "labeled" functions are considered. Moreover, if emphasis is put on "plain" functions, there is no more concept of a surjective function. So, emphasizing on "plain" functions, or giving the same weight to both definitions may confuse most readers.

You assert that you have found a logical error in the article (more exactly in textbooks that follow the same approach). You must be more explicit and open a discussion about this alleged error.

As far as I know, the "plain" definition is used only in mathematical logic and related area. This seems the opinion of the authors of Wikipedia article, as this definition appears only in § In the foundations of mathematics and set theory and § In computer science (with the mention of lambda calculus). Nevertheless, these mention could deserve to be expanded, maybe by adding a new section for comparing and discussing the two definitions. But, again, only one definition must appear in the main part to the article. D.Lazard (talk) 09:37, 16 February 2022 (UTC)[reply]

The "plain" definition is the one used in all 25 references cited for that variant in the proposal, covering many branches of science. Please look at them before making unfounded statements. None of these sources mention a codomain at all, since it is not needed for that variant. What you claim to be "impossible" (in a non sequitur) is in fact entirely normal. The logical errors in the current Wikipedia definition were clearly explained in my comments on these talk pages last month (see here). Not looking at the references and disregarding logical analysis is the main cause of confusion in these discussion pages. Boute (talk) 10:06, 16 February 2022 (UTC)[reply]

First of all, I want to thank

WP:EPSTYLE

).

I also agree with Boute that the definition in the article, while (I believe) trying to explain the concept of labeled function, fails to do so properly. I suggest to change at least section Function_(mathematics)#Definition to 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".)

As for the discussion about "plain" and/or "labeled" version, I am, however, inclinded towards a presentation that is "abstract", i.e. representation-independent, as long as possible. The concept of a function is pretty much abstract, and can be understood, and applied, independent of its "implementation" in set theory. As an analogy: the concept or ordered pair is usually introduced by "specifying" just its properties; after that, remarks on possible implementations may, or may not, follow. (The notion of "specification" and "implementation" are borrowed from formal verification in computer science.) In category theory and in universal algebra, (labeled) functions are used, under the name "morphism" and "homomorphism", respectively, without referring to the set implementation. Also the "dynamic" aspect of a function, illustrated by File:Function_machine2.svg, fits well with an abstract description (e.g. "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.

Nevertheless, both plain and labeled implementations should be presented in the article, with their interrelations and discrepancies. I'm not sure about how to achieve this. Maybe, the section Function_(mathematics)#Relational_approach could be split into "Relational approach (plain)" and "Relational approach (labeled)", and maybe even accompanied by "Algebraic approach" (mentioning homomorphisms and morphisms as primitives - an expert should comment on this)? - Jochen Burghardt (talk) 13:12, 16 February 2022 (UTC)[reply]

I agree with Jochen Burghardt that, ideally, we should define a function by its properties rather than by its representation as a (functional) set/class of pairs (FSOP). This is why item A.7 in my proposal separates the two. However, as I mentioned in the "rationale", the article should serve the interests of the readers rather than our own preferences. A central issue is the order of presentation. There are many reasons for starting with the plain variant. It is the simplest, and most represented throughout the sciences, so readers will encounter it most often. Composition is defined most generally (for any pair of functions), which explains the absence of the labeled variant in calculus texts. It supports the easiest way to describe the labeled variant (as a triplet with a plain function as one of its components). I have not yet heard any arguments for starting with the labeled variant, which seems backwards. In any case, the definition should be properly referenced. The current Wikipedia formulation is incorrectly attributed to Halmos, who really says on page 30

"If $X$ and $Y$ are sets, then a function from (or on) $X$ to (or into) $Y$ is a relation $f$ such that $\mathrm {dom} \,f=X$ and such that, for each $x$ in $X$ there is a unique element $y$ in $Y$ with $(x,y)\in f$ ."

Clearly the plain variant. For defining the labeled variant it does not suffice adding something like "Given sets

X

and

Y

". These sets must be attached explicitly to the function in the definition, as in a 1975 paper by Hilary B Shuard : " A function $f:A\rightarrow B$ consists of two sets $A$ and $B$ and a rule which attaches to each $x\in A$ exactly one $f(x)\in B$ .". This is clearly the labeled variant. With any formulation for this variant, the extra complication of attaching the set

B

to the function must be justified to the satisfaction of the discerning reader. No such justification can be found in the literature, apart from the circular "to satisfy a certain axiom", which shifts the burden of justification to the axiom. Aside: even category can be generalized by omitting the disjointness axiom for hom-sets (Mac Lane's book, last sentence page 27), which eliminates the need for codomains and frees composition from unnecessary restrictions (this evidently works for the axiomatic formulation of category theory; defining a category as a graph is more rigid since it requires unique endpoints for each arrow). Even so, the article we are discussing is about functions, not category theory, so we should not add exta sources of confusion. Boute (talk) 09:01, 17 February 2022 (UTC)[reply]

Your opinions as "complication", or "limiting" below, or "burden", are just your opinion and not objective qualities of the two definitions. The choice of going for the set theoretical definition (what you call plain variant) or for the morphism (labeled) has advantages or disadvantages depending on what you want to do with the concept of function. Lets also note that a large volume of literature defines function and doesn't really pay attention to the distinction. Even in your quote from Halmos, which you consider "clearly the plain variant", you see him say "or into Y". That doesn't make sense. A function defined as a relation looses the "into Y information". With the definition of function as a relation, there is no notion of surjectivity and codomain. Thatwhichislearnt (talk) 15:54, 1 February 2024 (UTC)[reply]

[As a matter of policy, I stop posting anything further here (lost effort), but will gladly reply to personal emails.]

If you read my earlier posts (open the green box in this Talk section), you will see plenty of arguments, not just an "opinion". Better still, send me an email and you will receive an itemized list justifying the term "burden". Boute (talk) 09:18, 14 February 2024 (UTC)[reply]

No thanks. Opinions about definitions are only philosophical, not math. It is not my interest. I also have not doubt that every argumentation about the supposed "burden" is only a lack of understanding of how viewing a function as a morphism, instead of as a relation, can help express information (the most immediate one being the codomain). Again, choices of definition depend on context. I suppose when talking to students in an introduction to calculus you would like to talk about "the sine function" and not pay attention to the codomain. They already have many other new theoretical detail to pay attention to. However, when doing algebraic topology you do want to pay attention to whether you are lifting a map across an inclusion function, for example. Thatwhichislearnt (talk) 15:03, 14 February 2024 (UTC)[reply]

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)[reply]

I noticed that Halmos's definition is different from the article. Why not go with his definition (copied below)?


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)  $\in$  f

174.112.98.128 (talk) 17:42, 10 May 2023 (UTC)[reply]

As you suggest, Halmos's definition would indeed be a good start. Apostol uses even fewer words: "a function is a set of ordered pairs no two of which have the same first component". Sets might be generalized to classes here. To answer your question: many wikipedia editors are rather adamant to include the notion "codomain" right from the start, apparently just because the word exists. Technically the codomain notion is very limiting in many ways (starting with function composition) and in textbook definitions it is a source of logical contradictions. Boute (talk) 07:36, 5 June 2023 (UTC)[reply]

It is actually the other way around. It is very common for books to start with the set theoretical definition of function, just by imitation, to then mindlessly consider the notions of surjectivity and codomain. What leads to contradictions is never the choice of definition, but not paying attention to the choice that was made. Also "function composition", the is no problem with either definition? If anything making the co-domain part of the function makes one conscious of inclusion mappings, instead of tacitly ignoring them. Thatwhichislearnt (talk) 16:06, 2 February 2024 (UTC)[reply]

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)[reply]

From what I can tell, the current definition has the domain and codomain as part of the specification of a function. If you change the domain and codomain, you get a different function. –jacobolus (t) 13:16, 12 February 2024 (UTC)[reply]

Look at the predicate after the verb "is" in "is an assignment of an element of Y to each element of X". If you change to

U\subset X

and

V\supset Y

, it is no longer "to each element of X". So, the domain is uniquely determined by what the the function is. The codomain is the one that can be argued is not determined by what the function is in that definition, since keeping X and changing to

V\supset Y

the function still is what it was before. I think the statement was kept somewhat informal, such that one can interpret either way regarding whether the codomain is or not part of the function. Thatwhichislearnt (talk) 14:28, 12 February 2024 (UTC)[reply]

The current (Wikipedia) definition does not say "is an assignment" but "assigns", which muddles things. But let's take your wording, because it clearly separates the definiendum ("A function from X to Y") from the definiens ("is an assignment etc."). The definiendum cannot make X and Y part of the function, even if you start with "Let X and Y be sets. A function from X to Y is an assignment etc.", Anyway, what I wanted to express with my brief addendum is my low expectation that a simple and logically tight definition will ever emerge in this article, unless someone honestly takes into account the actual literature, starting with the many references I provided 2 years ago. In the current article, even the reference to Halmos is mistaken. I'll sign off now, and am curious about whether any progress will be visible in 2026. Boute (talk) 01:11, 14 February 2024 (UTC)[reply]

That's not muddling, and it's not a definition. It's saying what a function does. Saying what a function is requires more assumptions, and the lead of an article like this is not the place for precise rigorous definitions. —David Eppstein (talk) 02:26, 14 February 2024 (UTC)[reply]

[As a matter of policy, I will not post anything further here (a waste of time), but will gladly reply to personal emails.]

As observed by David, the current text is indeed not a definition, but this is precisely the reason why it fully deserves being called "muddling". Even if one is against rigor in the lead of an article, one should not accept a situation that causes self-contradictions or ill-defined concepts that require extensive provisos and repair work afterwards. The choice is this: do we serve the readers, or some hidden agenda (self-interest, intellectual investment in one's own research or published books etc.)? Googling "criminal algebraists-axiomatisators") reveals Arnold's healthy warning against the latter attitude. Serving the readers is best achieved by starting with simplicity without sacrificing precision. An excellent principle is separation of concerns (Dijkstra), in this case: defining function without starting with "from X to Y; such qualifications can be introduced afterwards. For instance, although it is not my preferred definition (it is too concrete), I quote from Apostol, Flett and various others: "A function is a set of ordered pairs no two of which have the same first component". Boute (talk) 09:08, 14 February 2024 (UTC)[reply]

The set theory definition of function is not the only definition of function. It is just a fact of life that at this current moment of time, there are two inequivalent definitions of function. One that determines the codomain, one that doesn't. Both determine the domain. An encyclopedia should inform on that. None of those definitions is more adequate than the other. It is just a choice depending on needs, the context, or preference. The set theory definition is older, the definitions that make the codomain part of the function fit better with the more recent push to present mathematics from a category theory point of view. At the end of the day, it doesn't really matter as long as one is consistent. Wikipedia only needs to mention both, select one for the rest of the article, and be consistent to it. Thatwhichislearnt (talk) 14:49, 14 February 2024 (UTC)[reply]

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)[reply]

An category theory from the mid to second half of the 20th. What does your post imply? I don't understand. I myself cannot improve the History section, if it needs to be improved. I don't have relevant literature. All I meant to say is that while some have personal affection for the set theory definition, it also has inadequacies when you move it to more modern contexts. Who knows if in 200 years everyone will be including the codomain or not as part of the function. There is little point in fighting about definitions. Thatwhichislearnt (talk) 16:16, 14 February 2024 (UTC)[reply]

@Thatwhichislearnt In reality, both definitions, i.e., the Bourbaki function (a triple: f=(X,Y,G)) and the function as a set of pairs (binary relation), are related in such a way that the latter definition is a restriction of the former to only the graph.

I just submitted this in my last edit HERE (discussion - here) - which was reverted by a group of people who refuse to fix the article, stubbornly clinging to its old version, their own view that a function is just a binary relation. Kamil Kielczewski (talk) 14:24, 12 March 2024 (UTC)[reply]

Note that both definitions are in use, and it is not that one definition is right and the other is wrong. One can work with either one just fine, as long as what one says about functions later is adjusted accordingly. Wikipedia is (or should be) a reflection of current primary sources. Because currently there are sources for both, probably Wikipedia should mention both. One thing that is definitely wrong in the article is the definition in the section Formal Definition. It is malformed. It is not a definition. It has the same structure as "A prime number with hophops, is a natural number that is greater than 1 and not a product of smaller natural numbers", where "with hophops" plays the role of the undefined "with domain X and codomain Y". It is a predicate that the purported definition leaves undefined. In that sense, I do agree that your edit was a better option. Thatwhichislearnt (talk) 15:14, 12 March 2024 (UTC)[reply]

@Thatwhichislearnt Note that both definitions are in use, and it is not that one definition is right and the other is wrong I 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)[reply]

@Thatwhichislearnt one more thing - as a justification for reverting (10:00, 11 March 2024) my edit, D.Lazard wrote

"Here is a clear consensus on the talk page against this change"

which seems not to be entirely consistent with the truth, as confirmed by your opinion that this edit is an improvement to the article. Kamil Kielczewski (talk) 15:43, 12 March 2024 (UTC)[reply]

I don't know when the revert happened in relation to other discussions in the talk page. I did see that some of the objections that were of mathematical nature were unfounded. There were objections on complexity that I could get behind. In my opinion, for the section Formal Definition, I would take a bit of complexity over what is written right now that is not a definition. Thatwhichislearnt (talk) 16:59, 12 March 2024 (UTC)[reply]

@Thatwhichislearnt I will also ask other participants what they think about my reverted change in the article (related to the formal definition). If it were possible to reintroduce it, after consulting with other authors, it would be a step in the right direction and a starting point for further improvements ( that you mentioned). Do you think that would be ok? Kamil Kielczewski (talk) 20:37, 12 March 2024 (UTC)[reply]

You are reading the first paragraph, which doesn't necessarily need to be precise. I quoted from the section called Definition, where one would expect to have more details. In this article, there is one further step going to the section called Formal definition. In both of those the definition of choice does determine the domain, and doesn't determine the codomain. That's all. Thatwhichislearnt (talk) 14:39, 14 February 2024 (UTC)[reply]

"A function on S"

A user edited recently and claimed that a "function on $S$ means a function $f:S\to S$ " which I know is not true. I know it just means $f:S\to Y$ with $Y$ being unspecified. It's very common to say for example "a real-valued function on $S$ ". However I added then that some authors define it as $f:S\to S$ , but now I question this. I could not find an example of this. What is your opinion?

@Monywillbethebest: do you have an example where it is used this way?--Tensorproduct (talk) 07:41, 7 September 2023 (UTC)[reply]

Fn.[4] in the lead of Homogeneous relation is a source for the analogous sitation on binary relations, a generalization of unary functions. - Jochen Burghardt (talk) 08:20, 7 September 2023 (UTC)[reply]

@Jochen Burghardt: Sorry, I don't understand what you mean. What is "Fn.[4]"? What does a "homogeneous relation" have to do with my question? Maybe you misunderstood my question. My question is just if there are actually authors that use the sentence "a function

f

on

S

" as a synonym for an endofunction

f:S\to S

. Because in analysis if one says for example "a function

f

on a manifold

M

" it does only mean

f:M\to f(M)

and not

f:M\to M

. And in real analysis "a function

f

on

S

" would be

f:S\to \mathbb {R}

. But maybe some authors do it differently, that is why I ask. I am not a native English speaker but for me the sentence should be "a function

f

on

S

to

S

" to define

f:S\to S

.--Tensorproduct (talk) 13:26, 7 September 2023 (UTC)[reply]

"Fn.[4]" means "Footnote [4]". As I said above, a binary relation is a generalization of a unary function. A homogeneous relation is a binary relation where domain and range coindice. It is therefore a generalization of an endofunction.

That is, for relations, the wording "on S" (as used in [4]) indiciates that both domain and range is S. - Jochen Burghardt (talk) 16:08, 7 September 2023 (UTC)[reply]

@Jochen Burghardt I am still confused about your answer. 1) you mean the reference or the footnotes? Because the footnote 4 is a statement about elementary function: "Here "elementary" has not exactly its common sense: ..." which I fail to see how that is relevant with my question. 2) You mean when the sentence is "a binary relation on

S

", then it should be interpreted as "a binary relation over

S\times S

? Is that what you mean? If so, then I would still argue that this is different from the statement "a function on

S

".--Tensorproduct (talk) 17:12, 7 September 2023 (UTC)[reply]

(1) There is only one footnote [4] in the article, and this is what I mean. (2) Yes, you got me right here. - Jochen Burghardt (talk) 21:18, 7 September 2023 (UTC)[reply]

Hello, sorry for the late reply, my laptop was sent for repair,

Yes, I saw this convention used in many places, and i did not know of it, which is why i thought to edit this since it would help others also (for example in my country, our standard books use this convention, also in olympiad books and questions, this is very often used) .... should i link the photos of books also? Monywillbethebest (talk) 06:47, 10 September 2023 (UTC)[reply]

@Monywillbethebest Well you can give me sources (book titles with page number is fine). Maybe some authors use it that way, but this is not a convention in mathematics. I don't know any books that use it that way. The sentence is "a function on/from $A$ INTO/TO $B$ " and one also says for example "a real/complex-valued function on $S$ ". For example in

measure theory

, if one says "a measurable function on S" that is understood as

f:S\to \mathbb {R}

and not

f:S\to S

. What do you mean by "in my country, our standard books...", are you talking about English books or other languages? And what level of mathematics? I think we should stick to university books and not lower levels.--Tensorproduct (talk) 08:43, 10 September 2023 (UTC)[reply]

Definition issues

[Rescued 2 recent posts into the middle of an old discussion, by restoring them here:]

While many authors ostensibly use the "labeled" definition (in which the codomain $Y$ is part of the function), I have observed that they are often inconsistent in their use of it. Is the codomain of the sine function the reals, or just the interval [-1_+1]? In the "labeled" view, either $sin:\mathbb {R} \rightarrow \mathbb {R}$ or $sin:\mathbb {R} \rightarrow [-1\_+1]$ , not both.
As for surjective, in the "plain" definition one defines " $f$ is surjective on a set $Y$ "; which is more conveniently stated as " $\mathop {\mathrm {Img} } (f)=Y$ , where $\mathop {\mathrm {Img} } (f)$ is the image of $f$ (which is uniquely defined by the set of pairs).--Jorge Stolfi (talk) 19:07, 15 November 2023 (UTC)[reply]

A mathematics article, more than any other, should start with a definition of its topic -- not merely with statements of some interesting properties. The definition of function as set of ordered pairs is the simplest that is mathematically precise, and is perfectly valid for all fields that use that concept -- even for those who would rather define it in some indirect or more abstract way. Yes, it requires the concept of "set" -- but it is impossible to write an encyclopedia article about "function" that would be mathematically correct and understandable by a reader who does not know what a "set" is. --Jorge Stolfi (talk) 19:07, 15 November 2023 (UTC)[reply]

[ Jochen Burghardt (talk) 21:10, 15 November 2023 (UTC) ][reply]

As I wrote in more detail on my talk page: Stolfi's attempts to install one specific definition from advanced mathematics into the lead of an article whose topic is already the subject of middle-school mathematics (in the form of graphing real functions) is a violation of

WP:NPOV in favoring one specific definition as the only definition when there are multiple definitions possible and in use. —David Eppstein (talk) 21:15, 15 November 2023 (UTC)[reply

]

Having restored the posts doesn't mean I'd agree with Stolfi. Having read the discussion at User_talk:David_Eppstein#Function_(mathematics), I'd just like to add that the 1st lead sentence is not just some "interesting property", but a pretty precise informal definition. - Jochen Burghardt (talk) 21:30, 15 November 2023 (UTC)[reply]

Thanks Jurgen for recovering my posts. While I don't have the energy to fight any more, let me refer to my comment on David's Talk page about the supposed precision of the word "associate". And then add this: Wikipedia is meant to be an encyclopedia, that is, a reference book -- not a textbook, or popularization book. Making people like mathematics (or any other science) is not one of its goals. Yes, articles should be written so as to be useful to the widest possible readership, which means they must avoid unnecessary jargon and technical concepts -- but they should be written for those readers who come here to find out what a word like "function" precisely means. Hardly anyone will come to this article looking for an "informal definition": they already have that, thank you...
All the best, --Jorge Stolfi (talk) 16:11, 18 November 2023 (UTC)[reply]

@Jorge Stolfi I recently made a change in the formal definition (HERE, application and discussion here) which was rejected. The new definition (triplet) I proposed was introduced in such a way that the old definition (binary relation) was placed in the "reduced formal definition" section, where it is mentioned in the introduction (along with citing) that both definitions are commonly used. What do you think about this change - do you think it is an improvement to the article and could be reinstated? Kamil Kielczewski (talk) 20:54, 12 March 2024 (UTC)[reply]

Several years ago I tried to edit the definition in this article, but all I achieved into a three-page discussion in this Talk page with an editor who insisted that the word "set" should be avoided because it reminds readers of high-school math which drives them away. I gave up when a third editor joined the discussion, arguing that it should be defined using category theory.
Then I swore I would never again try to edit a math article here in Wikipedia.
I should have stood by that oath.
All the best, Jorge Stolfi (talk) 10:03, 18 March 2024 (UTC)[reply]

Speaking of which, I find it somewhat curious that there is no mention in this article of type theory and the way that functions are formalized in type theory. —David Eppstein (talk) 21:22, 15 November 2023 (UTC)[reply]

definition discussion continued

I agree with Stolfi's comment that it makes sense to define a function as a set of ordered pairs such that each first element is paired with exactly one second element. The comment is that this kind of definition runs contrary to the notion that this page is aimed at a middle school audience. This seems to contradicted by the fact that in sentence one of this article, we are talking about sets X and Y as being the domain and codomain of our function.

