Talk:Function (mathematics)
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Some old talk has been removed. Hopefully none of the following references it; if it does, you can look an old version.
Brianjd 09:49, 2004 Nov 7 (UTC)
Could you archive talk, rather than just cutting it out? Charles Matthews 12:04, 7 Nov 2004 (UTC)
See Talk:Function (mathematics)/archive.
Brianjd 03:06, 2004 Nov 9 (UTC)
SUMMARY: Every dyadic function can be represented as a dyadic relation. However, not every relation can be represented by a function. Dyadic relations include: mere associations, functions and series.
1. DYADIC RELATIONS Dyadic relations (xRy) or R(x,y) are predicates about relationships of two objects [Pe33] [Mad91]. (x > y), (x loves y), (x includes y), (x friend-of y) and (x son-of y) are examples of dyadic relations [R&W10]. x and y represent individual values. The set of x values is the domain and the set of y values, the co-domain. x and y can be tuples of degree n (n-tuples) but they continue being individual values. E.g.: a point of coordinates, a full name or an address. Sometimes, x and/or y has specific role. For instance, given (x references y), x is the referencing object and y, the referenced object. Each R relation can be viewed as a class {(x,y)|xRy}. Then, R definition as (x loves y)is called "intension" (or functor) of R [Dea93]. x and y can be substituted by individual values and they are the arguments of the functor. Each pair of ordered values (Romeo,Juliet) is an instance of (x loves y). The set of such instances is the extension of R. Each instance is a member. The number of members is called cardinality. The adjective of dyadic relation is "relative" [Pei33]. 2. RELATIONS AND FUNCTIONS Functions [y=f(x)] are monadic operations upon zero or more objects giving another object. Therefore, relations (xRy) being propositions are not functions. A dyadic relation [xRy] is a fact between two existing objects. But in a dyadic function: a new object (y) is calculated using existing object (x). Propositions of the form xRy are called functors (a word similar to functions) because arguments (a word similar to variables) are substituted by individual objects. At the moment of the substitution, the functor became a proposition; which, in turns, can be <absurd> or not ; <meaningful> sentences can be <falsable> [Pop59] or not; being, finally, the <falsable>, <true> or <false>. Nothing of that is applicable to functions. However, every dyadic function can be represented as a dyadic relation: Generating the set of pairs of the class of related values. But not every relation has an implicit transformation (or operation or algorithm) that can be represented by a function. Social convention is the driving force of the persistency of 'CA' ->- 'California', 'NY' ->- 'New York', etc. Note. Calling to relations without operation, "mere associations", Russell understands that dyadic relations include: mere associations, functions and series [R&W10]. BIBLIOGRAPHY - [Dea93] Deaño, A., Introducción a la lógica formal, Alianza ed., 1993. - [Ham02] Hammer, E., "Peirce's Logic", The Stanford Encyclopedia of Philosophy (Winter 2002 Edition), //plato.stanford.edu/archives/win2002/entries/peirce-logic/ - [H&L65] Hughes & Londey, The Elements of Formal Logic, Methuen, 1965. - [Mad91] Maddux, R. D., "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations" in Studia Logica 50(3/4): 421-455, 1991. //www.math.iastate.edu/maddux/>>origin2.ps - [Pei33] Peirce, C. S., "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic." in Memoirs of the American Academy of Sciences 9: 317-78. Reprinted in Peirce, 1933. - [R&W10] Russell & Whitehead, Principia Mathematica, 3 vol., Cambridge University Press (1910, 1912, 1913). 2nd ed., 1925 (Vol. 1), 1927 (Vol. 2, 3). Abridged as Principia Mathematica, Cambridge University Press, 1962. [Enrique Villar; mailto:evillarm@capgemini.es]
I removed this:
- Suppose the domain X is the set of all married men and the codomain Y is the set of all married women. The formula f(x)=the wife of x is clearly not a function. Given a married man a, f(a) may either not unique or not exist. In mathematics, it is not a good idea to write down such a fuzzy notation. One way to get around is to consider the formula as the subset of all (husband, wife) pair in X×Y. Clearly, if an explicit formula for f(x) is really a function, we can still construct the set of pairs f; so nothing is lost by this definition.
Since it's such a confusing example. The real reason it's not a function is that the relation "married" is obviously too wide - if it means the set of all men who are currently married and the set of all women who are currently married in the usual sense, then we would expect it to be a function - each man is married to exactly one woman. On the other hand, if it means ever married, then it's not. Or we were to take a realworld sample of men who claimed to be married, meaning that they had multiple wives... etc. The final sentence is repeated below under the advantages of the set theoretic approach. Chas zzz brown 10:52 Feb 26, 2003 (UTC)
Thanks for removing this paragraph. I am just too lazy to do it myself. :-P The writer of this confusing example
From the main page, the first graphic has this text: "This is not a function in usual sense because the element 3 in X is associated with two elements a and b in Y (Condition 1 is violated). It is a multivalued function." But element 3 in X is actually associated with elements b and c in the diagram.
