Talk:Operation (mathematics)

Notes & Queries

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)

I find it confusing that the second sentence in this article is "Operations, as defined here, should not be confused with ... arithmetic operations on numbers" yet the first example of operations are arithmetic operations and the figure is of arithmetic operators. I am not an expert so I will not edit myself, but leaving it to the experts to clarify this. I believe the wording should be something like "should not be confused with the more specific terms ... which are specific types of operations"? — Preceding unsigned comment added by 128.138.65.242 (talk) 16:51, 28 January 2016 (UTC)

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 × … × X_k → Y. The sets X_j 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)

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:

Tutorial Articles and Standard Articles in WikiPedia

Tutorial Article	Standard Article
Introduction to special relativity	Special relativity
Introduction to general relativity	General relativity
Introduction to quantum mechanics	Quantum mechanics, Mathematical form of QM
Set	Set theory, Algebra of sets
Naive set theory	Axiomatic set theory, Alternative set theory
Function (mathematics)	Function (set theory)
Binary relation, Triadic relation	Relation (mathematics)
Graph	Graph theory
Group, Elementary group theory	Group theory
Ring (mathematics)	Ring theory
Field (mathematics)	Field theory (mathematics)
Vector space	Linear algebra
Topology	Topological space, Algebraic topology
Introduction to topos theory	Topos theory
Boolean logic	Boolean algebra
Information	Information theory
Computer science	Theoretical computer science
Computation	Theory of computation
Computing, Computable function	Computability theory
Recursion	Recursion theory
Relational database	Relational algebra, Relational model
Biology	Theoretical biology
Music	Music theory
List of basic dance topics	List of dance topics

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)

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)

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 X_i 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)

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)

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)

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.

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., $A^{\alpha }\to A$ 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)

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)

"Power of codomain": clarify please

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)

1. The definition cited refers to "two binary operations ... on L" which means two operations on L to the power two, so it is certainly using the term "operation" in the sense referred to, but it does not make any particular point of the fact that it is doing so, and does not indicate that "operation" implies this, so it is an inappropriate reference, and I shall remove it.
2. Yes, "power of the codomain" does mean "the cartesian product of the output set with itself (N times)", and I have edited the article to make this explicit.
3. I am not sure what "a function does NOT qualify as an operation on this basis" means: does it mean something which can be considered as an operation, but only if one does not take the word "operation" in the given restricted meaning? I shall put such an example into the article. JamesBWatson (talk) 15:02, 14 December 2009 (UTC)

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)

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)

Generality of definition (original research?)

"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? — Kallikanzarid^talk 20:27, 20 December 2010 (UTC)

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)