A simple way to understand functions as sets of ordered pairs in which each input has a unique output is to state it as such and then give some simple examples:

{(1,2),(2,4),(3,6)} is a function. Each input is paired with only one output.

{(1,2),(2,2),(3,2)} is a function.

{(1,2),(1,4)} is not a function because the input of "1" has two different outputs.

{(x,y)|y=2x, x is any real number} is a function and is commonly referred to as f(x)=2x. This function has infinitely many ordered pairs, one for each number (input value) in the real number system.

This kind of introduction would be much more digestible as an introduction to the notion of functions than what is on the page now. This kind of treatment is even acknowledged under the subtopic "Total, Univalent Relation", though it is buried in such complex language that a new learner to the topic will find it hard to digest the simple ideas that lay within the discussion. This hardly complies with the notion that this is a topic commonly found in middle school. 2600:1700:2EC7:1C80:0:0:0:3D (talk) 21:52, 26 December 2023 (UTC)[reply]

The definition of "function" in the article is limited and inaccurate. For example it only includes one independent variable. Something like the following is much better:

"A mathematical function is a rule that gives value of a dependent variable that corresponds to specified values of one or more independent variables. A function can be represented in several ways, such as by a table, a formula, a graph, or by a computer algorithm." Rjdeadly (talk) 15:11, 2 January 2024 (UTC)[reply]

"A function of more than one variable" is merely a form of words for referring to a function for which the domain is a set of ordered pairs. JBW (talk) 16:31, 2 January 2024 (UTC)[reply]

It is not clear from the article nor from the definition of Domain (which is obscure in itself for the lead) that it can contain more than one variable, and all the examples given show only one, which is misleading. The lead should be accessible. Then it goes on to say "functions were originally the idealization of how a varying quantity depends on another quantity" so clearly it is talking about one variable. Rjdeadly (talk) 18:56, 2 January 2024 (UTC)[reply]

If you mean that it is not clear from the article that the variable which is the argument of a function may be an element of a set of ordered pairs (or in other words that the domain of a function may be a Cartesian product) then that is true, and it might be helpful to mention that somewhere. However, there is no fundamental difference between a function which maps elements of a set of ordered pairs to somewhere and a function which maps elements of any other kind of set to somewhere, and the failure to mention that Cartesian product are not forbidden to be domains of functions does not make the definition "inaccurate". JBW (talk) 20:05, 2 January 2024 (UTC)[reply]

There is a meaningful conceptual difference (and you can come up with definitions of function where the difference matters in a technical way as well, if you try). In this article we should describe how people think of functions, not only a specific (relatively recent) formal definition given in one or another source.

The key here is the idea of the dimension or degrees of freedom of the domain. We should somewhere in this article discuss this topic, including infinite-dimensional examples such as "operators" or "higher-order functions" or which take functions as an element of the domain or codomain. –jacobolus (t) 20:32, 2 January 2024 (UTC)[reply]

I'm not sure that going into such depth would be helpful; it might be just another of the many cases where making an article more complete from the point of view of a mathematician serves to make it less accessible to most readers of the encyclopaedia. However, I do agree that the concept of a function of more than one variable should be mentioned, in a form accessible to most readers. I have posted some content about this, but I think I did a very bad job of it, so I shall have another look at it, with a view to improving it. JBW (talk) 20:44, 2 January 2024 (UTC)[reply]

OK, I have now made an attempt at a better description. However, any improvements from other editors wll be welcome. JBW (talk) 20:52, 2 January 2024 (UTC)[reply]

Maybe it would be helpful to make a new article at

tuples, etc. –jacobolus (t) 01:53, 3 January 2024 (UTC)[reply

]

@Jacobolus: That may be a good idea. That way the information about that formalism would be available to read, but would leave this article to describe the basic idea in a way less likely to be confusing to a typical reader. A fundamental problem with coverage of mathematical topics in Wikipedia is that Wikipedia is not structured in graded chapters, moving from an elementary standpoint to a more advanced one, so there's a conflict between the desirability of making articles accessible to an elementary readership and the desirability of providing more than just a very elementary coverage. Segregating a more advanced treatment into a separate article is not an ideal solution, but it can be useful in some cases, and this may be one such case. (Incidentally, I think my first posts in this section were really unhelpful, and I regret having made them.) JBW (talk) 14:36, 4 January 2024 (UTC)[reply]

I have changed the target of

main article}} template. D.Lazard (talk) 15:55, 4 January 2024 (UTC)[reply

]

Presently, multivariate functions are firstly defined in a very short section § Functions of more than one variable, which, strangely, is the sixth and last subsection of § Notation. Section § Multivariate function appears very late in the article, and this may be problematic for many readers who are interested in general definitions rather than in technical details. So, I suggest to move § Multivariate function as the first subsection of § Definition, and to clean up accordingly the section and the remainder of the article. D.Lazard (talk) 15:55, 4 January 2024 (UTC)[reply]

Overall the notation section seems somewhat bloated with material that isn't necessary so close to the top of the page, and ends up being somewhat distracting. I wonder if we can summarize this information into a couple paragraphs sufficient to explain the notation used elsewhere in the article, and split the rest into a separate "notation for functions" article or possibly move it down toward the bottom somewhere. –jacobolus (t) 18:52, 4 January 2024 (UTC)[reply]

More generally, the whole article is presented in a awfully pedantic way. For example, the definition in terms of relations requires from its beginning the knowledge of Cartesian products and of the terminology of the theory of relations. I have started to fix this. For the moment, I have started to fix section § Definition, I have renamed the section on relations as § Formal definition, and added an historical explanation of the coexistence of two apparently very different definitions. D.Lazard (talk) 11:55, 6 January 2024 (UTC)[reply]

"function from the reals to the reals" edit

I have made an edit that corrects an incorrect statement, only to see that change be undone so that the statement is once again incorrect.

The section in question contains the following:

". . .one might only know that the domain is contained in a larger set . . . However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval."

The function {(1,2),(2,4)} is a function that falls under this "function from a the reals to the reals" as each number in the domain and range is a real number. However the domain of this function does not contain a non-empty open interval. I edited this section to say ..."but only that the domain is a subset of the real numbers." This edit was undone twice, each time to the formerly incorrect statement.

I assume that someone with some mathematical knowledge is looking after these edits and don't understand why this error has been insisted upon. 2600:1700:2EC7:1C80:0:0:0:3D (talk) 22:16, 26 December 2023 (UTC)[reply]

I can only assume that the statement is intended to refer to some specific use of the word "function" in analysis; clearly it is not true if we take "function" in a purely algebraic meaning, as it depends on a topology on the reals. Perhaps D.Lazard can clarify this, since he both posted the text originally and restored it after you had reverted it. JBW (talk) 21:55, 28 December 2023 (UTC)[reply]

It is explicitly said that this paragraph is about the use of the term "function" in mathematical analysis. Nevertheless, I have slightly changed the formulation for not excluding your example. However, nobody would call your example a "function from the reals to the reals", but rather a function from the integers to the integes (or to the reals). D.Lazard (talk) 23:21, 28 December 2023 (UTC)[reply]

The phrase "a function from the reals to the reals" is used as verbal shorthand in casual conversation, to make a distinction between real variables and complex variables. In writing a technical article, a careful mathematician is more likely to write "a real valued function of a real variable". Now, as mentioned above, technically {(1,2),(2,4)} is a real valued function of a real variable, just as Texas is technically a plot of land on which one could build a house. But a mathematician would naturally call {(1,2),(2,4)} a set of ordered pairs of natural numbers. More generally, the use of "real" suggests that the domain and codomain are larger than the rationals and smaller than the complex numbers, though it only explicitly says the latter: no complex numbers.

Here in Wikipedia, we have to be even more careful than someone writing a math article for a refereed journal, because we have readers who are professional mathematicians and readers who think they know all about mathematics because they took Algebra I in high school. It seems to me the paragraph is now well written as it stands.Rick Norwood (talk) 16:04, 4 January 2024 (UTC)[reply]

There are 27 "often"

Someone really liked to write "often". At the moment there are 27 of them in the article. Most, if not all, don't really add anything. Also, source for those claims of frequency? Even if sourced, I bet the justification of the claim in the source would be questionable, if any. Thatwhichislearnt (talk) 19:26, 31 January 2024 (UTC)[reply]

Many different conventions exist about mathematical functions, so the article often [sorry!] needs to speak about conventions adopted by a majority, or a large community. I'll rephrase some occurrences where the adopting community is not that large, imo. - Jochen Burghardt (talk) 13:26, 1 February 2024 (UTC)[reply]

Done: reduced the count to 17, for now. - Jochen Burghardt (talk) 13:39, 1 February 2024 (UTC)[reply]

"These conditions are exactly the formalization of the above definition of a function."

This sentence in the section about the formal definition is, strictly speaking, incorrect.

While "above definition" that it refers to contains the notions of "domain" and "codomain" of a function, the set theory definition as a relation, looses the notion of codomain.

One cannot tell what was the codomain, or talk about concepts like surjective, from the definition of function as a relation.

The article should warn as early as possible that there are two non-equivalent definitions of functions in common use. Many books, specially Calculus books, can be seen mixing the two, defining functions as relations and then later considering the notion of surjectivity. If I remember correctly Apostol's is one that takes care of distinguishing the two concepts. Thatwhichislearnt (talk) 19:44, 31 January 2024 (UTC)[reply]

The sentence "A function is uniquely represented by the set of all [and a pair follows]", while correct for the set theory definition (as a relation) is in conflict with surjectivity and codomain being a properties or not of a function, that appear later in the article. Thatwhichislearnt (talk) 20:50, 31 January 2024 (UTC)[reply]

"every mathematical operation is defined as a multivariate function"

This claim is incorrect and contradicts the linked article on [mathematical operation]. Nulladic and unary operations are not necessarily multivariate functions. Either the "every" should not be there, or link to a more restrictive concept than mathematical operation. Thatwhichislearnt (talk) 21:08, 31 January 2024 (UTC)[reply]

I added a note explaning that n (number of parameters) may also be 0 or 1. - Jochen Burghardt (talk) 13:51, 1 February 2024 (UTC)[reply]

You edit didn't change the erroneous sentence. That whole section is a bit delicate, since the notion of *multivariate* can be objective (when the nature of the objects of the domain are tuples) or subjective (the domain can consist of tuples, but the function is still viewed as of a single variable). I think the note blurs the distinction. That can cause more confusion instead of less. It also introduces the confusion between a constant as nullary operation and a constant function. That nullary operations are constant is incidental, due to there being only one null-tuple. But not all constant functions are nullary. Thatwhichislearnt (talk) 16:47, 1 February 2024 (UTC)[reply]

The sentence is not erroneous; and my footnote was intended to clarify that. A multivariate function can have 0, 1, 2, 3, ... parameters, according to the given definition.

"Constant function" is a notion from (e.g.) calculus, and is to be distinguished from "constant" used in formal logic (see e.g. First-order_logic#Non-logical_symbols: "Function symbols of valence 0 are called constant symbols"). Jochen Burghardt (talk) 16:46, 3 February 2024 (UTC)[reply]

The sentence is trivially erroneous, since not all mathematical operations are defined as multivariate functions. Period. And in your second paragraph you are preaching to the choir. That is what I was pointing out. Thatwhichislearnt (talk) 21:31, 4 February 2024 (UTC)[reply]

I am unconvinced that this "definition" is sufficiently generally recognised to be stated as a fact. Who says it is defined in that way? What

reliable source indicates that this is the usual definition? Unless one can be provided, this statement does not belong in the article. JBW (talk) 12:16, 5 February 2024 (UTC)[reply

]

Possible typo? Only I don't know enough about maths to say for sure...

Hi. New here. Thanks for your patience.

Quoting the section headed 'Definition' in paragraph 3:

" A function f, its domain X, and its codomain Y are often specified by the notation f:X -> Y In this case, one may write x |-> y instead of f(y). "

I've added the bold emphasis to the f(y) as it seems to me that this could be a typo: shouldn't this be y = f(x)? Like I say in the title, however, I don't know enough about maths to say for sure. Seems odd though, especially as everything relating to the function upto this point in the article is regarding the function f mapping x to y, so why is the function f now mapping y to, presumably, some other codomain? Really aware of my ignorance on this subject (hence why I'm reading the article in the first place...)

Anyway, if someone more knowledgable than me can speak to this, then great.

--

Also, on an unrelated point, I feel like there must be a more elegant way to ref. to a section of wikipedia than the rough quote/approximation I've used above. Does anyone know an article that provides guidance for how to use these talk pages that includes information on how to ref. to specific parts of a wiki article? Thanks. Prfect23 (talk) 07:32, 16 February 2024 (UTC)[reply]

Fixed

To refer to a section of the article you may use the template {{

alink}}; for exanple, your above concern refers to section § Definition

. To quote a part of text, you may display the source of the article by clicking on an edit button (or on the button "read the souce" if you have not the edit rights on this article); then select and copy the part of text to be quoted; and then past it in the edit window of the talk page.

I'll post on your talk page a "welcome" template with many useful links. D.Lazard (talk) 10:31, 16 February 2024 (UTC)[reply]

Formal definition - replacement

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

the concept of relationship was introduced unnecessarily (it has its uses, but here it is off topic)
a function seems to be a fragmented entity (we actually don't know what it is) that has some realtion and may be domain and codomain
in old definition is written: "function is ... a relation" - if so we will not be able to say whether the function is surjective or not - because we must have information about the codomain (relation not contains such information) (actually, based on the old definition, it is not clear whether a function is just a relation or maybe "something" else that also includes a domain and codomain)

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 $f=\{X,Y,W\}$ is a Set consisting of following elements:

domain $X$ that is any set
codomain $Y$ that is also any set
graph $W\subseteq X\times Y$ being a set of pairs such that $\forall x\in X,\exists !y\in Y:(x,y)\in W$

where $\exists !$ means "there is exactly one"

Notation

In traditional notation we usually separate domain and codomain definitions from graph e.g.: "The function $f\colon X\to Y$ is given by the formula $f(x)=\ ...$ " (formula represents graph W). By new definition this notation just describes following set: $f=\{X,Y,\{(x,f(x)):x\in X\}\}$ . 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:

$f\colon \mathbb {R} \to \mathbb {R} ^{+},f(x)=x^{2}$ , by definition it is: $f=\{\mathbb {R} ,\mathbb {R} ^{+},\{(x,x^{2}):x\in \mathbb {R} \}\}$
$g\colon \mathbb {R} \to \mathbb {R} ,g(x)=x^{2}$ , by definition it is $g=\{\mathbb {R} ,\{(x,x^{2}):x\in \mathbb {R} \}\}$ )
$h\colon \mathbb {R} ^{+}\to \mathbb {R} ^{+},h(x)=x^{2}$ , by definition it is: $h=\{\mathbb {R} ^{+},\{(x,x^{2}):x\in \mathbb {R} ^{+}\}\}$
$k\colon \mathbb {R} ^{+}\to \mathbb {R} ,k(x)=x^{2}$ , by definition it is: $k=\{\mathbb {R} ^{+},\mathbb {R} ,\{(x,x^{2}):x\in \mathbb {R} ^{+}\}\}$

distinction:

f is not g because g has 2 elements (domain = codomain) and f has 3 elements (so f and g must be different sets)
f is not h because h has 2 elements
f is not k because f contains element that contains pair (0,0) - the k not contains element with such pair
g is not h because it not contains $\mathbb {R} ^{+}$ element .
g is not k because k has 3 elements
h is not k because k has 3 elements

properties that can be easily derived from the new definition:

f is surjective (note that if you consider only graph (relation) then you can't check this for f and g)
h is a bijection
k is injective

Conclusion

New definition:

defines functions as a set - a basic mathematical concept - and not as a fragmented "something"
explicitly defines a codomain which allows us to determine whether the function is surjective
does not introduce redundant concepts (relation - Occam's razor)
is clean, intuitive and simple

— Preceding unsigned comment added by Kamil Kielczewski (talk • contribs) 13:14, 23 February 2024 (UTC)[reply]

Please sign your posts on the talk page with four tildes (~~~~). Also, since your post is answered, you must not modify it; instead, you should open a new post after the answer.

The new formal definition you suggest is incorrect, since in a set, the elements are not ordered. So, if

\{X,Y,W\}

would be a set, one could not distinguish between the domain and the codomain. Moreover, the phrasing "a function

f=\{X,Y,W\}

is ..." is incorrect, as nobody writes a function this way.

Even if it would be corrected, your definition would consist essentially in replacing the definition "A binary relation between two sets X and Y is a subset W of the set of all ordered pairs (x, y) such that

x\in X

and

y\in Y

" by the formula

(X,Y,W).

This may be shorter to read, but makes certainly reading more difficult. Note also that in all formulations, the domain and the codomain are parts of the definition of a function; so, your main concern is wrong, that the given definition would not allow defining a surjection. D.Lazard (talk) 14:53, 23 February 2024 (UTC)[reply]

1. You write " if {X,Y,Z} would be a set, one could not distinguish between the domain and the codomai" - this is false statement. Set {X, Y, W} (which is simpler structure than (X,Y,W)) where W is subset of

X\times Y

, contains enough information to deduce what element is graph, what is domain and what is codomain (I even give some "edge" examples which shows that). This is NOT NOTATION. This is DEFINITION - precise, avoiding redundancy, using the simplest possible concepts - but still concise and short.

2. "nobody writes a function this way." - untrue. Usually everybody use traditional handy notation - but not confuse notation with definition. Counterexample: e.g. my topology professor sometimes use this definition instead traditional notation to show some sophisticated things.

3. As I wrote in the proposed change, the old definition is formulated in such a way that it is not known what a function actually is (it is fragmented "something") - which only proves that the old definition is simply weak and leads to errors (because even in the old definition itself it is written: "A function with domain X and codomain Y is a binary relation R between" (this means that function is binary relation - but binary relation not contains information about function codomain - so this is false/confusing statement - (4 examples I give also shows that binary relation ("graph") is not enough). With old definition - the reader is left with a puzzle: function is binary-relaton or not - it is somehing more?

4 You try to mix old definition witch new and write "This may be shorter to read, but makes certainly reading more difficult." - no. Just look on old definition, and look on new definition separately (dont't mix them) - the new one is simpler, more clean and more intuitive (because whe use concept of set of pairs in direct way and set simple intuitive condition to that set). My proposition is to replace old definition with new one (remove old content of "Formal definition" except first pharagraph which can be optionally moved to the end of previous section - for clarity - to not to disturb the definition section witch meta info) - so no update old definition; no mix them; Optionally below new definition we can add explanation how to read quantifiers symbols using "natural language".

5. Apart from the other arguments I presented in the proposal to which you did not respond, I would like to emphasize that Introduce binary-relation insead using pairs from

X\times Y

directly is obviously unnecessary complication. This actually makes old definition essentialy more difficult.

Kamil Kielczewski (talk) 15:37, 23 February 2024 (UTC)[reply]

I see no references to published sources in your long posts in this thread. Are you proposing to replace the sourced definitions in our article with

original research? That's a non-starter. —David Eppstein (talk) 18:18, 23 February 2024 (UTC)[reply

]

No - because definition in "Formal definition" section in article is not sourced. Above proposition is also no original research. This is old concept of define function as some set. My proposition is to replace unclear, complex fuzzy old definition with simple, clear and complete definition. Kamil Kielczewski (talk) 18:30, 23 February 2024 (UTC)[reply]

"This is old concept of define function as some set": you must provide a source that defines a function as a set.

"complex fuzzy, simple, clear and complete"". This is your own opinion. It has no value for Wikipedia, if you do not provide a reliable source asserting the same.

"Set

{X, Y, W}

where W is subset of

X\times Y

": this is nonsensical since the sets

{X, Y, W}

and

{Y, X, W}

are equal, while

X\times Y\neq Y\times X.

In other words with your definition, alleged to be clear, the domain and the codomain are confused. D.Lazard (talk) 20:46, 23 February 2024 (UTC)[reply]

1. "This is your own opinion. It has no value for Wikipedia, " this is again false statement. Actually I give arguments above - you ignore them. If that arguments are wrong - give counter argument. However I will provide proof that old definition is not complete and fuzzy - step by step:

definition states: "A function with domain X and codomain Y is a binary relation R between X and Y"
top part of definition defines what is binary relation: "A binary relation between two sets X and Y is a subset of the set of all ordered pairs... $X\times Y$ " - this means that R is set (of pairs).
Ok so acording to this definition function f is set R which looks as follows $f=R=\{(x,y):x\in X\}$ where $\forall x\in X,\exists y\in Y,\left(x,y\right)\in R$ and $(x,y)\in R\land (x,z)\in R\implies y=z$ .
but there is a problem - if only set $f=R$ is given then we can not deduce codomain Y (because is not guaranteed that all elements from Y will be used in binary relation R) e.g $f=\{(x,x^{2}):x\in \mathbb {R} \}$ is suriective or not? It depends of what is te codomain of f... . If e.g. $Y=\mathbb {R} ^{+}$ then it is suriective, if $Y=\mathbb {R}$ is not suriective. But this definition that f is R not provide such information about Y...
so this definition that function f is set R is not comlete (we lose information about codomain when we define function in that way).
however, if someone state that function is something more than R because it also has domain and codomain - then this is fuzzy part: is funtion is binary relation R or not (not= something more than R) ?

2. "you must provide ..." - As you can see, old definition is very poor and week - actually old definition is wrong and does not meet the standards of modern mathematics (which can be verified by everyone - by above proof). I hope above proof is clear - However, if you have any doubts in above proof, indicate where. My goal is to remove wrong definition by good definition.

