Talk:Operation (mathematics)

message: Can an operation be defined without reference to its domain?
For example, it's easy to define addition with respect to real numbers; but then we can find other domains in which an apparently similar operation can be defined, e.g. (i) the natural numbers and (ii) the complex numbers.  One example is a subset of our original domain; the other is a superset.  Furthermore, we find that their mappings of operands to outputs are similarly a subset and a superset (respectively) of the mapping of addition on the real numbers.
This leads me to conclude that the operations are all in some way equivalent.  Is there a special term for this?
It's difficult to say which of the three domains discussed above gives the 'natural' definition of addition.  No doubt we could think of more subsets and supersets of these domains and find further equivalences.  I am now tempted to say that there is some process called addition which has meaning independent of any domain.
I'd be grateful for some discussion of this.
Major tom3 (talk) 20:32, 6 January 2012 (UTC)
thread_title: Notes & Queries

message: JA: Since this is a type of question that periodically arises, I am placing the just reverted version of the article here for further discussion.  Jon Awbrey 14:22, 23 June 2006 (UTC)
thread_title: Reverted Version

message: Operation is an elementary mathematical notion. It is often introduced in kindergarten.  It deserves an elementary treatment in Wikipedia.  If there is a need for an advanced article it is the one that should have a distinctive name. Per WP:NAME: "Generally, article naming should give priority to what the majority of English speakers would most easily recognize, with a reasonable minimum of ambiguity, while at the same time making linking to those articles easy and second nature."  In this case I see no need for two articles.  Your so called "standard" treatment merely adds the notion that an operation can have more than two arguments.  And it is completely opaque to anyone but a specialist, who doesn't need it in the first place. I am restoring my edits.--agr 15:07, 23 June 2006 (UTC)
replies: text: Hurry back, I can't wait to read why the above (which I heartily agree with) implies that an article intelligible to, perhaps, 2% of the English-literate world is superior to one understandable by the rest and which includes the same information. Mind you, there certainly are subjects too technical for non-specialists to ever understand (though often a sentence or two in the intro can at least place the subject in some context) and others where a satisfactory introduction would be too long and therefore needs to be in a separate article. This is not one of them. --agr 16:20, 23 June 2006 (UTC)
thread_title: Response

message: JA: Placing this material here for distribution to introductory articles.  Jon Awbrey 17:15, 23 June 2006 (UTC)
replies: text: I agree with agr. Paul August ☎ 19:11, 23 June 2006 (UTC)

text: I agree with agr.  What is the objection to making the article accessible, at least at the beginning?  A basic principle of expository writing is to start simple and get complicated later.  Zaslav 08:57, 25 June 2006 (UTC)
thread_title:  Removed material 

message: The general definition given only covers finitary operations; is there some reason why, given the title "general", it does not include  non-finitary operations? I would replace the given product with the product over an arbitrary set I of sets Xi with i in I; the arity is the cardinality of I, or more generally the set I itself. While I is usually an ordinal or cardinal, it need not be. For example, the most natural definition of the determinant of a given size is as an operation over elements indexed by the set nxn, rather than elements indexed from 1 to n2. Magidin 18:34, 23 June 2006 (UTC)
thread_title:  General definition 

message: Hmm, I read JA's call for a third opinion on Wikipedia talk:WikiProject Mathematics.  However, I'm not quite sure what the first two opinions are.
I think it might help if both parties were to repeat their opinion, and maybe include
*a draft of what they think the lead paragraph/sentence should look like (to make absolutely clear what they think this article should be about)
*what they want to happen with the information they don't want in the article
RandomP 19:46, 23 June 2006 (UTC)
replies: text: See the Removed Material section above for what I added (several editors have improved on it since). I do not propose removing any information from the article as it was before. --agr 21:21, 23 June 2006 (UTC)

text: See Wikipedia talk:WikiProject Mathematics for my comments on Relation. --agr 21:21, 23 June 2006 (UTC)
thread_title: Third opinion

message: I suggest creating operation (elementary mathematics), function (elementary mathematics) and relation (elementary mathematics) which would have content aimed at the primary-school/secondary-school/high-school  level. This might solve the edit-warring over these articles. linas 00:22, 2 July 2006 (UTC)
thread_title:  Create introductory articles to the topic! 

message: What is the difference between an operation and a function? Jim Bowery (talk) 17:50, 26 January 2008 (UTC)
replies: text: Every operation is a kind of function, but not every function is an operation. Specifically, an operation on a set A is a function from a power of that set to the set itself, i.e., A^{\alpha}\to A for some ordinal \alpha. But many functions are not operations, for example if the domain is not a power of the codomain, then it cannot be an operation. Magidin (talk) 20:14, 26 January 2008 (UTC)
replies: text: The most general definition of operation that the current version of the article eventually(!) offers is synonyous with function (map, mapping) as far as I can see. Magidin's more restricted definition appears as an alternative usage though. (I've added note to this effect to the article.) Dependent Variable (talk) 20:29, 7 November 2010 (UTC)
thread_title: Operation vs Function

message: The article claims "Often, use of the term operation implies that the domain of the function is a power of the codomain", and I see this reasserted in this talk page, above.
The reference cited is available at the provided link as a PDF, and as far as I can tell the mentioned "definition 1.1" does not speak to this point... so the first thing is that this point needs a citation.
Second, I, for one, don't understand what this assertion means, especially with multiple input domains.  Is "power of the codomain", as used here, the cartesian product of the output set with itself (N times)?  
At any rate, could we have an example of a case where a function does NOT qualify as an operation on this basis? Thanks Gwideman (talk) 17:47, 12 December 2009 (UTC)
replies: text:  Thanks James for looking at this.  On your point 3 -- yest that's exactly what I was looking for. Gwideman (talk) 22:27, 14 December 2009 (UTC)
replies: text:  Ooops, it looks like we were both led astray. EmilJ has pointed out that the Definition 1.1 referred to is the one in Chapter 2, not the one in Chapter 1.  This does appear to speak to the point, though it refers to a f as an n-ary operation as mapping from An to A, where An is the set of n-tuples from A.  OK, so since A plays a role as the codomain of f, and the domain is An, then I suppose the description in this article is true. However, the description seems to proceed from output to input in a somewhat mystifying way. I think what I understand from this is that if one starts with a set A, then an n-ary function on A draws its input (domain) from An, and produces output (codomain) within A.  Gwideman (talk) 22:55, 14 December 2009 (UTC)
thread_title:  "Power of codomain": clarify please 

message: "Thus, in the most general sense given here, operation is synonymous with function, map and mapping, that is, a relation..."
This makes so little sense to me! It's not even that relations and mappings are different things (at least in my non-specialist opinion). It's that such a general definition is completely pointless and it will harm the development of the article just like it was with operator (mathematics)! Moreover, there is just one source referenced, and it marks the conventional definition. Am I witnessing original research running rampant? — Kallikanzaridtalk 20:27, 20 December 2010 (UTC)
replies: text: The first is a fair point; the comment should really direct readers to the more specific articles for the separate notions, rather than the current phrasing. The second is not so much a complaint about "original research", but rather about lack of citations. I'm adding a "References needed" template, but I don't really have time to do a rewrite right now. Magidin (talk) 17:18, 28 January 2016 (UTC)
thread_title:  Generality of definition (original research?)