Talk:Function (mathematics)/Archive 13

"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?

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


 * 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)
 * 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 $$f:S\to \R$$. 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)
 * "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)
 * @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)
 * (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)
 * 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)
 * @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 $$f:S\to \R$$ 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)

"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)
 * 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 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)
 * 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)

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)

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)


 * 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)
 * "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)
 * 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)
 * 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)
 * 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)
 * 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)
 * 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)
 * Maybe it would be helpful to make a new article at Multivariate function instead of redirecting that title to Function of several real variables. Such an article could discuss the way a function of k variables can be formally thought of as a function of a "single variable" whose domain is a set of k-tuples, etc. –jacobolus (t) 01:53, 3 January 2024 (UTC)
 * 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)
 * I have changed the target of Multivariate function to Function (mathematics). I have no clear idea whether a separate article is needed. In any case, the new target section must be more visible (nearer to the beginning), and, if a separate article is created, a summary must be kept here, with a main article template. D.Lazard (talk) 15:55, 4 January 2024 (UTC)

Presently, multivariate functions are firstly defined in a very short section, which, strangely, is the sixth and last subsection of. Section 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 as the first subsection of , and to clean up accordingly the section and the remainder of the article. D.Lazard (talk) 15:55, 4 January 2024 (UTC)


 * 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)
 * 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, I have renamed the section on relations as , and added an historical explanation of the coexistence of two apparently very different definitions. D.Lazard (talk) 11:55, 6 January 2024 (UTC)

"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)


 * 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)

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)
 * 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)
 * ✅: reduced the count to 17, for now. - Jochen Burghardt (talk) 13:39, 1 February 2024 (UTC)

"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)
 * 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)
 * 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)
 * 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: "Function symbols of valence 0 are called constant symbols"). Jochen Burghardt (talk) 16:46, 3 February 2024 (UTC)
 * 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)


 * 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)

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)


 * To refer to a section of the article you may use the template alink; for exanple, your above concern refers to section . 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)
 * I'll post on your talk page a "welcome" template with many useful links. D.Lazard (talk) 10:31, 16 February 2024 (UTC)

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 set of pairs, that 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, []
 * C.C.Pinter, A Book of Set Theory, New York 2014, ISBN: 0-486-49708-9], p. 35, []
 * A. Nesin, "Foundations of Mathematics I, Set Theory", Istanbul 2004, p. 35, []
 * R. Mayer, 2007, Math 111 Calculus 1, Reed College, p. 67 (58) []
 * R. Schultz, Mathematics 144 Set Theory, California 2012, p. 63, []
 * 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)


 * 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)
 * 1. : If I misunderstood something - then explain to me precisely - Does 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. : 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. : Do you want to introduce another concept to formal function definition - such as type - which dispels the ambiguities of the old definition?
 * 4. :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)
 * 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)
 * 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)
 * 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)
 * 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
 * }
 * class NewFunctionType {
 * domain: set ;
 * codomain: set ;
 * graph: map;
 * }
 * 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)
 * 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)
 * 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)
 * 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)
 * 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.
 * - 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)
 * 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 $\pi$, 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)
 * 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)
 * 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)
 * 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)
 * 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 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)
 * 14. 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)
 * 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)
 * 16. 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)
 * 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)
 * 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)
 * 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)
 * 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 $$ [...] The graph of a function is defined as follows: If $f$ is a function from $A$ to $B,$ then the graph of $f$ is the class of all ordered pairs $(x, y)$ such that $$ [...]
 * 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)
 * 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)
 * 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)
 * 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)
 * 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)

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)= 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)


 * 12. - no, based only on the graph, we cannot reconstruct the codomain. (details and example in section "Why old definition is bad?" above)
 * 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)

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 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)


 * 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)
 * Sorry, I forgot to sign. I fixed it with the correct time stamp.
 * In the article, it is written: . 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)
 * 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)
 * As you can see - the old definition is ambiguous/fuzzy. Kamil Kielczewski (talk) 15:16, 1 March 2024 (UTC)

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:

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)


 * 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)
 * 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)
 * @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)
 * 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)
 * No. Your pdf say (page 1157) (Burbaki Triple is proposed "new definition"):
 * (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:
 * 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)
 * 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)
 * @jacobolus I'm afraid that 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)
 * You wrote . 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)
 * @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:
 * 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
 * In the last comment, you wrote
 * Which is a complete falsehood because I never wrote anything like that. Kamil Kielczewski (talk) 20:30, 3 March 2024 (UTC)
 * 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)
 * Moreover, such a way of conducting dialogue - quoting your commen (16:42, 3 March 2024) :
 * 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)
 * @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:
 * Univalent relation:
 * These conditions are exactly the formalization of the above definition of a function.
 * To instead say something like:
 * A binary relation $(X, Y, f)$ consists of three parts, a set $f \subset X \times Y$ called the domain, a set $X$ called the codomain, and a set ordered pairs $Y$ such that $f \subset X \times Y$ is an element of $R$ and $R$ is an element of $X$, a subset of the set of all such ordered pairs, the Cartesian product $Y$
 * A function is a binary relation that is:
 * total, meaning that for each $(x, y)$ in $x$, $X$ includes at least one pair $y$ for some $Y$ in $X\times Y.$, and
 * univalent, meaning that each first element of a pair in $x$ is unique, so if $X$ and $R$ are pairs in $(x, y)$, then $y$.
 * –jacobolus (t) 20:48, 3 March 2024 (UTC)
 * 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)
 * The point would be to be explicit enough to forestall complaints from extremely literal readers. –jacobolus (t) 22:18, 3 March 2024 (UTC)
 * Here is an example source, Meyer (2005) "Binary Relations", course notes for MIT 6.042J/18.062J:
 * A binary relation, $Y$ consists of a set, $R$ called the domain of $(x,y)$ a set, $(x,z)$ called the codomain of $R$ and a subset of $$ called the graph of $R,$
 * –jacobolus (t) 22:27, 3 March 2024 (UTC)
 * The triple (X,Y,G) in the Bourbaki definition is sufficiently explicit from a formal point of view. If you claim it is not, justify your assertion.
 * Meyer in this PDF does not use binary relation to define function - he only notes that a function is a certain binary relation. Similarly, we should use a simple definition of a function and in the "additional information" section, we can add that a function is a special case of such defined binary relation. Just as even numbers are not defined through being elements of some group (group theory) but directly (and possibly, additional information is added that even numbers with addition are a special case of a group).
 * Adding a binary relation unnecessarily complicates the definition, which may cause, for example, high school students to have difficulty understanding it.
 * I understand that the definition you propose is your original idea (personal invention) - ? Kamil Kielczewski (talk) 06:14, 4 March 2024 (UTC)
 * Jacobolus' proposal is fine, except for an omitted word and a too long first sentence. I would rewrite the first paragraph as:
 * A binary relation  consists of three parts, a set $$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)
 * 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)
 * 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)
 * 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 $A,$ called the domain, a set $R,$ called the codomain, and a set $B,$ of ordered pairs $R,$ such that $A \times B$ is an element of $R.$ and $X$ is an element of $Y$, with each element $f$ of $(x,y)$ appearing exactly once as the first element of a pair. The set $x$ 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 $X$ with $y$ in $Y$ and $x$ in $X$, the Cartesian product $f$.
 * 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 $(x,y)$, 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)
 * 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)
 * 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)
 * 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)
 * The function's graph is a subset of the set of all ordered pairs $x$ with $X$ in $y$ and $Y$ in $X \times Y$, 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 ⇭⇭⇭, 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)
 * 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)
 * 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)
 * 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)

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 [] 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)


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