3. "this is nonsensical since the sets {X, Y, W} and {Y, X, W} are equal, while

X\times Y\neq Y\times X

" - again you write false statement. If you read new definition, then it is clear that domain is denoted by X, codomain is denoted by Y and W is subset of

X\times Y

. The order of elements in set f={W,Y,X} doesn't matter. If I wrtie

A=\{X,Y\}

and

B=X\times Y

where

X=\mathbb {R}

and

Y=\mathbb {N}

you will also say that set A and B are nonesense?

4. If you have still doubts then I will expalin one example - how to determine domain and codomain for function defined as follows:

f=\{\mathbb {R} ^{+},\mathbb {R} ,\{(x,x^{2}):x\in \mathbb {R} ^{+}\}\}

: 1. You detect that W is

\{(x,x^{2}):x\in \mathbb {R} \}

because there is no other set of pairs. 2. You detect that first element of each pair is positive real number, and all positive real numbers are used - also you see that

\mathbb {R} ^{+}

is element of f - so

\mathbb {R} ^{+}

must be domain. 3. So the last element of set f which is

\mathbb {R}

must be codomain Y. It is clear that f is not surjective. So based on informations included in new definition of f we can write it using traditional notation

f\colon \mathbb {R} ^{+}\to \mathbb {R} ,f(x)=x^{2}

(but remember that it is only notation - not definition) - so new definition of function f contains ALL informations about this function.

5. In the other hand, if we try to define abouve funtion

f\colon \mathbb {R} ^{+}\to \mathbb {R} ,f(x)=x^{2}

(this is only notation - dont confuse it with definition) according to old definition - it will be following binary relation (which is set of pairs which old definition states):

f=\{(x,x^{2}):x\in \mathbb {R} ^{+}\}

. Analysing this set we can determine that domain is

\mathbb {R} ^{+}

but we can not determine what is codomain Y since we know that binary relation not allways use all elements of codomain. If someone see f dedined using only old definition and say that f is suriective or even it is bijection, - he will make mistake. That person based only on old definition of f will be unable to rewrite it coretly using traditional notation. Kamil Kielczewski (talk) 22:06, 23 February 2024 (UTC)[reply]

You wrote:"so acording to this definition function f is set R which looks as follows

f=R=\{(x,y):x\in X\}

where

\forall x\in X,\exists y\in Y,\left(x,y\right)\in R

and

(x,y)\in R\land (x,z)\in R\implies y=z

."

This a wrong rewrite of the definition. The correct translation of the definition in terms of formulas is: a function from

X

to

Y

is a set

R

such that

{\big (}R\subseteq \{(x,y):x\in X,y\in Y\}{\big )}\land {\big (}\forall x\in X,\exists y\in Y,(x,y)\in R{\big )}\land {\big (}(x,y)\in R\land (x,z)\in R\implies y=z{\big )}

As you are discussing a definition that is not the one given in our article, everything that you wrote after the above quotation is possibly correct (this is not my opinion) but is not relevant to a discussion on the content of the article.

If you prefer you may replace "a function from

X

to

Y

is a set

R

such that" with "a function is a

triple

(X,Y,R)

of sets such that". Personally, I find that the latter formulation is unnecessary pedantry.

D.Lazard (talk) 15:23, 24 February 2024 (UTC)[reply]

(I will continue points/threads numeration from my previous commment)

6. "This a wrong rewrite of the definition" - this is at most a misunderstanding (I assume that formula after word "and" is "one statement" (inside brakcets), and I assume it is obvious of context of our discussion that R is subset of

X\times Y

- My oversight - I should have written in a more precise manner.). Taking this into account, we have described the same set R. But this is not crucial and does not change the correctness of the further argument. The key point in that proof was that R (and f) in old definition are only SET OF PAIRS - this cause the problem witch old definition.

7. "As you are discussing a definition that is not the one given in our article" - this is a false statement. I am discussing the definition given in the article.

8. "everything that you wrote after the above quotation ... is not relevant to a discussion on the content of the article." - this is a false statement. I divided my last comment into 5 separate usually independent points - you cannot assume that if one point had an error, all of them are wrong or off-topic. If any of these points are incorrect, present your arguments - unless you agree with them.

9. "a function is a triple" - as I wrote earlier, your proposal to use triple (X,Y,R) is an ill-conceived, overly complicated structure. A function can be defined using only a simple set {X,Y,W} due to special structure of graph W (because we can always distinguish the graph W from the domain X and the codomain Y due to the axiom of regularity) . Maybe instead of R or W, we should use just G (graph) which is a more descriptive symbol. Look on example presented in point 4 in my last comment. If you think this is not true - show me a counterexample: a set consistent with the new definition that can be read in an ambiguous way - as two different functions written in traditional notation. And show me a function written in traditional notation that cannot be written as a set consistent with the new definition (in similar way I show you counter example in point 5 in my last comment for old definition)

10. Moreover, introducing the additional concept of "binary relation" is an unnecessary multiplication of entities. Explain to me - what does binary relation bring to the definition of a function besides complication? We can suffice with a set of pairs instead. A good definition should be as simple as possible. Kamil Kielczewski (talk) 19:59, 24 February 2024 (UTC)[reply]

Per

WP:DISENGAGE, only the last point deserve to be answered. My answer to this last point is that Wikipedia is neither a textbook nor a collection of mathematical monographies. For details see WP:Here to build an encyclopedia and WP:What Wikipedia is not. D.Lazard (talk) 19:10, 25 February 2024 (UTC)[reply

]

11. "Wikipedia is neither a textbook nor a collection of mathematical monographies" -

So this is your argument for using the clearly redundant and complicating definition of "binary relation" (instead the direct "set of pairs")? You provided two links - I didn't find any justification in either of them for using unnecessary complications in simple mathematical definitions. Explain this.

12. In light of the link you provided "What Wikipedia is not" it follows that the current incorrect definition cannot be replaced by your hastily invented, not fully thought-out (personal invention) concept of using the triplet (X,Y,R). Especially since there are simpler, known concepts, for example, {X,Y,W}.

13. (re 10) "only the last point deserve to be answered" does not constitute a substantive argument for point 9, i.e., your personal proposal to use the triplet f=(X,Y,R) (which is overly complicated structure) instead of f={X,Y,W} (which is simpler but sufficient structure).

14. The old incorrect formal definition should be completely removed and replaced with a modern, simple, correct definition. Kamil Kielczewski (talk) 20:43, 25 February 2024 (UTC)[reply]

@Kamil Kielczewski – have you tried to figure out what other sources aimed at a broad audience of non-specialists use as a definition for "function"? (for example, general-purpose encyclopedias, introductory textbooks, expository survey papers for scientists or historians of science, semi-technical popular math books, etc.). As a general matter (I haven't examined this in great detail), in my opinion this article should ideally give a high-level survey of the most common conceptions of function typically employed by various kinds of mathematicians, each phrased in the most accessible way practical and thus hopefully at least somewhat legible for e.g. high school students and laypeople. I think we should shy away from excessively formal or technical details of various concepts, and should try to focus on the broad idea and its basic implications rather than picking on the pedantic edge cases of specific definitions. YMMV. –jacobolus (t) 02:04, 26 February 2024 (UTC)[reply]

15. I don't entirely agree with you. Mathematics is a specific field where there is no room for compromise - if something is wrong (which can often be easily demonstrated by showing a counterexample), it should not be accepted as correct - and mislead others. At the beginning of this thread (at the top), I proposed replacing the existing incorrect formal definition of function with a correct one that is simpler than the old one

16. The most challenging aspect of the new definition is

\forall x\in X,\exists !y\in Y:(x,y)\in W

(old definition also use quantifiers), but we can add an explanation in natural language on how to read the quantifiers: for every x in X, there exists exactly one y in Y such that the pair (x,y) belongs to W. ). The new definition, due to its simplicity, is easier to understand than the old one

17. I agree that publishing various good solutions in one article is ok. Kamil Kielczewski (talk) 06:25, 26 February 2024 (UTC)[reply]

Just a comment regarding "wrong definition". Note that, as long as a definition is properly written in the language in question and defines something, there is no such thing as "wrong definition". You can judge a definition regarding whether it is convenient or inconvenient to express the ideas and theorems that you want, but not as being wrong. What can be wrong are theorems that make use of a definition. It is my personal preference the definition of functions as morphisms in Set. If I have to define it in a class, I choose that one, but there is nothing wrong with the definition as a relation. I do enjoy, for example, the notation that can be used in the latter, and that Halmos enjoyed to use of writing

(a,b)\in f

. There are always tradeoffs. What is wrong is those Calculus textbooks that define functions as a relation and then go to talk about codomains and surjectivity. Thatwhichislearnt (talk) 16:26, 11 March 2024 (UTC)[reply]

@D.Lazard You did not address points 11, 12, 13 and 14 (from my earlier comments) - I understand that you agree with them. If not, then present substantive arguments. Kamil Kielczewski (talk) 09:41, 27 February 2024 (UTC)[reply]

I disagree with everything you wrote and I have absolutely no obligation to respond to any of your injunctions ("present substantive arguments"). As said above, I disengage from this disruptive discussion. Please, consider the first sentence of this post as my answer to all your future posts on this talk page. D.Lazard (talk) 11:06, 27 February 2024 (UTC)[reply]

Your disagreement is just your private opinion - and I'm asking for arguments.

So what we do with current wrong old formal definition of function? (what was shown above).

Unless you want to leave it and mislead people Kamil Kielczewski (talk) 12:27, 27 February 2024 (UTC)[reply]

@Kamil Kielczewski If you want to persist in this one, you should (1) distill your criticism down to the shortest length you can make it, and (2) try to find existing published sources discussing this rather than basing it on your own logical reasoning, and link us to those sources. –jacobolus (t) 15:41, 27 February 2024 (UTC)[reply]

Regarding "The new formal definition you suggest is incorrect, since in a set, the elements are not ordered" Think again. One of the three is a subset of the Cartesian product of the other three such that the projection to the first component gives which one is the domain. The axiom of regularity ensures that which of the three must be the relation is uniquely determined. This in turn determines which is the domain. There is also no problem if the domain is equal to the codomain, which gets expressed as a set of two elements, instead of three. Strictly speaking there is no problem. The only problem is the unnecessary complexity. Thatwhichislearnt (talk) 15:46, 11 March 2024 (UTC)[reply]

My two cents. I do agree with the objections about sourcing. Even in those sources that look at functions as morphism, they don't define them in this way. It is a fact that there are in current use two definitions of functions that do define two different concepts: The set-theoretic definition as relations, and the definition as morphisms in the category Set. The article should mention the existence of the two, and pick one to carry for most of the rest, for consistency. I do think that the definition as relation is probably the more appropriate, only because it is still the most quoted. It is my opinion that that will change overtime, as literature updates, and people that have preferences for the older concept die out, but Wikipedia will also change overtime, when that happens. I do agree that the wordy definition in the lede is ambiguous, and that it does not allow to determine/define the codomain given the function (as it is the case with the set-theoretic definition). The supposed "mathematical objections" as either a set of domain, codomain and relation, yes are not correct. Lazard just doesn't have proper understanding of set theory. Thatwhichislearnt (talk) 16:12, 11 March 2024 (UTC)[reply]

The sentence "the functions are the morphisms of the category of sets" is not a definition of functions; it is the definition of the morphisms of the category of sets, and do not give any information on functions. Nevertheless, there are several nonequivalent definitions of a function that are in common use. Two of them are already in the articles (partial functions and multivariate functions are called "functions" in many texts). Another one is the definition of a function in lambda calculus, which does not carry any concept of a domain and a codomain.

Maybe you understand set theory better than me, but there is another thing that I do not understand. This is what is "wordy and ambiguous" in the first paragraph" (this is the unique place in the lede where a function is defined): 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.

D.Lazard (talk) 17:15, 11 March 2024 (UTC)[reply]

I agree with this change. The main locus of the dispute here seems to be how we are given its domain and its codomain: are they required to be part of the set-theoretic representation of a function, in order to make the correspondence between functions and the sets that represent them bijective, or can they be taken as metadata separate from the representation itself? I think your wording is appropriately agnostic on this point. —David Eppstein (talk) 17:41, 11 March 2024 (UTC)[reply]

Being agnostic will just keep a strong flow of people who notice the ambiguity and try to fix it or complain about it. It is better to be explicit and precise, in the appropriate section (Formal definition) and explicitly pick a choice to follow most of the article. Note, that the choice that you are implying, with the codomain as "metadata" is not what is done in the set-theoretical definition of function as a relation. While I personally prefer "codomain as metadata", there is lot of literature that either doesn't do that, or doesn't realize that is not doing that. Thatwhichislearnt (talk) 19:16, 11 March 2024 (UTC)[reply]

No. It is not the job of Wikipedia to make a choice, in situations for which mainstream scholarship has not made a choice. See

WP:BALANCE, part of one of Wikipedia's core policies on maintaining a neutral point of view. —David Eppstein (talk) 21:15, 12 March 2024 (UTC)[reply

]

You are misreading what I said. "explicitly pick a choice to follow most of the article" means for most of the article. In the section on Formal Definition, both choices in current use must be explicitly stated. As opposed to the nonsense that is right now written in that section. Currently what is written, has the same structure as "A prime with hophops, is a natural number that is greater than 1 and not a product of smaller natural numbers", where "with hophops" is playing the same role as "with domain X and codomain Y". In other words, it is not a definition of anything. Thatwhichislearnt (talk) 21:55, 12 March 2024 (UTC)[reply]

And that is the main reason people keep coming over and over trying to fix it. Because what it is said in English, when its formalization is extracted, is a non-definition. It leaves the predicate "with Domain X and codomain Y" undefined. It is not "agnostic", nor educative, nor "balanced". It is just badly written. Thatwhichislearnt (talk) 22:00, 12 March 2024 (UTC)[reply]

I am not using "the functions are the morphisms of the category of sets" to define, but to refer to a definition, given that the variants do not have single name that I can use to call them over and over. The sentence in green, the one in the lede, while OK-ish for the beginning of an article in Wikipedia, is also not a definition of anything. What is "assigns", for example. Likewise, the first sentence in the section Definition uses "is an assignment". Again, OK-ish for that location, but also not a definition. What is "assignment"? My biggest problem is with the one that purports being the formal definition "A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions", specially when combined with the sentence further down that says "These conditions are exactly the formalization of the above definition of a function." The first, when it says what the function "is" immediately makes the Y un-recoverable from what the function is, a relation. Then the sentence "These conditions are exactly the formalization of the above definition of a function." is just false. The intended definitions in all the OK-ish sentences from before, are not exactly what the formal definition expresses.

I entirely understand the intension of the "A function with domain X and codomain Y". It is being used with the intension of serving as the set on which a comprehension is restricted, in restricted comprehension. The set

A

, in

B=\{x\in A\mid \phi (x)\}

. It just doesn't work, because it hasn't been defined yet what is being "with domain X and codomain Y". Then when the "is" comes in the sentence, it makes it even worse. I don't mean that the formal definition has to be entirely formal. It is fine if written in English, but the English must imitate the stricture of the axiom of comprehension.

One way to see it more clearly, compare with some other definition in which the restricted comprehension is written properly. For example prime numbers "A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers.". The predicate of being larger than 1 and not a product of smaller natural numbers is restricted to the set of natural numbers. The set of natural numbers, has been previously defined. Some irrational real numbers satisfy the predicate, but we are not calling them prime, because the definition is only for some natural numbers. Note also, how given a prime number we can still say that it is a natural number.

Compare also with an improperly written definition that has the same structure, but uses some other words: "A prime number with hopshops is a natural number that is greater than 1 and cannot be written as the product of two smaller natural numbers". It hasn't been defined what "with hophops" means and one cannot recover it from the predicate that is after the "is". Thatwhichislearnt (talk) 18:17, 11 March 2024 (UTC)[reply]

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)[reply

]

Formal definition - remove old, introduce new

Why old definition is bad?

The old definition says (I quote): "A function with domain X and codomain Y is a binary relation" and "A binary relation between two sets X and Y is a subset of the set of all ordered pairs... $X\times Y$ ". This means, therefore, that a function f is a subset of a set of pairs, that is $f\subseteq X\times Y$ , such that there is no guarantee that all elements from Y are paired in the set f. This causes the definition to be ambiguous (and thus incorrect), as illustrated by the following example:

$f:\mathbb {R^{+}} \rightarrow \mathbb {R} ,f(x)=x^{2}$

$g:\mathbb {R^{+}} \rightarrow \mathbb {R} ^{+},g(x)=x^{2}$

according to the old definition $f=\{(x,x^{2}):x\in \mathbb {R^{+}} \}$ and $g=\{(x,x^{2}):x\in \mathbb {R^{+}} \}$ so $f=g$ which is not true because codomain for $f$ is $\mathbb {R}$ while for $g$ is $\mathbb {R} ^{2}$ . The function $g$ is a bijection while $f$ is not. But in light of the old definition, it is impossible to determine whether a function is a

surjective

(and therefore a bijection) because it has lost information about the codomain Y.

New definition

The function $f=(X,Y,G)$ is an ordered triple consisting of the following elements:

a domain $X$ that is any set
a codomain $Y$ that is also any set
a graph $G\subseteq X\times Y$ being a set of pairs, such that $\forall x\in X,\exists !y\in Y:(x,y)\in G$

Explanation: the graph $G$ is such a

for every element

x

from

X

there exists exactly one

y

from

Y,

such that the pair

(x,y)

is in the set

G

(meaning this pair is a "point" on the function's graph).

Sources:

Bourbaki, Théorie des ensembles, Paris 1970, ISBN 2-903684 003-0, p. 64, [[1]]
C.C.Pinter, A Book of Set Theory, New York 2014, ISBN: 0-486-49708-9], p. 35, [[2]]
A. Nesin, "Foundations of Mathematics I, Set Theory", Istanbul 2004, p. 35, [[3]]
R. Mayer, 2007, Math 111 Calculus 1, Reed College, p. 67 (58) [[4]]
R. Schultz, Mathematics 144 Set Theory, California 2012, p. 63, [[5]]
University of California, Berkeley, https://ptolemy.berkeley.edu/eecs20/sidebars/sets/functions.html
Wikipedia https://en.wikipedia.org/wiki/Codomain (it mention about triple)

Why new definition is better?

Is simpler because it does not introduce the unnecessary concept of binary relation (actually the graph G in the new definition is a binary relation, but this information can be given in a different section outside of the formal definition)
Unlike the old definition, the new one uses quantifiers in a simpler and more direct way that reflects the intuitive property of a function's graph.
Because in the new definition, the domain and codomain are explicitly indicated, there is no problem with determining whether a function is a surjection/bijection.
The new one has sources provided, unlike the old one

Examles in new defnition

The functions f and g, which demonstrated the ambiguity of the old definition, are distinguishable for the new one, that is, $f\neq g$ (the triples differ in the second element - for $f$ it is $\mathbb {R}$ and for $g$ it is $\mathbb {R} ^{+}$ ):

$f:\mathbb {R^{+}} \rightarrow \mathbb {R} ,f(x)=x^{2}$ by new definition: $f=(\mathbb {R^{+}} ,\mathbb {R} ,\{(x,x^{2}):x\in \mathbb {R^{+}} \}$

$g:\mathbb {R^{+}} \rightarrow \mathbb {R} ^{+},g(x)=x^{2}$ by new definition $g=(\mathbb {R^{+}} ,\mathbb {R} ^{+},\{(x,x^{2}):x\in \mathbb {R^{+}} \}$

Notes

As users @JochenBurghardt and @Jacobolus suggested - I found sources for the new and distilled the problems of the old definition (I do it in new thread to to avoid the distraction of old records)
I propose that in the "Formal definition" section of this article, the first paragraph (historical sketch) be moved outside of this section (e.g., directly before it) to focus the reader's attention, and the old incorrect definition should be completely removed and replaced with the new one.
Initially, I insisted on using the set {X,Y,G} instead of the triple (X,Y,W), but firstly, the sources use the triple, and secondly, using the set would require reconstructing the domain and codomain from the graph G, which could necessitate certain additional assumptions, for example, the axiom of choice... Therefore, although the structure of the set is simpler, the triple allows for a greater generality of functions, which offers more benefits.
A function is a central element of modern mathematics, therefore the formal definition should be correct.
Do you have any questions or is there anything else you would like to know or discuss?

Kamil Kielczewski (talk) 00:19, 29 February 2024 (UTC)[reply]

I think you misunderstand the current text. The formal definitions section does not define "a function" to be a set of pairs. It defines "a function with domain X and codomain Y" to be a set of pairs. If you like, you can think of this as meaning that there is a type X→Y of functions from X to Y, and that the inhabitants of this type are sets of pairs. They are not triples, because in describing an inhabitant of type X→Y it would be redundant to specify X and Y themselves. You could plausibly instead define a type of all functions, using triples as you suggest, and some authors do this, but this leads to difficulties elsewhere: for instance, composition of types X→Y and Y→Z is a well-defined operation, but there is no "composition" operation on all pairs of functions.

Taking a step back, much of the heat and confusion in these discussions comes from an attitude that there can be only one correct definition of a function and that we should present only that one definition and pretend that the others don't exist. That attitude violates

WP:NPOV. We should describe with appropriate balance the different approaches that have been used, and not try to decide here which one of those is best. —David Eppstein (talk) 02:56, 29 February 2024 (UTC)[reply

]

1. "I think you misunderstand the current text. (...) It defines "a function with domain X and codomain Y" to be a set of pairs": If I misunderstood something - then explain to me precisely - Does "to be a set of pairs" mean something more than a set of pairs? So, what exactly set are we talking about? And how does this solve the problem of determining whether a given function, using the old definition, is a surjection? Please refer to the presented counterexample.

2. " but there is no "composition" operation on all pairs of functions": I see no problem in defining the composition of two functions defined as triples. Where exactly do you see a problem here? (maybe use some example to show the problem).

