Talk:Operation (mathematics)
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[edit] Notes & Queries
Jon Awbrey 15:38, 20 February 2006 (UTC)
[edit] Reverted Version
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)
In its simplest meaning in mathematics and logic, an operation combines two values to produce a third. Examples include addition, subtraction, multiplication, division, and exponentiation. Such operations are often called binary operations. Other operations only involve a single value, for example negation (changing the sign of a number), inversion (dividing one by the number) and taking a square root. These are called unary operations.
Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the concatenation operation, performing the first rotation and then the second.
Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined is called its domain. The range of values that can be produced is called the codomain. For example, in the real numbers, the squaring operation only produces positive numbers.
Operations can involve dissimilar objects. A vector can be multiplied by a scaler and the inner product operation on two vectors produces a scaler. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
An operation is often called an operator, though other users of the term may reserve it for more specialized uses. The values combined are called operands, arguments or inputs, and the value produced is called the result or output.
General definition
An operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k (the number of arguments) is called the arity of the operation. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y.
Often times, use of the term operation implies that the domain of the function is a power of the codomain.
JA: By way of guidance, I am appending a table of data that that I collected the last time I encountered this issue. It illustrates the various ways that previous editors have dealt with the shift from tutorial to standard articles in several other settings. Jon Awbrey 14:34, 23 June 2006 (UTC)
[edit] Table. Tutorial Articles and Standard Articles in WikiPedia
JA: Here is a sample of data gathered on the division between introductory level and advanced level articles, as the distinction is currently drawn on a de facto basis in WikiPedia:
[edit] Response
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)
JA: Wikipedia is an encyclopedia. Its purpose, according to a current fundraising appeal, is officially portrayed in the following manner:
Imagine a world in which every person has free access to the sum of all human knowledge. That's what we're doing.
JA: Fundamental principles of truth in advertising and fundraising behoove us to take the slogan "sum of all human knowledge" rather seriously.
JA: Gotta go for now, time for lunch. Will discuss the implications of this mission statement in a little bit. Jon Awbrey 15:45, 23 June 2006 (UTC)
- 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)
JA: The word "standard" in the context of an article titled Operation (mathematics) and placed under the WP Category of Mathematical Logic means that the concept of an operation is here intended and treated in the way that it is understood by those who use the concept in mathematics and mathematical logic. The article was created because there was recognized to be a definite need, in part stemming from the recurrent need to refer to the concept in other articles, to cover the concept of Operation at just this level of logical exactness and mathematical generality. I know this as a matter of history. It is ill-advised for you to attempt to recategorize the article under the category of Elementary Mathematics.
JA: If you visit the generic-disambiguation page for Operation, you will see that there are the following articles where your introductory material might fit better, if not already adequately covered there, for instance, Unary operation and Binary operation. There is also a page for Algebraic operation that currently redirects to Operation (mathematics), that might be recycled to your purpose, but I cannot guarantee that editors with a stake in algebra will not say many of the same things that I am saying here. Still, it might be worth a try.
JA: Again, I invite you to scan the table of style models for how this issue has been handled in a de facto way in other settings. Jon Awbrey 16:58, 23 June 2006 (UTC)
[edit] Removed material
JA: Placing this material here for distribution to introductory articles. Jon Awbrey 17:15, 23 June 2006 (UTC)
In its simplest meaning in mathematics and logic, an operation combines two values to produce a third. Examples include addition, subtraction, multiplication, division, and exponentiation. Such operations are often called binary operations. Other operations only involve a single value, for example negation (changing the sign of a number), inversion (dividing one by the number) and taking a square root. These are called unary operations.
Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the concatenation operation, performing the first rotation and then the second.
Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined is called its domain. The range of values that can be produced is called the codomain. For example, in the real numbers, the squaring operation only produces positive numbers.
Operations can involve dissimilar objects. A vector can be multiplied by a scaler and the inner product operation on two vectors produces a scaler. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
An operation is often called an operator, though other users of the term may reserve it for more specialized uses. The values combined are called operands, arguments or inputs, and the value produced is called the result or output.
"Operation" is an elementary term. The disambiguation page says "An operation or operator in mathematics. See unary operation, binary operation, arity." Someone helping their kid with his or her homework would likely end up here. So that is the right audience. Again according to policy WP:NAME: "Generally, article naming should give priority to what the majority of English speakers would most easily recognize..." And of course the term belongs in Category:Elementary mathematics. If there is a need for a specialized page for the mathematical logic community (and I fail to see why since they are using the ordinary meaning), a proper name for such an article might be "Operation (mathematical logic)." --agr 17:28, 23 June 2006 (UTC)
- I agree with agr. Paul August ☎ 19:11, 23 June 2006 (UTC)
- 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)
It is incorrect to say that "operation" and "operator" are synonyms. See my explanation at Talk:Operator. Zaslav 08:57, 25 June 2006 (UTC)
[edit] General definition
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)
[edit] Third opinion
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)
- 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)
JA: It might be a good idea to look at the format eventually settled on at Relation (mathematics), by way recycling some of the NP-hard-achieved consensual experience finally arrived at there. Everybody here wants mathematical topics to be more accessible to everybody. But there is also a need for quick refreshers and notation fixers that can be referred to in other articles. Again, I think that all of this calls for some kind of generic strategy, perhaps even aided by a standard template. Jon Awbrey 20:01, 23 June 2006 (UTC)
- See Wikipedia talk:WikiProject Mathematics for my comments on Relation. --agr 21:21, 23 June 2006 (UTC)
[edit] Create introductory articles to the topic!
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)
[edit] Square Root = Unary?
Isn't the square root the same as x^1/2 (x to the power of one half)?
So it's not unary, you really need two arguments.
Or can you simply define an unary operation as a binary that comes with an embedded argument?
If it is so, that's kind of confusing.
At any rate, I think it would be nice making that clear in the article.
I would do it myself, except I can't explain it in fancy encyclopaedic jargon, so it's better leaving that for someone more competent.
- Exponentiation is binary. The square root is unary. They are two different functions, although one can be expressed in terms of the other. It's just like negation, -x, which is unary, even though it's equal to a binary subtraction with a fixed argument, 0-x. —David Eppstein 05:26, 7 March 2007 (UTC)
- Thanks for the info, David. I never thought negation could also been seen in these terms, because I usually think of negation in the propositional calculus context.
[edit] Operation vs Function
What is the difference between an operation and a function? Jim Bowery (talk) 17:50, 26 January 2008 (UTC)
- 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.,
for some ordinal α. 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)