- Yes. Don't be shy, change it! - Patrick 00:27 Apr 10, 2003 (UTC)
The one thing missing from this article was an intro that ahh actually explains what the article is about ;-) - David Gerard 00:08, Mar 23, 2004 (UTC)
The recent addition on YX notation is in the wrong register - too advanced. Charles Matthews 20:34, 6 Jul 2004 (UTC)
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Even and Odd functions
Can someone put something in the article that tells the difference between even and odd functions? I came to Wikipedia to look it up, and I can't seem to find it there. --pie4all88 01:30, 27 Aug 2004 (UTC)
See Even and odd functions. I added the link. Donar Reiskoffer 06:28, 27 Aug 2004 (UTC)
- Great. Thanks a lot, man. --pie4all88 20:53, 27 Aug 2004 (UTC)
Category
This article is in the category Category:Set theory. While totally correct to be included there and not in a supercat of Set theory, I'd like this article to be included directly in Category:Mathematics (also). This because so many use functions without having the slightest clue of Set theory, and that many kinds of functions are in Cat:Maths but not in Set theory (the exponential function, for example). ✏ Sverdrup 12:55, 25 Sep 2004 (UTC)
Old discussion from the talk-page of "function" (Talk:function)
On the talk page for "function" (Talk:function), there is some material that should belong on this talk page instead. Pehraps it could be moved here or archived in this talk-page's archive. Ae-a 11:25, 9 Feb 2005 (UTC)
Computer Science vs. Math
A lot of articles on Wikpedia seem to have problems when they confuse mathematical concepts with Computer Science concepts. This definition is confused in this and other manners, and I don't seem to be able to fix the first part, which starts badly. I've deleted the word "unique" where I could, and pointed out the example which contradicted it anyway.
The way I learned about functions (as a Math student) is this:
A function is a MAPPING from some SET A to some other SET B, such that if you pick an element from set A it takes you to some element in SET B.
That's it. There's no guarantee of uniqueness at the destination (that's a special type of function).
- In this encyclopedia, a multivalued function is not a function (or, it is a set-valued function of codomain equal to the power set of the "original domain"). This is commonly accepted concensus. On the other hand, things like "random" functions are functions, but with somehow "hidden" parameters (i.e. depending on what is called "global variables" in Computer science; including here: random seed, system time and entropy,...). — MFH: Talk 14:32, 20 May 2005 (UTC)
A function which takes multiple parameters is perfectly OK, it's just mapping from some n-dimensional space (each parameter is a dimension). You don't need to get complicated. f(x,y) is simply a function from R^2 or whatever.
f(ANYTHING) = 0 is a function. It satisfies the above definition. It is a counter example to every definition on the page. I deleted two examples that contradict the items on the page, e.g. determinism. Lemonade sales CORRELATE with temperature, but, since they are not deterministic, are not a FUNCTION of the weather.
There are lots of different special properties functions can have that are useful to know about, probably the most important (Mathematically) are homomorphisms, isomorphisms, one-to-one, and onto functions. None of these are mentioned. Most of the functions Computer Scientists are interested in are down the boring pathological end of things (i.e. they lose information).
functions that are not functions
I think the introduction should be a little less categorical in the first phrase (delete "unique") and/or maybe cite the (counter-)example of multivalued functions. Also, rather than (or: before) citing "operators" right here, I think a link to partial functions could be useful.
Concerning the latter, I doubt that Dirichlet (cited in "History") used the term "function" as here, as in Europe (at least France and Germany), "function" usually means "partial function", and "map/mapping" (application / Abbildung) means "total function".
In some sense, this is not an objection to WP's use of "function", but of "domain", which imho should mean the set of points where they are defined - i.e. play the role analogue to "range" and not to "codomain" (whose corresponding notion should be something like "origin" or "departure set").
To justify my opinion, I refer to Google search results for "with dense domain" and "its domain is dense", e.g. (put into quotes as it stands to avoid irrelevant results). This usage (i.e. "let f:X->Y be a function from X to Y with (= defined on its) domain D(f)⊂X") is the most common one according to my experience.
PS: and maybe (even more) links should be provided to make it clear that this article is not about function (programming). — MFH: Talk 15:11, 20 May 2005 (UTC)
- It's a bad idea, in this case, to begin with the history. 'Function' probably is still often used for relations of non-functional kind; but I don't think any professional mathematician would see that as more than abuse of language. Charles Matthews 15:26, 20 May 2005 (UTC)