3. " composition of types X→Y and Y→Z is a well-defined operation": Do you want to introduce another concept to formal function definition - such as type - which dispels the ambiguities of the old definition?

4. "these discussions comes from an attitude that there can be only one correct definition of a function and that we should present only that one definition and pretend that the others don't exist. ":A function is a central concept in mathematics. If we adopt the old definition, we will have a problem with determining surjections and bijections (which is an important property of functions - often used). Therefore, okay - we might leave the old definition if you want - but in some additional section named, for example, "Alternative formal definitions" where these definitions will be presented along with indicating their limitations. What do you think about this? Kamil Kielczewski (talk) 07:14, 29 February 2024 (UTC)[reply]

Among the functions from X to Y, the surjections are the ones for which every element of Y is in the image of the function. There is no counterexample. You are describing a function from X to Y, and a function from X to Z, that happen to be described by the same sets of pairs. But they are inhabitants of different types, so as long as one specifies the type, there is no ambiguity. Do you, perhaps, think that mathematics is an untyped language, like Javascript, where questions like "is 1 a member of 1/2" are meaningful? —David Eppstein (talk) 07:26, 29 February 2024 (UTC)[reply]

5. From what you're saying, it follows that a definition of type must be added to the definition of a function, and information about its type must be added to the function itself. So, are you suggesting that a function should be such a pair: a graph and a type?

I would like to point out that in the new definition, there is no such problem, i.e., a function is simply a triple, and there is no need to define any "type". Kamil Kielczewski (talk) 07:35, 29 February 2024 (UTC)[reply]

I think that in actual mathematical discourse every mathematical object carries around a type. The type is not the thing; it is what kind of thing it is. When you use one of your triples to describe a function, you are thinking of it as having the type of function. You can apply it, you can compose it, you can do the things that you do with functions. You cannot add it to an integer because it has the wrong type. If you carried out the set-theoretic definition of integer addition to the set-theoretic encoding of a function as a triple and the set-theoretic encoding of a natural number (or the different encodings of the same number as an integer or a rational number or a real or a complex), you would get garbage, not anything meaningful like the pointwise addition with a constant integer-valued function. That does not mean that a function is really a quadruple (function-type,domain,range,graph): it is a triple, but a triple that you are using as a function. In exactly the same way, when I use the bare graph of a function to describe it, I am thinking of it as having a more constrained type: the type of a function from X to Y, for some X and Y. It can be applied, but only to elements of type X. It can be precomposed, but only with things whose type is a function from Y to something else. It still cannot be added to integers, because it has the wrong type. You are writing as if there is only one kind of mathematical object, and maybe to a foundational set theorist there are only sets, so you don't need to say what type of thing anything is (it is a set). But most mathematicians are not foundational set theorists and we need to describe mathematics as it is actually used, not just how it is encoded to people who want to encode it as only one kind of thing. —David Eppstein (talk) 08:15, 29 February 2024 (UTC)[reply]

Not quite, because if we define a function solely by its graph G (thus according to the old definition), we lose information about the codomain, and we will need to convey this information additionally (outside of the definition). The triple definition automatically incorporates this important information.

If you like referring to programming languages, then consider this:

class OldFunctionType {

graph: map<object, object>

}

class NewFunctionType {

domain: set<object>;

codomain: set<object>;

graph: map<object, object>;

}

In the (pseudo) "type" NewFunctionType, you simply have more information about the function that allows you to determine if it is a surjection. In OldFunctionType, we've lost this important information. Kamil Kielczewski (talk) 08:57, 29 February 2024 (UTC)[reply]

If you encode a function solely by a triple, you lose the information that it is a function and not just a triple of arbitrary things. How is that any different? —David Eppstein (talk) 15:58, 29 February 2024 (UTC)[reply]

8. It differs quite significantly: Assume someone knows both the old and new definition - in that case:

when given a triple, they will be able to determine if it is a function and whether that function is a surjection.
when given only a set of pairs, they will also be able to determine if it is a function, but will not be able to determine if it is a surjection.

Indeed, in mathematics, it is a common situation where someone analyzes a certain object and recognizes, for example, that it is a group, or a function, etc. It is not necessarily the case, as you say, that one must know a priori what a given object is, because by knowing various definitions, they can determine this themselves. Kamil Kielczewski (talk) 18:13, 29 February 2024 (UTC)[reply]

False, incorrect, misleading, and wrong. It is not a common case that someone examines a certain set, discovers "oh, this set happens to match the set-theoretic encoding of a complex number according to the scheme used by [reference]", and starts doing complex arithmetic on that set. One recognizes that something is a group not from its encoding, but from its behavior. Set-theoretic encodings do not capture behavior. Types do. —David Eppstein (talk) 18:18, 29 February 2024 (UTC)[reply]

10. When I said "it is a common situation in mathematics," I of course meant it from my own private perspective - this is my private opinion - I didn't intend to mislead anyone.

"One recognizes that something is a group not from its encoding, but from its behavior." - no - whether something is a group or not is recognized by examining if it meets the definition of a group. (and similarly with recognizing functions, graphs, etc.)

For example, if I receive a set

H=(R,R,\{(x,x^{2}):x\in R\})

- then I can determine that it meets the (new) definition of a function - therefore, it is a function.

Similarly, if someone receives a set

E=\{x:x=2k,k\in \mathbb {Z} \}

with an addition operation - then they can determine that it meets the definition of a group - therefore, it is a group - and consequently, they can use the known properties of the group in further considerations of (E,+).

Therefore, good definitions are quite critical. Kamil Kielczewski (talk) 20:52, 29 February 2024 (UTC)[reply]

To you, there appears to be no distinction between "a set that meets the set-theoretic encoding of a function" and "a function". I imagine that, in the same way, to you, there would be no distinction between "a set that meets the set-theoretic definition of a real number" and "a real number". So if I gave you a set F encoding the group multiplication function

f

for some specific encoding of the Klein 4-group, and I gave you a different set P encoding the real number

π

, and you happened to notice that F could also be interpreted as an encoding of a real number, you would happily add F+P as real numbers and think that the result was meaningful? —David Eppstein (talk) 23:31, 29 February 2024 (UTC)[reply]

11. I have the impression that we are straying from the topic. Let's go back:

A function is either a surjection or not - either this can be determined based on the definition or not. Notation is one thing and definition is another. Notation derives from the definition. In traditional notation, we might write a function like this:

f:R^{+}\rightarrow R^{+},f(x)=x^{2}

according to the old definition, this function would be such a set: $f=\{(x,x^{2}):x\in R^{+}\}$
according to the new definition, this function would be such a triple: $f=(R^{+},R^{+},{(x,x^{2}):x\in R^{+}})$

as you can see, the old definition narrows a function down to its graph G - and this causes - that a person who looks only at such defined functions - in the case of the old definition will not be able to reconstruct the codomain and determine whether it is, for example, a surjection.

If we do not accept some definition of a function - then the traditional notation

f:X\rightarrow Y,f(x)=...

has an unclear meaning - then we do not know what a function is. Therefore, the definition is very important. Kamil Kielczewski (talk) 00:15, 1 March 2024 (UTC)[reply]

I have the impression that we are not straying from the topic, but rather that despite all my attempts at probing what you think you mean by a definition, all you can do is to respond with exactly the same assertions that you already added to the discussion long ago. That is to say, my attempts to bring you into a dialogue rather than a monologue have failed. Since dialogue is necessary for building consensus, and consensus is necessary for making change to Wikipedia articles, I think we should consider the conversation over and leave your proposals as having failed to gain consensus. —David Eppstein (talk) 01:02, 1 March 2024 (UTC)[reply]

" despite all my attempts at probing what you think you mean by a definition " 13. This is off-topic. In this discussion, we are not concerned with what I have in mind as the definition - we are only examining the definitions themselves - both old and new - in a precise way.

You also introduced some concept of a "type" - but you didn't define it precisely - and you want to discuss it. Kamil Kielczewski (talk) 06:37, 1 March 2024 (UTC)[reply]

It seems to me that fundamentally this is about a misunderstanding, on your part @Kamil Kielczewski, of the intended meaning of the definition currently in the article, based on what seems to me like an excessively literal and narrow reading, which you keep insisting is the only possible interpretation even as several people including a number of professional mathematicians tell you that is not quite right. Disclaimer: I am not an accredited professional mathematician, just an interested amateur.

I could certainly believe that some improvements / clarifications could be made to the definitions given here and their explanations. For example, I think it might be helpful to explicitly mention in § Formal definition that this definition defines a function based on its graph, the way Pinter does in the document you linked.

But I don't think demanding that everything in mathematics is precisely and only a set is the most neutral or reader-helpful change. Indeed if anything I would like to see more emphasis and more extended discussion about the ways functions are understood differently at different times / in different contexts. –jacobolus (t) 01:38, 1 March 2024 (UTC)[reply]

14. " It seems to me that fundamentally this is about a misunderstanding, on your part @Kamil Kielczewski, of the intended meaning of the definition currently in the article, based on what seems to me like an excessively literal and narrow reading " This is another argument in favor of the necessity to remove the old definition (or place it in a different section). A mathematical definition should be as simple as possible, precise, clear, and unequivocal, rather than vague, incomplete, and ambiguous. Note that the new definition does not have these kinds of ambiguities. Kamil Kielczewski (talk) 06:49, 1 March 2024 (UTC)[reply]

Let me be more precise: when typical anticipated readers come to

A function f from a set X to a set Y is an assignment of an element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function. [...] However, it cannot be formalized, since there is no mathematical definition of an "assignment". [...] set-theoretic definition [...] A function with domain X and codomain Y is a binary relation R between X and Y

they understand X and Y to be intrinsically part of the specification of the function. The text directly implies that every function has a particular domain ("the domain of the function") and codomain. The language is not ambiguous, and is cemented by the repeated use of these phrases which make it clear that the domain and codomain are intrinsic to the function. (There are some disclaimers included about how sometimes the domain and codomain are only implicitly specified for convenience.)

The issue is that you are taking one phrase out of context. You are unpacking "is a binary relation" to "is a subset of the set of all ordered pairs ..." and you are then discarding the first part of the sentence of the definition (before "is") and also discarding the immediately following sentence. This makes for a strained and excessively pedantic (mis)reading.

Some authors directly state that the function "is" a triple

(X,Y,f)

with

f\subset X\times Y

instead of stating that a function from

X

to

Y

is

f\subset X\times Y

. In practice these two variants are equivalent. The difference is that the "triple" formulation is a bit more cumbersome to read but is perhaps more aligned with the pedagogical structure of an introductory set theory course / with the aesthetic preferences of some specific authors. –jacobolus (t) 16:59, 1 March 2024 (UTC)[reply]

16. "the domain and codomain are intrinsic to the function" I completely agree with this.

Therefore, a binary relation cannot be equated with a function. This is because binary relation does not always contain complete information about the codomain.

Thus, the notation from the section "Formal definition" with content:

"A function with domain X and codomain Y is a binary relation R ..."

is incorrect/false statement, as f is not a binary relation.

At most, the graph G of the function f can be defined as a binary relation.

Simply put, a binary relation is too poor a structure/concept to fully describe a function (meaning the domain, the codomain and the graph). I also wan Kamil Kielczewski (talk) 21:19, 1 March 2024 (UTC)[reply]

The reason why people keep complaining about that sentence in green (and others like it) not actually making X and Y intrinsically part of the function, is because it is malformed. It is easy to see, when you compare it with the structure of the axiom of comprehension. It is, of course, fine to write in English, but still adhering to the main features of the axiom. In | this message below, in the last two or three paragraph there is a bit more detail. The fact that the sentence in green, and others like it in the article are malformed definitions, become even clearer if one replaces some of the objects in it by some other words, while keeping the structure of the sentence. I did that in the last paragraph of the linked post, last paragraph. Thatwhichislearnt (talk) 19:34, 11 March 2024 (UTC)[reply]

Here's Mayer's definition:

Let

A,

B

be sets. A function with domain

A

and codomain

B

is an ordered triple

(A,B,f),

where

f

is a rule which assigns to each element of

A

a unique element of

B.

The element of

B

which

f

assigns to an element

x

of

A

is denoted by

f(x).

We call

f(x)

the

f

-image of

x

or the image of

x

under

f.

The notation

f:A\to B

is an abbreviation for "

f

is a function with domain

A

and codomain

B.

We read "

f:A\to B

" as "

f

is a function from

A

to

B.

"

This seems to me effectively the same as the definition currently in the article, but slightly more pedantic. It is almost exactly identical to D.Lazard's comment, If you prefer you may replace "a function from

X

to

Y

is a set

R

such that" with "a function is a

triple

(X,Y,R)

of sets such that", except with "set" replaced by "rule" (such a rule could be formally defined as a set). –jacobolus (t) 04:48, 29 February 2024 (UTC)[reply]

7. Mayer uses a triple (A,B,f) to define function, but I do not see him defining what a rule f exactly is but he wrote "f is a rule which assigns to each element of A a unique element of B", it looks that a rule can be expressed like graph G in new definition, and there is no need to introduce the additional concept of a "binary relation" R (which D.Lazard refer to). The graph G can be interpreted as a kind of relation, but this is additional information that can be placed outside the 'Formal Definition' section. Kamil Kielczewski (talk) 08:11, 29 February 2024 (UTC)[reply]

Here's Pinter's definition:

A function is generally defined as follows: If

A

and

B

are classes, then a function from

A

to

B

is a rule which to every element

x\in A

assigns a unique element

y\in B

; to indicate this connection between

x

and

y

we usually write

y=f(x).

[...] The graph of a function is defined as follows: If

f

is a function from

A

to

B,

then the graph of

f

is the class of all ordered pairs

(x,y)

such that

y=f(x).

[...]

Since a function and its graph are essentially one and the same thing, we may, if we wish, define a function to be a graph. There is an important advantage to be gained by doing this—namely, we avoid having to introduce the word rule as a new undefined concept of set theory. For this reason it is customary, in rigorous treatments of mathematics, to introduce the notion of function via that of graph. We shall follow that procedure here.

As far as I can tell this is nearly precisely the same as the current definition here. –jacobolus (t) 05:03, 29 February 2024 (UTC)[reply]

Pinter's note defining a function as a "rule" and then explicitly pointing out that it can also be defined as the graph (set of ordered pairs) might not be a bad approach though. I wouldn't be against that. (Though I also don't think it's necessary.) –jacobolus (t) 05:07, 29 February 2024 (UTC)[reply]

6. On page 52 (on my proposal, 35 is the wrong page number) Pinter's pdf (below of section "2 FUNDAMENTAL CONCEPTS AND DEFINITIONS") we see:

2.1 Definition A function from A to B is a triple of objects (f,A,B) , where A and B are classes and f is a subclass of A × B with the following properties. (...)

Pinter generalized the concept of a function as triples to classes (top of page 53) - in practice, however, sets are usually used, therefore the new foundational definition should remain unchanged and be based on sets, and information about generalization to classes can possibly be added in a separate article section. Kamil Kielczewski (talk) 07:54, 29 February 2024 (UTC)[reply]

What do you think is the difference between "binary relation" and "subclass of

A\times B

"? From what I can tell the latter description is nothing more or less than the definition of the former. I still don't understand what the hangup is here. –jacobolus (t) 15:10, 29 February 2024 (UTC)[reply]

9. Short answer: "binary relation" is a redundant and unnecessary concept that adds nothing to the definition of a function - except for complication ("let's not introduce unnecessary entities" Occam's razor). Introducing this concept only muddies the waters and distracts the reader's attention.

Long answer: through an exaggerated similar analogy - would the following definition of even numbers be reasonable: "Let's start with the fact that a group is a non-empty set G and an operation...(here the definition of a group). Even numbers are elements of a group, whose set is of the form

\{x:x=2k,k\in \mathbb {Z} \}

"

I know, it's not a perfect analogy - but doesn't it highlight the absurdity of introducing the concept of a group in order to define even numbers? Although even numbers with addition do form a group, defining the concept of a group in order to define the concept of an even number is redundant, unnecessary, and only obscures the main definition we are concerned with.

Similarly (though not identically), in the case of the old definition of a function - introducing the concept of a binary relation (even though a function is such a relation, just as even numbers with addition form a certain group) only introduces unnecessary complication and obscures the basic definition - that is, the definition of a function.

The additional information that a function is a certain binary relation should be placed in a different section - outside the formal definition of a function.. Instead of introducing the definition of a binary relation, the Cartesian product should be used directly as in the new definition of function. Kamil Kielczewski (talk) 20:29, 29 February 2024 (UTC)[reply]

It seems to me that the essential fact about functions is that they are either 1) a rule, written out in words, that allows you to change an input to an output or 2) a set of ordered pairs, with each input paired with a single output. The advantage of the second idea is that it is more mathematical and less linguistic. But since nobody can ever actually produce even a countably infinite subset of the set of ordered pairs of y=x^2, we actually fall back on the rule: multiply the input by itself. As for the common ordered-triple definition, since we can reproduce (A, B, f) by saying that f is a set of ordered pairs, and A is the set of all first elements of the ordered pairs, and B is the set of all second elements of the ordered pairs, the ordered-triple definition does not seem to add anything to the concept.

And, while we are on the subject, if the function has n inputs and m outputs, then the order pair is an n-tuples paired with an m-tuple. For example, an element of f(x,y)=<x, y, x+y> is the ordered pair ((x,y),(x,y,x+y)).

Using the sources, we should either define a function as a set of ordered pairs or as a rule giving a unique output for any given input. Anything more will only confuse readers. Either of these definitions is simple, reliable, referenced, necessary, and sufficient. Rick Norwood (talk) 17:38, 29 February 2024 (UTC)[reply]

12. "As for the common ordered-triple definition, since we can reproduce (A, B, f)" - no, based only on the graph, we cannot reconstruct the codomain. (details and example in section "Why old definition is bad?" above)

"the ordered-triple definition does not seem to add anything to the concept " what about determining whether a function is a surjection (and bijection) or not?

I also wrote a few other positive aspects of the definition based on the triple in section "Why new definition is better?" above. Kamil Kielczewski (talk) 00:43, 1 March 2024 (UTC)[reply]

All the definitions referred to in this discussion (including the definitions in the article) agree that a function consists of a domain, a codomain and something else that associates exactly one element of the codomain with every element of the domain. This "something" may be called a "rule", a "process" or an "assignment", but the exact term is unimportant since these terms are not mathematically defined. More formally, the "something" is a set of pairs whose first elements belong to the domain and the second ones belong to the codomain; this set of pairs having the property that it contains exactly one pair for each element of the domain. The only difference between these various definitions lies in the wording and in the common mathematical terms and notations that are used (set-builder notation, binary relation, Cartesian product, etc.). As the readers of the article may have very different backgrounds, the definition of these auxiliary terms must be recalled, but those terms that are commonly used in sources must be kept in the article, for not confusing people that learnt a definition using these terms. This is what has been done in § Formal definition for binary relation and Cartesian product.

So, there is no reason to continue discussing on the replacement of a "old" definition by a "new" one, when the two definitions are essentially the same. IMO, this thread must be closed. D.Lazard (talk) 10:40, 1 March 2024 (UTC)[reply]

D.Lazard You didn't sign off on the last changes (above summary), but I was able to determine from the change history that you made that entry.Please sign your comments. My answer to your summary is below:

15. The discussion is ongoing - it's too early to summarize it. Especially because:

as I described in the proposal in section "Why old definition is bad?", the old definition is problematic. (I agree that the old definition can be moved to another section, for example, "Alternative Formal Definitions"). No counterarguments have been presented to dispute this.
both definitions differ (the old definition narrows a function down to the graph G from the new definition)
moreover, the new definition is based on simpler concepts than the old definition (no binary relation is used, and also quantifiers directly illustrate the concept of a graph), and readers familiar with the old definition should not have any problems understanding the new one.
furthermore, we must also think about new readers for whom the new definition, due to its simplicity and directness, will be easier to understand.

Kamil Kielczewski (talk) 11:12, 1 March 2024 (UTC)[reply]

Sorry, I forgot to sign. I fixed it with the correct time stamp.

In the article, it is written: A function with domain

X

and codomain

Y

is a binary relation

R

between

X

and

Y

that .... This means clearly than the domain and the codomain are parts of the definition. So, this very long section is devoted only to discuss your wrong interpretation of this sentence, namely that the codomain would not be included in the definition. Your "new" definition is strictly equivalent with the one of the article. So the above wall of text is totally useless, and continuing this way is WP:disruptive editing. D.Lazard (talk) 14:47, 1 March 2024 (UTC)[reply]

No. "A function with domain X and codomain Y is a binary relation R between X and Y that..." - this is a false statement because, by its very definition, a binary relation does not always contain the entire codomain. So we have two possibilities:

either the function f is not a binary relation,
or the function f is a binary relation, but in that case, it cannot contain full information about the codomain Y.

As you can see - the old definition is ambiguous/fuzzy. Kamil Kielczewski (talk) 15:16, 1 March 2024 (UTC)[reply]

jacobolus and D.Lazard I see that both of you are essentially circling around the same theme ("Why old definition is bad?" in my proposal) - therefore, I will give a separate response maybe as a kind of summary of the above discussion.

Quoting jacobolus: "the domain and codomain are intrinsic to the function"

To dispel any doubts, let me provide an analogy: "A Kuratowski pair is a set of the form (a,b)={{a},{a,b}} for any a (named first element) and b (named second element). Then, the function f along with its domain X and codomain Y is a Kuratowski pair such that the first element is X and the second is Y."

