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)
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)