We can probably easily agree that the above definition is not good, because such defined Kuratowski pair is an inappropriate, too impoverished structure to store full information about the function - specifically, such defined Kuratowski pair does not store information about the graph of the function. This is obvious - right? And similarly with the binary relation in the old definition - it does not store information about the codomain - hence, it is an incomplete/bad definition (though a mention of it can be placed in a section like "Alternative definitions" as it is a widely used simplified definition - but rather of the function's graph than the function itself).

jacobolus and D.Lazard, you haven't responded to my earlier explanations regarding the fact that a binary relation is too impoverished a structure to fully describe the function f - therefore, I understand that you agree with this and all doubts have been dispelled. If not - then please provide arguments under this comment.

I also remind you that my conclusion concerns the section of the article titled "Formal Definition" - in this context, there must be clarity and distinctness (which the new definition provides) and there can be no accusation of purism here. The formal definition must be formally correct. Kamil Kielczewski (talk) 6:08, 3 March 2024 (UTC)

@Kamil Kielczewski I gave up because it does not seem like you are listening to what other people are telling you here. I don't see any point in repeating the same few comments back and forth ad infinitum. –jacobolus (t) 05:11, 3 March 2024 (UTC)[reply]

In the old definition, a binary relation is unequivocally defined as a set of corresponding pairs - such a set does not contain full information about the codomain - none of you above have shown that this is not the case. You have not provided formal arguments that demonstrate this. Kamil Kielczewski (talk) 05:17, 3 March 2024 (UTC)[reply]

Replying to a message saying you are repetitive and unresponsive by being repetitive and unresponsive is...special.

WP:IDIDNTHEARTHAT may be in play. —David Eppstein (talk) 07:08, 3 March 2024 (UTC)[reply

]

@Kamil Kielczewski You might find the discussion of "dogmatism" vs. "contextualism" in the conference paper Mirin, Weber, & Wasserman (2020) "What is a Function?" to be useful (it turned up in a web search). This paper draws I think an overly sharp distinction between these definitions (while allowing that many authors are ambiguous), but includes: "With dogmatism, we can insist that one of the two definitions is the right definition, argue that textbook writers and other researchers should use this definition, and regard those who do not act in accordance with this definition as being mathematically sloppy or incorrect. [...] The alternative approach, contextualism, is to declare that there is no universal definition of function, but rather that the definition of function depends on context." And goes on to point out various places where it is convenient to ignore or change various features of the function definition to suit the situation. –jacobolus (t) 10:39, 3 March 2024 (UTC)[reply]

jacobolus Your pdf "What is Function" is quite good - thanks for it.

Ok - lets read again last paragraph from your comment from 16:59, 1 March 2024 (UTC)

"Some authors directly state that the function "is" a triple

(X,Y,f)

with

f\subset X\times Y

instead of stating that a function from

X

to

Y

is

f\subset X\times Y

. In practice these two variants are equivalent. (...) The difference is that the "triple" formulation is a bit more cumbersome to read (...)"

No. Your pdf say (page 1157) (Burbaki Triple is proposed "new definition"):

It’s worth emphasizing that a Bourbaki Triple function is a different sort of object than an Ordered Pairs function;

(on page 1158, the examples shown illustrate the significant, structural differences between both definitions.). I'll add that I think the definition of a function is rarely used in practice (except in teaching), rather one relies on some notation, for example,

f:\mathbb {R} \to \mathbb {R} ,f(x)=2x

.

Let this serve as an explanation for you as to why I repeated the same thing again about the binary relation.

But getting back to the topic of your last comment - regarding the use of several definitions in the article - I fully support this idea (as I expressed, for example, in the comment at 6:08, 3 March 2024 - because, as suggested by your excellent pdf:

In the mathematics literature, both definitions of function are common.

However, since the old definition (Ordered pairs) is a reduction of the new definition (Bourbaki) only to the graph - I suggest using the new definition as the main definition in the "Formal definition" section. The old definition could be placed in the "Simplified formal definition" section.

What do you think about this? Kamil Kielczewski (talk) 19:43, 3 March 2024 (UTC)[reply]

This Wikipedia article and many book authors are implying something equivalent to what these authors call the "Bourbaki triple definition" when they phrase their definition like "A function from X to Y is a set of ordered pairs (x, y) ..."; that is, the X and Y are part of the definition. Bourbaki's style is to make sure that everything is as explicit as possible, but it has the drawback of being more cumbersome to state and talk about. (We've already gone through this repeatedly, so you don't need to repeatedly insist that my claim above is wrong. After this I'm about done with this.)

A stricter "ordered pairs" definition per se would be something more like "A function is a set of ordered pairs with no duplicated first elements." Full stop, with no X or Y involved. Under such a definition, of which some examples can certainly be found in the literature, the domain is not a separate part of the function's definition but is implicit as the set of first elements, and the codomain is not part of a function at all, only the image, which is also implicit. –jacobolus (t) 20:02, 3 March 2024 (UTC)[reply]

@jacobolus I'm afraid that ""A stricter 'ordered pairs' definition (...) Full stop, with no X or Y involved"" and the old definition (binary relation) are one and the same.

In other words, there is no such thing as two definitions based on the set of pairs (x,y). Therefore, one cannot say that there is a definition of the set of pairs containing X and Y, and a separate definition of the set of pairs not containing X and Y. It is one and the same definition - there are not two.

I know this might be a cognitive shock for you, as until now you thought it was different.

But try to do some research in this direction to see for yourself. Kamil Kielczewski (talk) 20:45, 3 March 2024 (UTC)[reply]

You wrote jacobolus and D.Lazard, you haven't responded to my earlier explanations regarding ... - therefore, I understand that you agree with this. Your understanding is definitively wrong. I wrote above that I strongly disagree with everything you could write on this subject, and we both have written that we stop discussing on this subject. If you continue this way, I'll ask administrators to block you for WP:Disruptive editing. D.Lazard (talk) 16:42, 3 March 2024 (UTC)[reply]

@D.Lazard Stop spreading falsehoods, and misleading readers!

Below, I will correct what you have made up:

I created two talks on the same topic: Formal Definition. This second talk stems from the first (as I mention in the "Notes" section) and is a continuation of it. In the first talk, in comment at 11:06, 27 February 2024 you wrote:

As said above, I disengage from this disruptive discussion.

But in the second thread on the same topic, you added your pseudo-summary, thus re-entering the discussion (contrary to your previous declaration) in the comment at 10:40, 1 March 2024

All the definitions referred to in this discussion (...)

In the last comment, you wrote

we both have written that we stop discussing on this subject

Which is a complete falsehood because I never wrote anything like that. Kamil Kielczewski (talk) 20:30, 3 March 2024 (UTC)[reply]

Sorry, "we" referred to Jacobolus and me. I thought that it was clear from the context. Also note that after having written "I disengage", I have not discussed the mathematical content of your posts. I have only discussed your disruptive behavior, especially when you misinterpreted my disengagement as an approbation (although I wrote that I disagree with everything that you can write on the subject). D.Lazard (talk) 21:48, 3 March 2024 (UTC)[reply]

Moreover, such a way of conducting dialogue - quoting your commen (16:42, 3 March 2024) :

"I strongly disagree with everything you could write on this subject"

Is unpleasant, hostile, and certainly not substantive - because it refers to the person and not the content.

You keep accusing me of disruptive editing - but in reality, it's you who are behaving that way. Kamil Kielczewski (talk) 22:26, 3 March 2024 (UTC)[reply]

@D.Lazard – how would you feel if we amended the 'formal definition' section from the current version:

A binary relation between two sets

X

and

Y

is a subset of the set of all ordered pairs

(x,y)

such that

x\in X

and

y\in Y.

The set of all these pairs is called the Cartesian product of

X

and

Y

and denoted

X\times Y.

A function with domain

X

and codomain

Y

is a binary relation

R

between

X

and

Y

that satisfies the two following conditions:

Total relation: $\forall x\in X,\exists y\in Y,\left(x,y\right)\in R$
Univalent relation
: $(x,y)\in R\land (x,z)\in R\implies y=z$

These conditions are exactly the formalization of the above definition of a function.

To instead say something like:

A binary relation

R

consists of three parts, a set

X

called the domain, a set

Y

called the codomain, and a set ordered pairs

(x,y)

such that

x

is an element of

X

and

y

is an element of

Y

, a subset of the set of all such ordered pairs, the Cartesian product

X\times Y.

A function is a binary relation that is:

total, meaning that for each $x$ in $X$ , $R$ includes at least one pair $(x,y)$ for some $y$ in $Y$ , and
univalent
, meaning that each first element of a pair in $R$ is unique, so if $(x,y)$ and $(x,z)$ are pairs in $R$ , then $y=z$ .

–jacobolus (t) 20:48, 3 March 2024 (UTC)[reply]

Question: What's the point of introducing the concept of a binary relation (as a triplet) instead of directly a triplet as in the Bourbaki definition? (not to mention that you might have trouble finding appropriate citations, because I've never seen such a definition anywhere...) Kamil Kielczewski (talk) 21:15, 3 March 2024 (UTC)[reply]

The point would be to be explicit enough to forestall complaints from extremely literal readers. –jacobolus (t) 22:18, 3 March 2024 (UTC)[reply]

Here is an example source, Meyer (2005) "Binary Relations", course notes for MIT 6.042J/18.062J:

A binary relation,

R,

consists of a set,

A,

called the domain of

R,

a set,

B,

called the codomain of

R,

and a subset of

A\times B

called the graph of

R.

–jacobolus (t) 22:27, 3 March 2024 (UTC)[reply]

The triple (X,Y,G) in the Bourbaki definition is sufficiently explicit from a formal point of view. If you claim it is not, justify your assertion.

Meyer in this PDF does not use binary relation to define function - he only notes that a function is a certain binary relation. Similarly, we should use a simple definition of a function and in the "additional information" section, we can add that a function is a special case of such defined binary relation. Just as even numbers are not defined through being elements of some group (group theory) but directly (and possibly, additional information is added that even numbers with addition are a special case of a group).

Adding a binary relation unnecessarily complicates the definition, which may cause, for example, high school students to have difficulty understanding it.

I understand that the definition you propose is your original idea (personal invention) - ? Kamil Kielczewski (talk) 06:14, 4 March 2024 (UTC)[reply]

Jacobolus' proposal is fine, except for an omitted word and a too long first sentence. I would rewrite the first paragraph as:

A binary relation consists of three parts, a set

X

called the domain, a set

Y

called the codomain, and a set

R

of ordered pairs

(x,y)

such that

x

is an element of

X

and

y

is an element of

Y;

the set

R

is thus a subset of the set of all such ordered pairs, the Cartesian product

X\times Y.

If we were at Binary relation, I would add that one gives usually the same name to the relation and to the set of pairs, but this is not useful here, since the name of the relation is not used. D.Lazard (talk) 21:26, 3 March 2024 (UTC)[reply]

Maybe we don't need to say "three parts", but just "consists of a set ...".

I wonder if it would be helpful to explicitly say that relation's name, the symbol R, is routinely used for either the relation or for its graph (or both), and that in practice the relation and the graph are often conflated. Maybe that was said well enough in the previous section. –jacobolus (t) 23:02, 3 March 2024 (UTC)[reply]

A large number of the binary relations I've dealt with on Wikipedia and elsewhere do not have three parts. They are relations where the left and right elements of each pair both belong to a single set. For instance, partially ordered set, preorder, and equivalence relation are all of this type. It would be silly to formalize these as triples and then constrain two of the elements of the triple to be the same. —David Eppstein (talk) 03:16, 4 March 2024 (UTC)[reply]

The same is often true of functions, but maybe more confusing to mention for binary relations. As an alternative, what if we put binary relation after the definition of function. Something like:

A function consists of a set

X

called the domain, a set

Y

called the codomain, and a set

f

of ordered pairs

(x,y)

such that

x

is an element of

X

and

y

is an element of

Y

, with each element

x

of

X

appearing exactly once as the first element of a pair. The set

f

is called the graph, and is often conflated with the function; sometimes a function is defined by its graph alone.

The function's graph is a subset of the set of all ordered pairs

(x,y)

with

x

in

X

and

y

in

Y

, the Cartesian product

X\times Y

.

A function is a special type of binary relation, an object defined in the same way as a function but whose graph is any subset of

X\times Y

, without the restriction that each element of the domain should appear exactly once.

Maybe this is more awkward though. And I didn't work 'total' or 'univalent' in. How important are those keywords to include here?

@David Eppstein I'd be glad to see you take a crack at copyediting the first few sections of this article, as I feel there are some places that are a bit awkward or could be tighter (or could be deferred, moved to a footnote, or scrapped), and you usually end up leaving prose pretty clean after you work through it. –jacobolus (t) 06:18, 4 March 2024 (UTC)[reply]

Your definition is starting to closely resemble Bourbaki's definition (and the one in my proposal), with additional informations. This is good. I would also explicitly add that a function is a triplet (X,Y,f) - exactly as Bourbaki defines it. There is no need to "reinvent the wheel" (and you will find many quotations with the triplet). Kamil Kielczewski (talk) 07:30, 4 March 2024 (UTC)[reply]

Yeah, and I think this version is probably worse than the version in the article already. But it's worth spitballing to see what ideas turn up. Writing clearly for a general audience of nonspecialists is hard, and Bourbaki in particular was quite bad at it. –jacobolus (t) 07:34, 4 March 2024 (UTC)[reply]

I happen to have the exact opposite impression that Bourbaki's definition is exceptionally clear, simple, and lucid. But that's a matter of personal opinion :) Kamil Kielczewski (talk) 07:38, 4 March 2024 (UTC)[reply]

Summary of the discussion and changes in the article

In the above discussion, as of the last week, there have been no new arguments - so let's summarize:

It was not shown that the surjectivity problem of the old definition (indicated in my proposal - above) does not occur.
No better alternative solution than the one indicated in my proposal was pointed out (the user jacobolus tried to fix the old definition, but ultimately arrived at the Bourbaki version (comment: 06:18, 4 March 2024), i.e., the one from my proposal.
Despite the problem with surjectivity, during the discussion it was indicated that in certain contexts the old definition might still be acceptable - which has been taken into account in the latest change in the article (look on excellent text [[6]] provided by jacobolus).

Therefore, since the current formal definition leads to confusion/misunderstanding among readers (regarding surjectivity) and no better alternative has been presented, I have taken the liberty to introduce a change in the article that addresses this issue - HERE. If you have ideas for further improvements, I invite you to substantive discussion and editing. Kamil Kielczewski (talk) 01:10, 11 March 2024 (UTC)[reply]

No, people still disagree with what you have said here. You are mischaracterizing my comments. I don't really want to substantively engage with this further, because I don't feel you are respecting the other participants here. –jacobolus (t) 02:03, 11 March 2024 (UTC)[reply]

@Jacobolus All the arguments you have presented so far regarding the current issue and the proposed solution have been challenged (if not, then point out the specific argument). Therefore, present further substantive arguments and/or constructive solutions. I am not claiming that the current solution is final - but we cannot stick to the problematic old definition.

We also cannot rely on private opinions or feelings, which is why substantive arguments are necessary. I would also add that the current change did not alter the old definition - it merely provided the appropriate context, which remedies the situation. The concept of introducing context was, after all, proposed by you (comment 10:39, 3 March 2024) Kamil Kielczewski (talk) 07:15, 11 March 2024 (UTC)[reply]

There is a clear consensus here that everybody but you disagree strongly with your arguments. Despite this, you tried to insert your personal view in the article. This goes against the fundamental rules of Wikipedia. This is the reason of my revert of your edit. D.Lazard (talk) 10:21, 11 March 2024 (UTC)[reply]

Also, you are asking for substantive arguments aginst your views. Many have been given and repeated here. Apparently you are not willing or not able to

hear them. D.Lazard (talk) 10:36, 11 March 2024 (UTC)[reply

]

But this isn't about agreeing or disagreeing - that falls within the realm of private opinions. One must present factual arguments.

"Despite this, you tried to insert your personal view in the article" This is untrue - because I provided the appropriate citations and arguments in proposal. It is exactly the opposite of what you write - you are the one forcefully pushing your own vision of this article even though it is incorrect (as demonstrated in the proposal) and you do not allow for the improvement of the article's quality. This is against the rules of Wikipedia.

"Many have been given and repeated here. Apparently you are not willing or not able to hear them"

That is again not true, because for every argument, I presented the appropriate counterarguments that showed their incorrectness.

And if someone, due to a lack of factual arguments, has nothing to reply with - and simply "disagrees because they feel like it" - then it goes beyond a substantive discussion. Mathematics is not a science about feelings. Kamil Kielczewski (talk) 10:55, 11 March 2024 (UTC)[reply]

I agree that, here, there is no place for feelings. Nevetheless, you feel that your conception of mathematics is not only correct, but also the only correct one; you feel that your arguments are correct even when they are challenged by others; you feel that other's arguments are feelings even when they are factual; you feel that a "substantive discussion" consists for you to repeat the same arguments at nauseam, and for others to accept your repeated arguments. Please, read and try to understand

WP:ICANTHEARYOU. D.Lazard (talk) 11:31, 11 March 2024 (UTC)[reply

]

"you feel that your conception of mathematics is not only correct, but also the only correct one;" - This is not true. The evidence for this is that in the article change, I kept the old definition (which I initially proposed to remove completely) - taking into account the arguments given by the other disputants (in the form of links to literature).

"you feel that your arguments are correct even when they are challenged by others" - This is not true. Well, specifically - point out which arguments (except keeping the old function) from the proposal, considering the above summary, were right challenged by the disputants - indicate specific comments. Or stop spreading falsehoods.

This is mathematics - here it is usually easy to verify the validity of the argument. Kamil Kielczewski (talk) 11:56, 11 March 2024 (UTC)[reply]

In mathematics, it is easy to verify the validity of a proof, but none of your arguments are proofs, except some that are based on a wrong interpretation of the definitions given in the article. D.Lazard (talk) 12:23, 11 March 2024 (UTC)[reply]

You were supposed to provide specific comments - you are not doing so - which means you admit to spreading falsehoods. Such statements undermine your credibility.

And what about the literature I quote in the article revision? According to you, the authors also misunderstand that the old definition (based on pairs/relations) is reduced to a concept equivalent to a graph in Bourbaki's definition, and that there is a problem with defining surjectivity in the old definition? Kamil Kielczewski (talk) 12:43, 11 March 2024 (UTC)[reply]

Total reset needed for this article

Outsiders watching the style of this discussion would probably think about a pond of frogs. When one person replies to another, three more jump in between; others exhibit adolescent behavior in explicitly refusing even to hear any technical arguments; still others wield Wikipedia rules instead of logic. All this is only to say that a total reset is urgently needed, and that restraint is needed to avoid derailing the entire project. I will set the example in exercising restraint, and add a brief note only when logic is all too severely violated. Otherwise I just observe the proceedings...

In any case, can we at least agree on the following points?

- The article should primarily serve the readers, not some personal hidden agenda.

- The current state of the art requires (at least) two conceptually different definitions.

- The simplest of these would be the best starting point.

- Any definition should be referenced accurately (immediately verifiable, no misquotes). The reason for insisting on this last point is not some bureaucratic consideration but the hope that in the literature of the past hundred years there is at least one formulation people are prepared to quote literally, even if it does not fit their own dream definition. Functions are not rocket science but high school material. Boute (talk) 05:14, 1 March 2024 (UTC)[reply]

You should try to make the same point without the insulting preface. Comparing people to frogs and calling them children is not an exemplary "exercise in restraint". –jacobolus (t) 15:47, 1 March 2024 (UTC)[reply]

No insult, but a general observation in empathy with potential outsiders, and apparently an effective wake-up call. Restraint here obviously means making only contributions that improve the paper. Look at the pages and pages wasted on an high-school level issue. What would outsiders think? Boute (talk) 19:41, 1 March 2024 (UTC)[reply]

I'm not clear what you're hoping for. I guess you want someone to answer the complaint politely one time and thereafter just let the critic talk to themselves / shout into the void without reply? –jacobolus (t) 20:27, 1 March 2024 (UTC)[reply]

Re "high-school level issue": a big part of the problem here is that this is not (only) a high school level issue. We need to describe functions at a level that can be understood by high school students, which is a high school level issue. But we also need to describe functions with the understanding gained from research into the foundations of mathematics, in set theory (well covered in this discussion), in type theory (not so well covered), and maybe also with some discussion of constructivism (currently not covered at all), at a level far above high school mathematics. The need to treat these things at multiple levels significantly adds to the difficulty of formulating an article that anyone interested in only a single level can be satisfied with. —David Eppstein (talk) 21:35, 1 March 2024 (UTC)[reply]

That's exactly why I suggested several non-equivalent definitions. However, the discussion got stuck at high school level issues. This must be resolved first before moving to the next level. Boute (talk) 12:12, 2 March 2024 (UTC)[reply]

To substantively address your comment though: I think everyone wants the article to serve readers (in particular including high school students and early undergraduates). I agree with you that it would be nice to immediately reference one or several sources about any definition(s) here, though I don't think an exact quotation is necessary to insert inline into the text, and making basic material look like a quotations can even be kind of confusing to readers (exact quotations in footnotes can be helpful though). –jacobolus (t) 16:37, 1 March 2024 (UTC)[reply]

Please don't distort clear messages. Good intentions need constant reminders. Evidently basic material should not look like quotations, but in a Wikipedia article a definition should be traceable literally to some source, because not observing this principle is precisely the cause of the endless repetitive discussions we are witnessing here. Don't believe that the literature of the past hundred years does not contain excellent candidate definitions (examples found by opening the hidden text box at the start of this discussion page). Enough generalities, make a solid proposal and submit it for discussion. Boute (talk) 20:19, 1 March 2024 (UTC)[reply]

Is the hidden "Outline for a new version of the article" supposed to be text to go directly into the article? I personally find it to be incredibly confusing and distracting, and not written in anything like the encyclopedic "house style" I expect for Wikipedia articles. I don't think it's suitable to go into a Wikipedia article as-is, but perhaps contains some useful ideas which could be turned to productive use; the sources themselves seem fine to cite. –jacobolus (t) 20:36, 1 March 2024 (UTC)[reply]

Logical error: it was clearly stated that the outline was meant as a resource to the other editors, and that integration into an article was left to them. Boute (talk) 02:26, 2 March 2024 (UTC)[reply]

Fair enough, and thanks for answering the question. I don't personally see a clear way to do that, and even each portion is rewritten as flowing encyclopedic prose, I find the organization confusing. Maybe someone else has concrete ideas of how to write some section(s) based on that, but I don't think you can reasonably expect someone to appear to do such translation for you. –jacobolus (t) 03:37, 2 March 2024 (UTC)[reply]

Your outline cites Apostol's Calculus. Here is what Apostol says:

The word "function" was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. It was later realized that Leibniz's idea of function was much too limited in its scope, and the meaning of the word has since undergone many stages of generalization. Today [1960s], the meaning of function is essentially this: Given two sets, say

X

and

Y,

a function is a correspondence which associates with each element of

X

one and only one element of

Y.

The set

X

is called the domain of the function. Those elements of

Y

associated with the elements in

X

form a set called the range of the function. (This may be all of

Y,

but it need not be.)

Letters of the English and Greek alphabets are often used to denote functions. The particular letters

f,

g,

h,

F,

G,

H,

and

\varphi

are frequently used for this purpose. If

f

is a given function and

x

is an object of its domain, the notation

f(x)

is used to designate that object in the range which is associated to

x

by the function

f,

and it is called the value of $f$ at $x$ or the image of $x$ under $f.$ The symbol

f(x)

is read as "

f

of

x

."

The function idea may be illustrated schematically in many ways. For example, in Figure 1.3(a) the collections

X

and

Y

are thought of as sets of points and an arrow is used to suggest a "pairing" of a typical point

x

in

X

with the image point

f(x)

in

Y.

Another scheme is shown in Figure 1.3(b). Here the function

f

is imagined to be like a machine into which objects of the collection

X

are fed and objects of

Y

are produced. When an object

x

is fed into the machine, the output is the object

f(x)

.

Although the function idea places no restriction on the nature of the objects in the domain

X

and in the range

Y,

in elementary calculus we are primarily interested in functions whose domain and range are sets of real numbers. Such functions are called real-valued functions of a real variable, or, more briefly, real functions, and they may be illustrated geometrically by a graph in the

xy

-plane. We plot the domain

X

on the

x

-axis, and above each point

x

in

X

wit plot the point

(x,y),

where

y=f(x).

The totality of such points

(x,y)

is called the graph of the function.

Leaving aside definitional disagreements about the range of a function, this seems overall like a nice introduction to me, which could quite plausibly be cited here and recommended to readers as a source to read. –jacobolus (t) 20:56, 1 March 2024 (UTC)[reply]

Possibly a good start, but which edition of Apostol's Calculus are you using? The second edition (Wiley, 1967) literally states on page 53:

" A function f is a set of ordered pairs (x, y) no two of which have the same first member. "

and goes on (same page) defining domain and range as, respectively, the set of first and second members.

Did Apostol (in your assumed quotation: edition? page?) really say in one paragraph that the range need not be all of Y and later on call Y the range? Sorry, but as this stands, it cannot be plausibly cited. Boute (talk) 02:53, 2 March 2024 (UTC)[reply]

This is the 2nd edition of Apostol, from a couple pages before the part you are quoting; it's essential context for the "formal" version, considering the intended audience of ~18 year olds. –jacobolus (t) 03:38, 2 March 2024 (UTC)[reply]

An informal paragraph on p. 51 in Apostol's Calculus indeed contains this error (a typo? better not speculate), which makes it unsuitable here. A Wikipedia article should contain no logical errors, a fortiori in an introductory context because setting misconceptions straight afterwards is always more difficult. As for the intended audience of ~18 year olds, I don't know whether this is a Wikipedia rule, but it is a reasonable working hypothesis. We also must assume a minimum of interest in mathematics (otherwise, why read the article?). In Apostol's time (1960s) set theory was unknown in most high schools. Nowadays this has changed, and Apostol's formal definition on p. 53 is perfectly accessible to that audience. After all, the first paragraph you quoted (from p. 50) assumes the reader knows about sets. The formal definition has the extra advantage of also appearing in other textbooks (e.g. Flett), and is less prone to the misconceptions caused by prematurely mentioning the set Y (hence the warning by Flett). So I think your suggestion to follow Apostol's path is worth pursuing. Selectivity is important. Boute (talk) 09:05, 2 March 2024 (UTC)[reply]

In other words, we should skip the gentle explanatory prose and cut straight to something unmotivated and inscrutable, so as to maximize novice readers' confusion? No thanks. –jacobolus (t) 09:40, 2 March 2024 (UTC)[reply]

"In other words"? Logic violation alert! Motivation is essential for any definition (even for an advanced audience, perhaps more). It must be done carefully, based on a lot of experience with and thought about how various different people think about mathematics. Boute (talk) 11:43, 2 March 2024 (UTC)[reply]

Sorry, that comment of mine was more dismissive than necessary, but let me try to be precise: I don't think we can transplant Apostol's discussion directly, but for his context of a calculus book I like the way he leads off with (1) a very brief history of the word (2) a mention that a function can be thought of as a formula, a machine, or a pairing, (3) some discussion of notation setting up common letters used later in the book, (4) a comment that functions might involve any objects but in calculus are usually real numbers, (5) [not quoted above] a couple of pages of concrete examples before trying to make the "formal" definition. These features let a reader who is not too familiar with functions ease into the idea.

It seems to me that you want to skip Apostol's 3 pages of leading context and jump straight to the "formal definition" part. That seems like a big mistake to me. –jacobolus (t) 16:14, 2 March 2024 (UTC)[reply]

This looks like a good plan, which most people will endorse. Apt considerations on the importance of justifying definitions can be found in Rogaway's paper [7]. I'm sorry for not being able to contribute anything to this article for several months (as explained after Lazard's post below). The resources and references under the "green slab" at the start of this talk page may suffice for now, but if any additional resources seem useful I can be reached by email. Boute (talk) 17:24, 2 March 2024 (UTC)[reply]

Chiming in as an outsider - this article is incredibly hard to read, and really serves no one but the maths elites.

I think the first thing that should be addressed is explaining "functions" in simple terms, without assuming people know what "of a set x to a set y" even means. This reads like mathematicians trying to one-up each other, not like an encyclopedia that is useful for anyone.

Here's my basic attempt, as a math-hating programmer : "A function is an equation that transforms a value (x) into another value (y), or describes X's relation to Y. A function of x {f(x)} is an equation including X, which can be assigned a value and solved. The solution to that equation, for a given x value, is often considered Y."

It may lack nuance that people with mathematics degrees want, but should the intro really be written to appease them? They already know what functions are. 47.55.178.193 (talk) 23:51, 22 March 2024 (UTC)[reply]

If it were true "serves no one but the maths elites". It definitely does not. Believe me. It is an embarrassment of an article. The article has a really really hard time defining precisely "of a set x to a set y". In my opinion, one reason why it is hard to read, is because it does not define this as explicitly as possible. Instead, the article makes one, two, three failed attempts at definition, that are not definitions, before it finally reaches a definition that is okay. No wonder it is hard to read. Nothing more confusing than an explanation that meanders on and on and on before getting to the point and missing it. Thatwhichislearnt (talk) 11:31, 23 March 2024 (UTC)[reply]

Keep up the good work, Jacobolus. I have nothing to add today, but two things have become clear. 1) Your views are the standard, referenced views. And 2) The people pushing their own views will argue 'til the cows come home. Rick Norwood (talk) 10:59, 2 March 2024 (UTC)[reply]

Fully agree, that's why I always insist on definitions that are literally traceable to reliable sources. Boute (talk) 11:46, 2 March 2024 (UTC)[reply]

I agree also with Jacobulus. As Wikipedia is not a text book, we must follow the dominant practice in mathematics and not the pedagogical simplifications that appear in some textbooks. Moreover, the ambiguity of the term "

image of a function" and "codomain

". We must not go back to the old ambiguity, which is induced when the codomain is not included in the definition of a function, at least because this would imply the rewrite of many Wikipedia articles. However, there are two cases where one considers functions whose domain (and codomain) are not well defined. The presentation of these two cases requires to be improved.

A subsection "Partial functions" of § Definition must be added, which should be a short summary of Partial function, and must say that in some contexts, typically calculus, "function" is commonly used for "partial function". Otherwise, reader may believe that the multiplicative inverse of the real funtion $x\mapsto x$ would not be a function.
A section must be added on the definitions of a function that are not based on set theory. These definitions are mainly used in
constructive mathematics, and do not include the domain and the codomain because the concept of a set is lacking. Examples of such definitions are lambda calculus and general recursive functions
. Personally, I do not know how functions are defined in constructive mathematics. The addition of such a section would imply to rewrite several sections of the article.

D.Lazard (talk) 12:22, 2 March 2024 (UTC)[reply]

In Bishop's masterpiece Constructive Analysis, a function from A to B is defined as a rule assigning to each a in A a unique f(a) in B, with the extra (constructivist) condition that "the rule must afford an explicit, finite, mechanical reduction from the procedure for constructing f(a) to the procedure for constructing a." The quoted part might be streamlined with the terminology from algorithmics. Boute (talk) 12:44, 2 March 2024 (UTC)[reply]

Since another deadline is coming up and I already exceeded the time I allotted myself for Wikipedia, Il sign off now, and will not be giving "replies" or input for several months. At this time I have nothing significant to add to the collection of resources and references I compiled a year ago at the start of this talk page. I wish everyone good progress in editing this article. Boute (talk) 17:02, 2 March 2024 (UTC)[reply]

Partial functions

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)[reply]

1. I do not see how partial functions would explain or solve anything in the above two threads "Formal definition ..." (e.g., the problem of surjectivity). Please clarify this.

2. You did not provide any citations

3. The 'Partial functions' section can remain in the article. It constitutes a separate thread." Kamil Kielczewski (talk) 06:12, 6 March 2024 (UTC)[reply]

§ Partial functions is not a subsection of § Formal definition; it is a subsection of § Definition, at the same level as § Formal definition and § Multivariate functions. D.Lazard (talk) 11:31, 6 March 2024 (UTC)[reply]

Riemann Hypothesis and partial functions

The section Partial Functions which has no source, contains this paragraph:

"Similarly, a function of a complex variable
is generally a partial function with a domain of definition included in the set

\mathbb {C}

of the

complex function is illustrated by the multiplicative inverse of the Gamma function

: the determination of the domain of definition of the function

z\mapsto 1/\Gamma (z)

is equivalent to the proof or disproof of one of the major open problems in mathematics, the Riemann hypothesis."

I think that it means the
Gamma Function
.
Also the meaning of the phrase : "the determination of the domain of definition" of the function $z\mapsto 1/\zeta (z)$ is not clear to me and also then meaning of "equivalent to the proof or disproof of" the Riemann hypothesis.
For example, does it mean that if we prove that the R.H. is false (by giving a counterexample), then we can (or cannot) "determine" the domain of definition of the function $z\mapsto 1/\zeta (z)$ "?

--Cbigorgne (talk) 17:38, 13 March 2024 (UTC)[reply]

Thank you for pointing this. I'll fix this. for the second point, I'll simply replace "equivalent" with "more or less equivalent". I think that it is a good way for making clear without entering in technical details that it not a true equivalence. D.Lazard (talk) 19:19, 13 March 2024 (UTC)[reply]

Formal definition (new version)

It seems that the above endless discussions were motivated by the fact that it was unclear for some readers whether the domain and the codomain belong to the definition of a function; also, some readers could be confused by the unmotivated introduction of relations. Therefore, I edited section § Formal definition for clarifying these two points. From now on, I suggest to discuss the new version only. D.Lazard (talk) 11:58, 13 March 2024 (UTC)[reply]

There are two attempts at definitions at this moment, in the section Formal Definition. Here is one observation on why the first of them is not yet a well-formed definition and it is not the same as the second. The second is fine.

The first attempt, begins "A function with domain X and codomain Y", but does not defines what "with [...] codomain Y" means for the object (the function) that this predicate is applied. That attempt then continues "is a binary relation R between X and Y", which defines the object as a relation ("is a binary relation"). This leaves the predicate "with codomain Y" still undefined. There is one way to understand the "with codomain Y" from "is a binary relation R between X and Y" and that is by interpreting the relation as an "embedded relation", the relation together with the inclusion in

X\times Y

. But let's face it, how many think automatically of relations as embedded? Interpreting it this way is just delegating the question to whether one understands relations as remembering their codomain.

The second definition is good. Now, one observation. By writing it, it means the article has sided with "functions know its codomain". While I personally tend to favor that one, it is also true that that is not what everybody does. There are authors that define functions as the relation alone. Someone above was reminding of Wikipedia needing to be balanced. In that light, something should be said. Thatwhichislearnt (talk) 13:30, 13 March 2024 (UTC)[reply]

Firstly "with codomain" is not a predicate, since a predicate must be a complete sentence or a well-formed formula; it is a

qualifier

that makes the definition more specific. Thus the sentence means that the sets that qualify the function are the same that qualify the relation. Indeed the definition of a binary relation involves two sets explicitly. Maybe some sources define "non-embedded" relations, that is, define relations betweeen arbitrary mathematical objects; this is not defined this way in Wikipedia or any source that I know. Moreover, this may cause many errors. For example, "equality" does not refer to a "non-embedded relation" since

x^{2}-y^{2}=(x-y)(x+y)

is true when "=" is considered as a relation between polynomials, and false when considered as a relation between expressions. D.Lazard (talk) 15:02, 13 March 2024 (UTC)[reply]

A qualifier has a corresponding predicate that is implied, in this case "x is with codomain Y". And exactly as I said, what the first attempt at definition does is pass the ball to whether relations carry codomain or not. They do not. They do not when you talk about the relation itself. They could, when the language de facto emphasizes talking about the relation embedded in a Cartesian product, like when saying "a relation over". Your example on that equation is saying nothing. The = in expressions is a symbol of a language and not a relation, until interpreted semantically. Also, while everyone abuses languages and writes that equation alone, there are implied quantifiers on the variables and restricted ranges over which those variables are quantified. To write informally one skips them, but to write formally one should add them. If one is going to think of a relation, there is no need for the embedding to know that the relations are different. The elements of one version are pairs of say real numbers, while the other are pairs of expressions. The relation alone is sufficient to tell. The only confusion in writing only that equation is that it hasn't specified any relation in particular. Thatwhichislearnt (talk) 15:34, 13 March 2024 (UTC)[reply]

Regarding "Maybe some sources define "non-embedded" relations, that is, define relations betweeen arbitrary mathematical objects; this is not defined this way in Wikipedia or any source that I know.". There is some confusion here. Here is a relation defined by comprehension

R=\{(x,y)\in A\times B\mid \phi (x,y)\}

. While the comprehension uses

A\times B

, the relation is the set

R

. The set R does not necessarily remember A or B, just like it doesn't remember multiple elements, or order, or anything that is not a property of objects of Set. When I say "non-embedded" is only to emphasize that, as opposed to using phrases like "a relation from A to B" or similar, that are implicitly (informally) talking about the triple (A,B,R). Note also, that the relation R does not change, if in the comprehension one uses larger

A',B'

for which no new elements satisfy

\phi

. It is the same relation, it is not the same triple (A', B', R).

We are talking about a section called Formal Definition. While the section doesn't need to be written completely formally, discussing about it we need to keep in mind what is the actual formalization that the English translates to. Thatwhichislearnt (talk) 16:11, 13 March 2024 (UTC)[reply]

A relation should also be defined to "remember" A and B, and should parallel the definition of function, so that "a function is a relation ..." remains a true statement. That was one thing I was getting at with attempts above. –jacobolus (t) 18:23, 13 March 2024 (UTC)[reply]

I totally agree with consistent definitions. But again, note that that should be explicit. Not everyone does that. Of the top of my head, I don't remember anyone doing that (the triple thing) for relations. I am not doubting the existence of such sources. I will need to check sources, and the Wikipedia article(s) on relation and binary relation would need those sources too. I have only seen relations being the set of pairs/tuples. Thatwhichislearnt (talk) 19:02, 13 March 2024 (UTC)[reply]

One that I have at hand is Halmos' Naive Set theory. In pages 26 and 27 you can see him being explicit about the distinction between considering a "relation" versus "a relation from ... to ..." (see end of page 27). Halmos counts among those who define functions as just the relation (and thus the graph) and relations as only the set. Thatwhichislearnt (talk) 19:16, 13 March 2024 (UTC)[reply]

Having separate definitions for "partial function" (with underspecified domain) vs. "function" (with unspecified codomain) vs. "function from X to Y" seems excessively pedantic to me, but I suppose could be described that way, with the note that even when not specified the word "function" still usually means "(partial) function from X to Y".

As for sources with this definition of relation, I'm not sure about textbooks but above I linked some course notes: Meyer (2005) "Binary Relations", notes for MIT 6.042J/18.062J:

A binary relation,

R,

consists of a set,

A,

called the domain of

R,

a set,

B,

called the codomain of

R,

and a subset of

A\times B

called the graph of

R.

–jacobolus (t) 19:59, 13 March 2024 (UTC)[reply]

I don't think one would be able to prove the "usually". I believe Wikipedia has some sort of policy/guidance of avoiding such claims regarding frequency, unless supported by a reliable source that makes such a claim. Again, count me among those who prefer having codomains, but lots of people also don't do that. I think more appropriate is just say that that "is the definition used for the rest of this article". Thatwhichislearnt (talk) 20:06, 13 March 2024 (UTC)[reply]

@Thatwhichislearnt I completely agree with you - both definitions (of triples - with codomain) and of the set of pairs (without codomain) - due to their ubiquity, should be described (which I did in in reverted change ). Kamil Kielczewski (talk) 19:07, 13 March 2024 (UTC)[reply]

I reverted your edit because you not provide appropriate citations and it is a "Personal invention". In article you wrote: "A function is formed by three sets" - You merged Bourbaki's version (HERE - which you arbitrarily withdrew witch comment "Here is a clear consensus on the talk page against this change" - without asking all discussion participants) with the old version of the definition. And I assume that this is the reason why you couldn't find appropriate citations.

In the edit you arbitrarily withdrew - there was a source (even several) (here) describing two commonly used definitions: Bourbaki's (triple) and the set of pairs (which have significant structural differences) - you created a third one by combining these two. Kamil Kielczewski (talk) 18:09, 13 March 2024 (UTC)[reply]

@David Eppstein I see that you have reverted (19:01, 13 March 2024) what I did (17:57, 13 March 2024) with the comment "tendentious" - please explain how this relates to the issues I have highlighted, namely, lack of sources and "personal invention" - unless you are in some kind of conspiracy with D.Lazard. Kamil Kielczewski (talk) 22:17, 13 March 2024 (UTC)[reply]

Please refrain from wildly speculative and unlikely ad hominem conspiracy theories. The explanation is (1) "tendentious": you keep repeating the same fixed idea of there being only one possible correct way of defining functions, and responding to any attempt to bring you into a wider view by repeating the same thing again.

WP:IDIDNTHEARTHAT. (2) "technical": this is supposed to be an article that can, to the extent possible, be read and understood by high schoolers. Formulating things as pedantically as possible using sets doesn't achieve that. —David Eppstein (talk) 00:34, 14 March 2024 (UTC)[reply

]

@David Eppstein You did not answer the asked question: what about the lack of citations and "personal invention" in the changes made by D. Lazard ?

The sentence "technical: ..." explains nothing - it is irrelevant to the question. Moreover, D. Lazard's formal definition is more complicated than the Bourbaki definition (reverted here) because it uses the redundant concept of a binary relation, so the argument about better accessibility for high school students is misguided. But this is a separate issue - and it has nothing to do with the question I asked and which you are avoiding answering.

P.S.1: "there being only one possible correct way of defining functions" I'll add immediately that you are seriously mistaken - because in this change, I explicitly added a separate "Reduced function definition" section with an appropriate introduction and preserved the old definition. So, there are two, not one as you claim.

P.S.2 Accusing formal mathematical definitions of being too pedantic is absurd. It's like faulting poetry for having multiple meanings. (not to mention that D. Lazard in his innovation also tried to use precise language - but I see that you are biased and fail to notice that.)

P.S.3: Your attempt to dodge the answer and responsibility by changing the subject is transparent. Such eristic tricks are contrary to substantive discussion. Kamil Kielczewski (talk) 06:47, 14 March 2024 (UTC)[reply]

@D.Lazard, besides the fact that you didn't provide citations and your proposal is a "personal invention" (as I demonstrated in the comment at 18:09, 13 March 2024 (UTC).) I've also noticed that it contains an internal contradiction. Specifically, at one point you define a function as a binary relation, that is, a set of pairs ""a binary relation between two sets X and Y is a subset of the set of all ordered pairs (...) A function... is a binary relation,"" and at another point, you state that ""A function is formed by three sets."" Therefore, there is a contradiction because on one hand, "a function is a binary relation so it is a set of pairs," and on the other hand, "a function is three sets X, Y, R." - contradicition! Kamil Kielczewski (talk) 08:04, 14 March 2024 (UTC)[reply]

Even though he is doing the reverse (introducing functions and relations that do not know codomains), you can see how Halmos still addresses the semantics of "relation from X to Y" and "relation in X" versus just "relation" in pages 26 and 27 of his Naive Set Theory. Perhaps it can be seen online in google books here, for those who don't have the book. He is still using (almost only) plain English, yet being formally precise. Note how to do give meaning to those two, the inclusion needs to be mentioned: "If R is a relation included in ..." Whether wanting to introduce functions with codomain or not, I think it is worth imitating the level of precision in Halmos' (which is again, is written in pretty much plain English). Thatwhichislearnt (talk) 19:58, 13 March 2024 (UTC)[reply]

Please don't revert without discussing first. If you don't understand how to write a definition, just recuse yourself from the article. The current imitates masters of exposition, like Halmos. There is nothing undefined there. Thatwhichislearnt (talk) 12:12, 14 March 2024 (UTC)[reply]

Note that is it not true that X and Y are undefined. They has been mentioned before. Compare with what you had written, in which "with domain X and codomain Y" was indeed undefined. Not X, not the Y, the "with" was undefined. Thatwhichislearnt (talk) 12:15, 14 March 2024 (UTC)[reply]

Whether X and Y are undefined

This section is to keep the discussion focused. User Lazard questioned whether X and Y are undefined in this version. I claim that they have been defined in the paragraph above the definition, while this version does leave the "with" in "with domain X and codomain Y" undefined (specially for the codomain Y, the domain can be strictly speaking recovered). There is essentially no difference in content and intension between the two versions, only this point. For comparison, refer to Halmos' Naive Set Theory, pages 26 and 27 (end of page 27) for how the "with" is given meaning. Thatwhichislearnt (talk) 12:33, 14 March 2024 (UTC)[reply]

I have only one question: why don't we use two simple, well-known definitions: Bourbaki's and the set of pairs for which there is abundant literature (and which are simple)? Instead, we're trying to forge our own new (complicated) definition based on D. Lazard's personal invention? Why "reinvent the wheel"? Kamil Kielczewski (talk) 14:10, 14 March 2024 (UTC)[reply]

The shield of keeping this Wikipedia simple. And that is fine. The definition can be done in any language. It can be done as close to English as any wants. It is better not to digress. Focus, in this section, on discussing whether X and Y are undefined (any more or any less than they are in the other version). And whether the other version has "with" still undefined. Thatwhichislearnt (talk) 14:16, 14 March 2024 (UTC)[reply]

And let me make it even simpler. When you define a new set, say by (restricted) comprehension $F=\{x\in A\mid \phi (x)\}$ , the set $A$ needs to be known to be a set from before, it is assumed to have been defined before. The $F$ is the one being defined in the comprehension. Likewise, when defining a function (which in set theory foundations, like everything, is a set) we have previously defined sets X and Y and from them one defines f. There, is never a question whether X and Y are defined. One could have had the same unfounded concern in this inadequate version. All the discussion(s) above, are not regarding X and Y themselves, but whether the meaning of "f with codomain Y" has been defined or not. That inadequately written version does not define "f with codomain Y" (in the first of its purported "formal definitions"). Thatwhichislearnt (talk) 13:08, 14 March 2024 (UTC)[reply]

A definition must be mathematically consistent. If a definition implies

X

and

X

without saying what they are, it cannot be consistent. More precisely, in a clause "something is defined as some_description", the only

free variables that could appear must appear in both "something" and "some_description". As your edit does not follow this basic principle, it make your definition nonsensical, and cannot be accepted in Wikipedia. D.Lazard (talk) 13:37, 14 March 2024 (UTC)[reply

]

In

\{x\in A\mid \phi (x)\}

the

A

is not a free variable. Likewise, the X and the Y in the definition are not. They are defined, they are some sets, that (you) in the previous paragraph(s) have named. If your "objection" were valid, your version would have exactly the same problem. The objections about your version are not even that. Have you already understood that the "with" is what is undefined? Thatwhichislearnt (talk) 13:46, 14 March 2024 (UTC)[reply]

And stop reverting when you have trouble understanding set theory. This is so basic that it should make you ashamed. Thatwhichislearnt (talk) 13:50, 14 March 2024 (UTC)[reply]

The onus is on you to get consensus support for your changes. Making personal attacks is not going to help you do that.

MrOllie (talk) 14:06, 14 March 2024 (UTC)[reply

]

Wait, but where was the consensus when he added his version, that does still leave undefined what nearly the entire talk page is dedicated to discuss. My edit, nearly the exact same content he added. He raised a new "concern" that even his edit would have, if it were at all founded. Thatwhichislearnt (talk) 14:11, 14 March 2024 (UTC)[reply]

None of that is a reason for you to edit war and make personal attacks.

MrOllie (talk) 14:15, 14 March 2024 (UTC)[reply

]

Ok. Sufficient digression. Would you like to add to the discussion in this section? Thatwhichislearnt (talk) 14:18, 14 March 2024 (UTC)[reply]

I broadly agree with D.Lazard's criticism of your change. Definitions should be as self-contained and self-consistent as possible. I also find the added paragraph (Viewing the relation...) to be ungrammatical and confusing, it doesn't add anything to the section and should be omitted.

MrOllie (talk) 14:39, 14 March 2024 (UTC)[reply

]

Do you think the "with" in "with codomain Y" has been defined in that version? What that paragraph in green adds is giving meaning to precisely that. Whether ungrammatical or not, while I disagree, I am perfectly happy with the grammar being fixed. Compare with authors that take the care of defining "with codomain Y", like Halmos A large volume of discussion in the talk page, throughout the years has been spent on the definition(s) in the article not properly defining the meaning of the phrase "with codomain Y". That last edit, still leaves it undefined. Thatwhichislearnt (talk) 14:50, 14 March 2024 (UTC)[reply]

@

MrOllie In my comment at 18:09, 13 March 2024 (UTC), I pointed out serious objections to D. Lazard's changes (lack of literature and "personal invention") - so I also do not understand what this consensus is based on since he introduces any changes he wants - and ignores all voices of opposition.He also prevents improvements to the article that are based on literature (e.g. here) - by providing arguments that are irrelevant or contradict the said literature. Kamil Kielczewski (talk) 14:21, 14 March 2024 (UTC)[reply

]

Making more personal attacks in a response to a warning about personal attacks is an unusual tactic. Kindly stop.

MrOllie (talk) 14:25, 14 March 2024 (UTC)[reply

]

What I wrote above was not a personal attack - only opposition to the actions of D. Lazard - it's not the same thing. And it's not unfounded what I write - I have mentioned some arguments. Kamil Kielczewski (talk) 14:33, 14 March 2024 (UTC)[reply]

Keep your 'opposition' civil and without personal comments. You are only harming your own position.

MrOllie (talk) 14:35, 14 March 2024 (UTC)[reply

]

In this version, I added to the formula to address your concern (which would exist also in the previous, but well). While also addressing the lack of definition of "with". Happy now? Any new concern? Thatwhichislearnt (talk) 14:38, 14 March 2024 (UTC)[reply]

@Thatwhichislearnt: I am only speaking for myself, but my assessment of the situation is that you are dragging other contributors into spending disproportionate amounts of time and energy solving a non-existent problem. I find @D.Lazard's version of the article better-written than yours and perfectly suitable for this kind of article; and despite your explanations above, I fail to see what is the exact problem you are trying to address (and apparently I am not the only one here).

Please try to convince us that there is a legitimate problem with the current version of the article — taking into account the fact that this is a Wikipedia article, not a foundational treaty on set theory. If you bring up legitimate points, I (and I am sure D.Lazard as well) will be happy to think about how they could be best fixed in a broad-audience article. If no-one share your concern that there is something really wrong that needs to be fixed in the article, consider the possibility that maybe you are splitting-hair and that is not worth the time investment.

Best, Malparti (talk) 15:04, 14 March 2024 (UTC)[reply]

@Malparti In my comment at 18:09, 13 March 2024 (UTC), I pointed out serious objections to D. Lazard's changes (lack of literature and "personal invention"). I also found contradiction - details in my comment: 08:04, 14 March 2024 . Kamil Kielczewski (talk) 15:09, 14 March 2024 (UTC)[reply]

Look, I am not dragging anything. If you look from the very beginning of the talk page to the bottom, and likely much older version, the topic of whether the definition properly determines the codomain keeps bringing people other than me, discussing that. The version that you such praise, still has the definition malformed, because the predicate "[f] is (or has the property) with codomain Y" is not defined in it (I am talking about the first of the two attempts at definition). I personally disagree that Lazard does a good job at mathematical exposition, but I am happy with any style as long as when the literal translation into formal set theory corresponds to a well-formed definition. One can write as plain English as anyone wants, while still maintaining a proper correspondence to formally written set theory. For example, Halmos pages 26 and 27 are pretty plain English. See them addressing exactly the meaning of "with domain X and codomain Y" and "from X to Y" in the last paragraph of page 27. Thatwhichislearnt (talk) 15:21, 14 March 2024 (UTC)[reply]

I just went to the history of the talk page, clicked "older 500" a few times, clicked on a version, and here, 2012 and people discussing that the definition does or does not define codomain. I would bet that there are earlier. Is it me dragging? The only way resolve that, is by being explicit on that meaning. It doesn't have to be complicated, it doesn't have to be abstract, or formal, just explicit. Of course, translatable into formal language, for correctness. Thatwhichislearnt (talk) 15:51, 14 March 2024 (UTC)[reply]

A la Halmos "with codomain Y" while close to previous version

Ok, here is a version that address "with codomain Y" still being undefined in the current version.

Given sets $X$ and $Y$ , a function can be defined to be a binary relation $R$ between $X$ and $Y$ that satisfies the two following conditions:
For every $x$ in $X$ there exists $y$ in $Y$ such that $(x,y)\in R.$

$(x,y)\in R$ and $(x,z)\in R$ imply $y=z.$

Viewing the relation $R$ together with the inclusion into $X\times Y$ allows to speak of the function with domain $X$ and codomain $Y$ or say that it is a function from $X$ to $Y$ . Some authors do this implicitly, some explicitly, and some view functions as only the relation itself.
Without referring explicitly to the concept of a relation, but using more notation (including set-builder notation), a function with domain $X$ and codomain $Y$ is formed by three sets, the domain $X,$ the codomain $Y,$ and the graph $R$ that satisfy the three following conditions:

$R\subset \{(x,y)\mid x\in X,y\in Y\}$

$\forall x\in X,\exists y\in Y,\left(x,y\right)\in R\qquad$

$(x,y)\in R\land (x,z)\in R\implies y=z\qquad$

The first paragraph introduces the function. A careful reader will note that this is the function viewed as only the set of pairs. The paragraph "Viewing the relation R ..." then defines the meaning of "with codomain Y". Well, also "with domain X", but a careful reader can note that already the first paragraph determines X from the set of pairs, if not the phrase "with domain X" itself. This second paragraph is imitating Halmos' Naive Set Theory last paragraph of page 27, which gives meaning to "with codomain Y" while still mostly in English and without much formal language. The concern of whether that second paragraph has some grammatical problems ... I am happy for it to be written in any other form.

The concern of whether X and Y are undefined. Sorry, but mathematically makes no sense. Please suggest a wording that you think "solves" that. I am happy with different wording and style. An exercise that everyone should do is to think about how does the English translates into a completely formal definition. The "Given X and Y", as is customary, translates to the $\forall A$ in the axiom of comprehension. Note that there is no antagonism between being precise and using simple wording. A simple, plain English wording can be assessed to be precise, when you can take the translation of each of its syntagmas into set theory and you get a valid formula. Thatwhichislearnt (talk) 15:07, 14 March 2024 (UTC)[reply]

I still do not understand what is the problem with the current version, and why this version would be an improvement... In fact, to me this version seems worse for several reasons:

I do not find it to be well-written: (1) "can be defined to be" is heavy; (2) it feels very awkward for me to introduce X and Y and then talk about "a function" instead of "a function from X to Y" (as is done, e.g, in Halmos's Naive Set Theory). A bit like comparing "Let r be a positive real number. A circle is [...]" instead of "Let r be a positive real number. A circle with radius r is [...]"
I find the paragraph "Viewing the relation R [...]" unnecessarily complicated and confusing.

For these reasons, I prefer the current version over the proposed new one. Since I still don't understand exactly what problem that new version is supposed to address (I guess it might have to do with whether the codomain is included in the definition of the function or not? If so, then to me that discussion is a waste of time and I'm not willing to invest any more time into it), I don't see a reason to change what is currently written. Malparti (talk) 15:48, 14 March 2024 (UTC)[reply]

Happy with any other wording of "can be defined".
The "Given X and Y", is the $\forall A$ in the Axiom_schema_of_specification#Statement. See how the $\forall A$ in it is before the $\exists B$ , which is the set that the axiom effective defines. That is the same structure. Again, happy to move it, or reword it. There are many styles of writing the $\forall A$ , including the customary omission, which in the very first version I used, but there was the unfounded claim that X and Y are left as free variables.
Note that Halmos defines "a function from X to Y" after having defined "relation" and having defined "relation from X to Y" (page 30). Regarding the previous point, I quote his page 30 "If X and Y are sets, a function from (or on) X to (on into) Y ...".
Finally, the version that you like is nearly the same as this. The only difference is that it still leaves undefined (in the first of the two version of definitions that it claims equivalent) the meaning of "with codomain Y". A subject that has kept bringing people to discuss since as early as 2012, and likely before. Leaving it it still under the rug, undefined, is just going to keep being (unnecessarily) a point of contention.
There is also the issue with the current version of the article regarding neutrality. The current version of the article (tries to) present functions as having codomain as part of the concept. That is not the only case in current use. Big guns of set theory, of functional analysis (and many introductory books that did it without paying attention), like to use the definition that does not carry the codomain. Per neutrality, the article should also make as little bit of effort in presenting that that position also exists.

Thatwhichislearnt (talk) 16:09, 14 March 2024 (UTC)[reply]

Formal definition - why not use the literature?

It seems that the current definition is an attempt to "reinvent the wheel" - let's note that:

In the literature there are two common definitions (more info here p. 1158) of a function: Bourbaki triple and as a set of ordered pairs (sometimes using a binary relation, though it is not necessary). This second definition is effectively Bourbaki's definition reduced only to the function's graph.
Literary definitions are well-thought-out, simple, direct, clear, short and unambiguous - we can introduce both of these popular definitions directly (especially since there is a relationship between them)
The above literary definitions usually do not have the inaccuracies that the above endless discussions (for many years!) are about.
The "Formal definition" section clearly informs the reader that formalism will appear here, so the reader knows what to expect from this section: a good, well-thought-out, clear, concise formal definition (and preferably simple).
In order not to distract the reader the "Formal definition" section doesn't even need to contain any comments on the definition (or history information) - these can be placed further on
Currently, the "Formal Definition" section is becoming something of a "monster" - it is long, complicated (e.g. too many quantifiers - it can be simplified), contains contradictions (details in the comment at 08:04, 14 March 2024 (UTC)) and is basically a personal invention (details in the comment at 18:09, 13 March 2024 (UTC)) that consists of combining Bourbaki's definition and the set of ordered pairs into one.
The current "Formal definition" section is definitely not easy to understand for a high school student
Because the current definition is a patchwork of two popular definitions, it will be difficult to find references in the literature - perhaps some will be found, but certainly not as abundant as for the two above common definitions.

Can someone explain to me why we don't use simple classic short literary definitions but invent our own?

What is blocking us, where is the problem?

PS: For example, my withdrawn edition introduces Bourbaki's definition with references to literature, and places the old definition based on a binary relation in the appropriate context. Kamil Kielczewski (talk) 16:39, 14 March 2024 (UTC)[reply]

Bourbaki wrote in French. I have not their book under hands but, as far as I remember, they do not define fonctions (functions), but applications (maps). The two terms are more or less synonyms in English but not in French ("a linear function between vector spaces" seems very odd when literally translated in French). Moreover, Bourbaki's preface says that their aim was to establish a strong basis for analysis, and that the chapters on analysis was not yet written (it is still not written). Since this article must cover functions in analysis as well as functions in set theory and algebra, Bourbaki's and Halmos' references must be considered with care, since both references are focused on the algebraic and set theoretic concept (Halmos' book is entitled Naive set theory). D.Lazard (talk) 18:45, 14 March 2024 (UTC)[reply]

Bourbaki's exposition setting everything up is extremely convoluted and painful to decode, because they want to define everything entirely symbolically with utmost precision. They usually use Roman letters for sets, but sometimes bold italic letters instead for a reason I don't quite understand. For someone who doesn't already know exactly what it's supposed to be saying, just reading the text takes repeatedly solving tricky logic puzzles to work out the meaning. I would expect deciphering the relevant definitions to take at least a half hour for a newcomer, maybe hours. Here's a link into the thick of it.

In English translation, Bourbaki define a graph to be a set of ordered pairs, use the term relation to mean more or less any meaningful statement about some list of terms, and the relation

R\{x,y\}

is said to have a graph

\mathrm {G}

if

(\forall x)(\forall y)R\iff {}\!

((x,y)\in \mathrm {G} ).

The projections (1 and 2) of the graph are the set of first/second elements of the pairs, also respectively called domain and range (what we call image here).

They then define a correspondence to be a triple

\Gamma =(\mathrm {G} ,\mathrm {A} ,\mathrm {B} ),

with source

\mathrm {A}

and target

\mathrm {B}

where the domain of the graph is a subset of the source and the range of the graph is a subset of the target.

A graph is called a functional graph if each element in the domain only appears as the first element of one pair, and a correspondence is called a function if its graph is a functional graph and if furthermore its source is the domain of the graph.

Then a mapping of $\mathrm {A}$ into $\mathrm {B}$ is defined as a function whose source (the domain of its graph) is

\mathrm {A}

and whose target is

\mathrm {B}

.

In other mathematical literature, any of "graph", "functional graph", "correspondence", "function", "mapping of

\mathrm {A}

into

\mathrm {B}

" might sometimes be called a "function", depending on the context. –jacobolus (t) 19:34, 14 March 2024 (UTC)[reply]

There is english translation of Bourbaki book "Theory of Sets", Springer, 1970 - page 81.

But it is not necessary to accept Bourbaki's definition in its original form (because he used quite specific formalism to express it), only in the contemporary one provided by other authors (bearing in mind only that the idea of the triplet comes from him).

If it turns out in the future that Bourbaki's definition is insufficient (which I think is, however, unlikely.), there is nothing to prevent supplementing or modifying the article.

Perhaps Bourbaki intended to use their definition only for analysis, but over time it turned out that the definition of the triplet is surprisingly general despite its simplicity. And maybe that's why it became widespread. Kamil Kielczewski (talk) 20:11, 14 March 2024 (UTC)[reply]

And the current section Formal definition, is not neutral, in the sense that there is no explicit mention on the definition of function as its graph (even though the first attempt at definition in it, that is what it unintentionally defines). The article on
binary relations#Definition
does a better job at neutrality. Unfortunately the article binary relations does exactly the same mistake of starting trying to define "binary relation R over sets X and Y" and then says it is a subset of $X\times Y$ . Hilarious. How to carry the information of "over X and Y" using an object from the most forgetful of categories? Magic! Thatwhichislearnt (talk) 20:19, 14 March 2024 (UTC)[reply]

In Finitary_relation#Definitions they did a better job. It doesn't have the issues in this article and the issues in binary relation. The owners of this article should probably imitate what is done in Finitary relations. There, there is a sentence, that like the one in this article forgets what is trying to define, but acknowledges it by saying "is given by", instead of "is". Then it issues a correction acknowledging that it is a correction with "may be more formally defined". Thatwhichislearnt (talk) 21:12, 14 March 2024 (UTC)[reply]

I think we could clear this up with a definition that includes the following information (but rewritten for clarity):

A binary relation R between X and Y consists of a set X called the source or domain, a set Y called the target or codomain, and a set of ordered pairs {(x, y)} such that x is in X and y is in Y, called the graph, which is a subset of the Cartesian product X × Y. A binary relation and its graph are routinely conflated, and some sources define the term binary relation to mean a graph; the symbol R can be used interchangeably to refer to either one, with meaning inferred from context.
A function f between X and Y is a binary relation for which each element x in the domain is the first entry of exactly one ordered pair in the graph. As a result, each x in X unambiguously indicates a specific corresponding y in Y, called the image of x under f and denoted f(x). The set of every y in Y which is included as the second entry of some pair in the graph is called the image of the function. As for binary relations, a function and its graph are routinely conflated, and some sources define the term function to mean a graph satisfying the same constraints, but without a defined codomain. The domain of a function can be inferred from its graph.
A partial function f between X and Y is a binary relation for which each element x appearing as the first entry of an ordered pair in the graph appears in only one pair, however not every element of X need appear in the graph. Sometimes a partial function is just called a function; this can be convenient because it saves the trouble of defining a more restricted domain where the function is defined.
A multivalued function f between X and Y drops the condition that each x in X can only have a single image under f.
A multivariate function f between (X₁, ..., X_n) and Y is a function for which each element of the domain is itself an n-tuple (x₁, ..., x_n).

I would skip the business about ordered triples of sets, which is a bit of set-theory bookkeeping not necessary for describing the concept. –jacobolus (t) 22:20, 14 March 2024 (UTC)[reply]

I agree with bullet (1), in general. In bullet (2) I think "relation" should not drop its surname "over X and Y". After all, that is the full name of that defined in bullet (1). If dropped, then there is the uncertainty for the reader: "Are we doing the conflation that that bullet (1) was saying, or not?". Bullet (3) is awkward, because it is using a concept that is muddied. A solution can be to start fresh, and define it in the style of the beginning of bullet (1): "A set X, a set Y, and a set of ordered pairs ...". In bullet (5), I don't think that that language is common, of referring to the "first part of the between" (what would be the domain) using a symbol for a tuple of sets. It is not what multivariate functions are. Anyway, multivariate functions are not important in the section of Formal Definition. It is just a label for some functions. Every n-variate function is also uni-variate, and k-variate for every k in between. But let's ignore bullets 3,4, and 5. The discussion is regarding the section Formal Definition. Thatwhichislearnt (talk) 23:19, 14 March 2024 (UTC)[reply]

I don't personally think we need to have "Formal definition" as a section heading. I would just have one top-level "definition" section, try to keep it moderately concise, and not divide it into subsections. I think it's worth including partial functions and multivalued functions there because they are sometimes called "functions" and it's helpful to readers to clearly describe them; multivariate functions can plausibly be set-theoretically formalized in multiple distinct ways some of which might not make them straight-forwardly "functions" per se. I'm not an expert on what the literature says about this though. (My bullet points were not intended to be wiki-ready definitions of any of these, and anything going into the article should be written carefully). –jacobolus (t) 23:31, 14 March 2024 (UTC)[reply]

I believe that the 'Formal definition' section is important and has its practical significance. For instance, individuals who are familiar with mathematical formalism and wish to recall or refine the meaning, can quickly find the necessary information in this section. This section also allows in a concise way to keep the remaining elements of the article in check, which should not contradict this section. Probably that's why many mathematical articles contain this section. 85.221.133.173 (talk) 07:53, 15 March 2024 (UTC)[reply]

I just think the article would be more readable if we left § Definition as a consolidated section without internal subsections, and tried to keep it as concise as practical. We should also try to shorten the following sections about notation if we can. This article spends way too long on technical minutiae near the top in a way which is distracting and unhelpful to less-technical readers, and not really very legible for anyone. –jacobolus (t) 08:01, 15 March 2024 (UTC)[reply]

I would skip the business about ordered triples of sets - I think we should not overlook definitions that are commonly accepted in the literature. 85.221.133.173 (talk) 08:02, 15 March 2024 (UTC)[reply]

There is a nice discussion about this general topic discussing the motivation and considerations involved with various definitions in:

Goldblatt, Robert (1984). "2. What Categories Are". Topoi: The Categorial Analysis of Logic (Rev. ed.). Amsterdam: North-Holland. pp. 17–36.

... Although the modified [triple (X, Y, R)] definition does tidy things up a little it still presents a function as being basically a set of some kind – a fixed, static object. It fails to convey the “operational” or “transitional” aspect of the concept. One talks of “applying” a function to an argument, of a function “acting” on a domain. There is a definite impression of action, even of motion, as evidenced by the use of the arrow symbol, the source–target terminology, and commonly used synonyms for “function” like “transformation” and “ mapping” . The impression is analogous to that of a physical force acting on an object to move it somewhere, or replace it by another object. Indeed in geometry, transformations (rotations, reflections, dilations etc.) are functions that quite literally describe motion, while in applied mathematics forces are actually modelled as functions. This dynamical quality that we have been describing is an essential part of the meaning of the word “function” as it is used in mathematics. The “ordered-pairs” definition does not convey this. It is a formal set-theoretic model of the intuitive idea of a function, a model that captures an aspect of the idea, but not its full significance.

–jacobolus (t) 23:14, 14 March 2024 (UTC)[reply]

That author is about to introduce diagrams (with its arrows), commutative diagrams, and categories. That is his whole point in that paragraph. The better way he has in mind, is to look at the entire category. That is the "dynamical quality" that he wants to bring up, the functions, the arrows, not in isolation, but living in a category (See after he defines pullback). That is beyond the scope of this article. Thatwhichislearnt (talk) 10:53, 15 March 2024 (UTC)[reply]

The quoted paragrapg does not talk at all about categories. I suppose that its aim is to motivate category theory. The last sentence of this quotation seems fundamental for understanding functions. I had something like this in mind when I wrote the first paragraph of section § Formal definition. D.Lazard (talk) 12:24, 15 March 2024 (UTC)[reply]

Just like if you take the first word alone, it doesn't talk about functions at all. Read the book. And if it is not interesting for a career of implementing other's algorithms, read at least until after he introduces pullback. The entire point of the author in that paragraph is "functions' dynamic nature is exposed when viewed within the (concrete) category where they live". Thatwhichislearnt (talk) 12:49, 15 March 2024 (UTC)[reply]

What a curious exegesis! D.Lazard (talk) 15:40, 16 March 2024 (UTC)[reply]

Yeah, things look like magic, when you know know how they work. They fly over your head. Thatwhichislearnt (talk) 19:33, 16 March 2024 (UTC)[reply]

Please avoid

personal attacks. —David Eppstein (talk) 21:11, 16 March 2024 (UTC)[reply

]

To be clear, I would recommend reading the text before the part I quoted, which I think does a good job of setting the context. I just didn't want to quote several pages of exposition into a comment here.

In any event, whether or not they use categories (or draw commutative diagrams), technical authors working with the concept of a 'function' don't really think of functions as static collections of sets, and a formal definition in those terms is, as Goldblatt describes, a "model" of the concept, but not really the thing itself. It is difficult to convey, especially to newcomers, that mathematical definitions are in many ways contingent and arbitrary, in the sense that many possible definitions could be chosen which describe the same structural relationships people care about, but with different incidental details of the construction, which people generally "black-box" and then ignore. The same phenomenon occurs all over mathematics: some choice of definition is necessary to make rigorous proofs, but the specific choice often doesn't really match the essence of the thing being defined. –jacobolus (t) 16:03, 16 March 2024 (UTC)[reply]

I recommend reading the book (which is about Topoi), and perhaps after, follow up with the book "Homotopy Type Theory: Univalent Foundations of Mathematics" to see how the former can be used as models of a different way to develop foundations of mathematics. Just taking out the meme "the ordered pairs definition does not convey the dynamic quality of functions" is meaningless if you don't say what **exactly** is it that you are missing, and what is the replacement that you think does a better job. By the way, it has little to do with the black-box image of input coming in and output coming out. The ordered pairs tell you everything you need to know in that regard. They tell you what can come in, and what comes out. What he is getting at has to do with what happens with the black-box when you plug things at the entrance and/or exit. That all one needs to know about it can be learned in this way. This is why he mentions it again after defining pullback. So, yes, you can drop the set of pairs, but keep the category (and more drop the category and keep all equivalent categories).

Inserting that meme in this article would be completely meaningless and unjustified, withing the context of this article. Or is the article going to start talking about categories, toposes and HoTT? Thatwhichislearnt (talk) 18:33, 16 March 2024 (UTC)[reply]

HoTT, maybe not, but it does briefly mention type theory, and the treatment of functions as

primitive objects in that theory rather than something derived from sets. I think that inclusion is appropriate. —David Eppstein (talk) 19:12, 16 March 2024 (UTC)[reply

]

Indeed, very appropriate. In an article that has only one of the set-theory foundations definition badly written, is missing the other, and doesn't have the language to even begin articulating what exactly are the mathematical features that one cannot express with the definition. And who knows what exactly will be written, if written by those who didn't even realize what the author was referring to. Thatwhichislearnt (talk) 19:53, 16 March 2024 (UTC)[reply]

The ordered pairs tell you everything you need to know in that regard: How you define

binary relations (7,9222 page views). D.Lazard (talk) 21:32, 16 March 2024 (UTC)[reply

]

How you define continuous functions, differentiable functions and analytic functions in terms of ordered pairs To talk about continuity, additional concepts/structures must be introduced - open sets/neighborhoods in the domain and codomain (thus, topology). The mere definition of a function (regardless of whether as a triple or just its graph) is not enough - but this does not mean there is a problem with the definition of the function. Simply, the concept of function continuity is inseparably linked with the concept of topology as well as the concept of a function - and both of these independent concepts must be used to discuss continuity. (PS: It should be noted that often in the literature, a given topological space is not defined explicitly. For example, in the set of real numbers, open sets are defined using balls with certain properties - which, as it turns out, implicitly specify the topology on R) Kamil Kielczewski (talk) 22:18, 16 March 2024 (UTC)[reply]

Re "regardless of whether as a triple or just its graph": this reminds me of the famous quote about having "both kinds of music": country and western. —David Eppstein (talk) 23:07, 16 March 2024 (UTC)[reply]

Wow, just wow. Beginning from continuity, differentiability, and analyticity none of them being properties of the function alone. Did you even think about your question for one second? Thatwhichislearnt (talk) 13:04, 18 March 2024 (UTC)[reply]

But functions have been introduced for dealing with continuity and derivative. Until the 19th century, all functions were continuous and differentiable. Presently most readers of this article are more interested in continuous functions than in relations see above statistic). So the beginning of this article must be useful for the majority of its readers. This article must not define continuity and differentiability (they belong to other article), but its beginning must be compatible with these definitions. This is not the case of the definition by ordered pairs and relations. The only usge of the definition through pairs and relations is to provides a formal version of the intuitive definition trough assignment, process, mappings, etc. You must know that most users of functions do not know other definition than the intuitive one, and this is generally sufficient for them. D.Lazard (talk) 13:53, 18 March 2024 (UTC)[reply]

Here's another paper to look at, Leinster, Tom (2014). "Rethinking Set Theory". The American Mathematical Monthly. 121 (5): 403–415.

JSTOR 10.4169/amer.math.monthly.121.05.403

.

In this take, 'functions' are a primitive concept with certain axioms applying to them, and 'elements' of sets can be defined as a higher level concept, in terms of functions.

The root of the problem is that in the framework of ZFC, the elements of a set are always sets too. Thus, given a set X, it always makes sense in ZFC to ask what the elements of the elements of X are. Now, a typical set in ordinary mathematics is

\mathbb {R}

. But ask a randomly-chosen mathematician, ‘what are the elements of π?’, and they will probably assume they misheard you, or tell you that your question makes no sense. If forced to answer, they might reply that real numbers have no elements. [...] The traditional approach to set theory involves not only ZFC, but also a collection of methods for encoding mathematical objects of many different types (real numbers, differential operators, random variables, the Riemann zeta function, ...) as sets. This is similar to the way in which computer software encodes data of many types (text, sound, images, ...) as binary sequences. In both cases, even the designers would agree that the encoding methods are somewhat arbitrary. So, one might object, no one is claiming that questions like ‘what are the elements of π?’ have meaningful answers. [...] The bare facts are that in ZFC, it is always valid to ask of a set ‘what are the elements of its elements?’, and in ordinary mathematical practice, it is not. Perhaps it is misleading to use the same word, ‘set’, for both purposes. [...] § The working mathematician’s vocabulary includes terms such as set, function, element, subset, and equivalence relation. Any axiomatization of sets will choose some of these concepts as primitive and derive the others. The traditional choice is sets and elements. We use sets and functions. –jacobolus (t) 23:34, 16 March 2024 (UTC)[reply]

The root of the problem is that in the framework of ZFC, the elements of a set are always sets too The source of what problem?

It seems that the ZFC axioms constitute the most popular foundation of mathematics, while other systems represent exotic niches for specialists. Kamil Kielczewski (talk) 00:03, 17 March 2024 (UTC)[reply]

All you are doing is introducing noise, and noise that is unfounded withing the context of this article, and in particular the section being discussed. Even the section the brings up type theory raises an objection and instantly resolves it within the framework of set theory. Again, what is being discussed is the section Formal Definition, and the issue that the two definitions in use are not mentioned, only one is mentioned (badly in the first attempt). What that section should do is four things: (1) mention the definition without codomain, explicitly noting that there are no codomains in it (2) mention the definition with codomains, explicitly mentioning that there are codomains in it (3) mention that both definitions are in use in literature (4) mention which of the two is the one that this article will use, at least predominantly and unless otherwise stated.

A random click at the history of this talk page showed me that as far back as 2012, this article has had an influx of people complaining that the article does a poor job regarding the definition. It will continue, as long as there all that ambiguity. And it is not that it is a difficult task. There are plenty of award-winning authors that present these points in clear plain English, that can be imitated. Thatwhichislearnt (talk) 13:17, 18 March 2024 (UTC)[reply]

For given so much emphasis on functions without codomain, you must provide examples of common functions without a codomain. All examples that I know have a codomain provided implicitly by the context or by the definition of the specific function. For example, the

square function on the reals is defined as assigning to each real number the product of this number by itself. As the product of two real numbers is always a real number, the codomain of this square function is implicitly the set of the real numbers. If you cannot provide an example where the codomain is not defined, at least implicitly, this mean that functions without a codomain do not deserve to appear at the beginning of the article. D.Lazard (talk) 14:20, 18 March 2024 (UTC)[reply

]

Open Halmos', open Apostol's, open Carson's. There are [Talk:Function_(mathematics)#A._Functions:_the_plain_variant nearly 20 different references listed above]. Those are the ones that do it as a deliberate choice. In addition, there are myriads of Calculus books (plus this Wikipedia article's Formal Definition's first attempt) that do it by mistake, because they don't pay attention, they define a concept that doesn't have codomain, even though later talk about them and define surjectivity. The discussion here is the section Formal Definition, not the beginning of the article. Thatwhichislearnt (talk) 14:54, 18 March 2024 (UTC)[reply]

And your request for examples is entirely non-sensical. Within the context of a book that chooses to define functions as the set of ordered pairs, all functions are examples of functions without codomain. All of them. This doesn't mean that one cannot talk about codomains, or the idea of surjectivity. Instead of talking about surjectivity (as a property of the function), they talk about "surjectivity onto Y". Mathematically, no idea fails to be representable, it is only a different formulation that some authors choose. So many other properties: continuity, differentiability, analyticity, are not properties of the function itself anyway. Thatwhichislearnt (talk) 15:19, 18 March 2024 (UTC)[reply]

And not even going to the literature, the article Finitary_relation#Definitions does an okay job regarding the definition. While Binary_relation has the same problem as this (in the first sentence of their "definition"), except that over there they do use consistently the full name of the object that they intend to be talking about throughout the whole article. Thatwhichislearnt (talk) 14:21, 18 March 2024 (UTC)[reply]

@Jacobolus of course, the information that in some systems not based on ZFC, functions can be considered as primitive concepts, can be placed in a separate section titled e.g. "Additional Information" (as a curiosity for someone who aspires to join the narrow group of specialists...). So, outside the "Formal Definition" section that we discuss in this discussion thread. Kamil Kielczewski (talk) 23:41, 18 March 2024 (UTC)[reply]

Formal definition - the structure of a triple

Based on the statement by @Jacobolus I would skip the business about ordered triples of sets, which is a bit of set-theory bookkeeping not necessary for describing the concept from the thread "Formal definition - why not use the literature", I would like to address the aspect of defining a function consisting of domain X, codomain Y, and graph G - in literature, one can encounter two main approaches:

the classic Bourbaki: a function is an ordered triple f=(X,Y,G)
modified: function f is "something" containing X, Y, and G

At point 2, I wrote "modified" because it seems that before Bourbaki, the literature did not consider functions as three elements X, Y, G, and it was he who introduced the use of this. Later, some other authors allowed themselves to bypass the structure of the ordered triple - nevertheless, I have not encountered a justification anywhere (but this may be because I have not come across the right sources - so if someone has an article/book from before Bourbaki where X, Y, G are used, please quote it).

An ordered triple defined as an e.g. appropriate Kuratowski pair, for example, f=(X,Y,G)= ((X,Y),G)= {...} allows us to treat the function as a set - here are the advantages of such a construction:

the function becomes an ordinary set, a well-known primitive concept (we do not multiply entities without necessity - Occam's Razor)
the definition of the function becomes precise and unambiguous
completeness - the triple is a sufficient structure to capture the concept of function - no need for more complex structures
the set f can be directly applied with the well-known and developed apparatus/formalism known from set theory
ease of working with functions at the definition level (we can use well-known operations on sets)

I would like to emphasize that the introduction of the triple structure by Bourbaki is not merely "bookkeeping" within set theory, but is a well-thought-out structure. A similar approach is applied in the formal definitions of other important mathematical concepts such as:: group, topological space, graph, etc.

However, with authors who did not use the triple structure, but define the function as "something" a separate entity, having domain, codomain, and graph, I have not found an explanation.

So, I pose this question: What advantages does the introduction of functions as entities, distinct from sets, bring? What does it contribute?

PS1: If it turns out that it does not contribute much, I would suggest using the structure of the triple in the article first (i.e., using the Bourbaki definition) and only secondly adding information that some authors use a definition in which the function is not a set but a separate entity containing X,Y,G (with citations). In this way, we communicate to the reader that there are different definitions based on the three elements: X,Y,G.

PS2: I assume that in the "Formal definition" section we will use Zermelo–Fraenkel set theory because it is the most popular, so the above considerations are only within this system. Though of course in another section of the article one can go beyond that. Kamil Kielczewski (talk) 06:52, 20 March 2024 (UTC)[reply]

The short answer to your question is

WP:DROPTHESTICK. —David Eppstein (talk) 17:03, 23 March 2024 (UTC)[reply

]

Good luck with that. Work badly done will keep bringing people complaining about it. As early as 2012 one can find people complaining about the same problem. The reason is as simple: A student finds a book where functions have codomains, and a book where they don't. Comes to Wikipedia to check which is which, and finds an article written so badly that does not clarify that both are used in the literature and it doesn't clearly state either one of the definitions.

And your "short answer" is disingenuous. Just a moment ago you were saying that claiming that "a set theory definition doesn't convey the dynamical nature of functions" is a phrase that belongs in this article. This claim is demonstrably false, when taken within the context and needs of this article, and extremely technical to properly justify. So much that even those who brough up the quote didn't have the tiniest idea what the quote was really talking about. Thatwhichislearnt (talk) 14:50, 24 March 2024 (UTC)[reply]

@David Eppstein I do not see an answer to the question asked either in the WP:TECHNICAL or in the comments. If you want to prove otherwise, please indicate the relevant quote and explain.

As for the definition of the tripe f=(X,Y,G), it is commonly used in literature (what has been noticed e.g. here p. 1158), so omitting it in the article would be a serious mistake. Doubts about this seem to have been mostly dispelled in the previous thread. In this thread, I just wanted to more broadly explain certain misunderstandings that arose in the course of that discussion. Kamil Kielczewski (talk) 06:52, 20 March 2024 (UTC)[reply]

Function evaluation

A mini edit war has started about the paragraph on function evaluation. This paragraph was confusely written, and I rewrote it boldy for clarifiation. Nevertheless, some problems remain, for which I am not sure of the best solution.

Although very sketchy, this paragraph is too long for the lead. I am not sure of the part of it that belong to the lead and of the best place for details.
Function evaluation redirects here. This deserves more than two lines in the lead. Maybe a specific article?
Very often, "function evaluation" refers more properly to "
computer algebra systems. Expression evaluation is a redirect to Expression (mathematics)#evaluation that I created recently. Nevertheless, the treatment of evaluation (mathematics)
remains insufficient and confusing in Wikipedia.

Although very common, the vertical-bar notation for evaluation is not clearly defined in Wikipedia. I added recently a description of it at
Expression evaluation, but this is still insufficient. This vertical bar notation, is a particular case of a restriction
, if one consider $x=a$ as an abbreviation of $\{x|x=a\}.$ Maybe, a description of the vertical-bar notation for evaluation could be added at Restriction.

In summary, fixing this paragraph of the lead requires more work and more discussion, for which the tag {{

cn}} is of no help. D.Lazard (talk) 17:39, 24 March 2024 (UTC)[reply

]

The lead section of this article seems like an inappropriate place for the only discussion of this "vertical bar notation" which is not used anywhere else in the article. It's confusing, distracting, and only tangentially relevant. It should be put somewhere else in the article or in another article. Readers aren't going to look for it here, and readers who come across it here aren't going to find it valuable in context. –jacobolus (t) 17:43, 24 March 2024 (UTC)[reply]

One problem with this vertical bar notation is that it's hard to find clear sources describing its history/use. My guess is that it evolved out of the related notation for the bounds of a definite integral, as found in introductory calculus books/courses. There's a discussion of the earliest use (under "bar notation") in https://jeff560.tripod.com/calculus.html –jacobolus (t) 18:20, 24 March 2024 (UTC)[reply]

The use of "often" and "commonly" are unsourced and unnecessary. The second sentence has not one, but two parenthetical clauses. Yikes! The third sentence, unnecessarily restricts itself to talk about function evaluation to the case when the expression "depends on x". This, "depends on x", is ambiguous as an intuitive notion and it hasn't been defined in the article as a formal concept. Thatwhichislearnt (talk) 13:35, 25 March 2024 (UTC)[reply]

I agree that {{cn}} is not sufficient to elaborate on the vertical bar and evaluation, however, I consider(ed) is necessary - I apologize for my part of the "mini edit war" (I didn't intend to fight). I also agree that the vertical bar needn't be adressed in the lead, or even shouldn't be adressed there. However, it should be explained somewhere (not sure whether inside this article). And I, too, have seen the notation only in computations of definite integral [and in the recent versions of this article's lead].

Defining the vertical bar based on restriction appears to be one possibility, which also nicely motivates the notation. Another possibility would be to define it based on substitution (logic), i.e.

e\vert _{x=a}

means the substitution application

e\{x\mapsto a\}

, or the evaluation of it. I'm not sure about an explanation of "expression evaluation". Maybe it should be based on (item "Functions" at) First-order_logic#Evaluation_of_truth_values, which defines evaluation of an expression in terms of evaluation of its constituent functions. (First-order logic doesn't care about how to obtain a function's result value for given inputs. If a function is defined by an expression in turn, its constituent functions (usually

+,-,*,/,...

) need to be evaluated, which leads to the standard arithmetical algorithms. However, a function may well be non-computable, and it may be impossible to obtain its result value. — In computer science,

ground confluent".) - Jochen Burghardt (talk) 20:33, 25 March 2024 (UTC)[reply

]

"Overriding (mathematics)" listed at Redirects for discussion

The redirect Overriding (mathematics) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 26 § Overriding (mathematics) until a consensus is reached. Tea2min (talk) 11:59, 26 April 2024 (UTC)[reply]

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