Talk:Function (mathematics)

3.2: Functions with multiple outputs is ambiguous as to the notation of mapping two inputs to a single output

The general example of the two part notation, in 3.1("Notation"), doesn't intuitively extend to functions with multiple outputs. I think an example of the notation in 3.1, of a function with multiple inputs and outputs would make that section(3.2) of the article more accessible. —Preceding unsigned comment added by 96.235.67.230 (talk) 07:31, 26 January 2011 (UTC)

In mathematics, the definition of a function is that it has a single output for a given input. If in a particular context multiple outputs are desired, then the codomain of the function is usually defined as the power set of some universal set, so that the output is a single set rather than a single number. If something has multiple outputs (such as "less than") it is called a relation rather than a function. On the other hand, a function may have different inputs with the same output. Such functions are not one-to-one, but are still functions. Thus f(x) = x^2 is a function, f(x) = plus or minus the square root of x is not a function. Rick Norwood (talk) 13:07, 26 January 2011 (UTC)

I though the section "Functions with multiple inputs and outputs" was pretty clear. Multiple inputs and outputs are simply treated as functions with a single input and a single output which are a cartesian product of the various inputs or outputs. There is a simplification for multiple inputs of removing a set of brackets but In what way is that ambiguous? There is a simplification for multiple inputs of removing a set of brackets for a tuple representing a cartesian product but they are both tuples. Dmcq (talk) 15:26, 26 January 2011 (UTC)

You are correct, the section was clear on the use of f(x,y)=[mapping here] style definitions. I'm interested in how the f:D->C, where f is a function with D domain and C codomain, style definition, and in particular the second half, the half that maps the relation, would be used to define a multi-variable function. It was unclear, to me, how the example of the two part notation exemplified in 3.1 would extend to multi-variable functions.

Well using the example of swap there which has domain R×R and codomain R×R you'd get using the notation in the section above

$\begin{align} swap\colon \mathbb{R} \times \mathbb{R} &\to \mathbb{R} \times \mathbb{R} \\ \left(x, y\right) &\mapsto \left(y, x\right) \end{align}$

though of course with other mappings one might have to use something more complicate instead of that second line. I suppose one could if one were being unhelpful even use one symbol for the pair as in swap(p) where p is (x,y), in fact I think most people would forget about the brackets round p unless it was an expression, and write the mapping as for instance p → (tail(p), head(p)). Of course many mappings can't be written straightforwardly in a line like that anyhow. Dmcq (talk) 12:32, 27 January 2011 (UTC)

As a more interesting example you could consider the integer divide function, where the range is the (quotient, remainder) pair. Formally this gives

$\begin{align} intdiv\colon \mathbb{Z} \times \mathbb{N} &\to \mathbb{Z} \times \mathbb{N} \\ \forall (x, d, q, r), d > 0 \implies \mathit{intdiv}\left(x, d\right) = \left(q, r\right) \\ \text{where}~ x = q \times d + r \and 0 \leq r < d \end{align}$

with apologies for the bad layout which I haven't yet learnt to handle. SamuelTheGhost (talk) 16:32, 27 January 2011 (UTC)

That clears up any question I had. Is that definition worth putting in the article, or should we leave it at this? —Preceding unsigned comment added by 108.10.194.223 (talk) 23:09, 27 January 2011 (UTC)

I don't think there's anything new there but if somebody thinks it worth sticking in some more explicit example of what it means, well I haven't any strong feeling about that which means I'm not doing anything but I wouldn't object if somebody else did. Dmcq (talk) 00:14, 28 January 2011 (UTC)

Current state of the lead paragraph:

The lead paragraph ought be briefer and more concise; together with the rest of the lead, it needs to beg the reader to hang around to learn more about the notion of a 'function'.

The lead paragraph could be a simple (incomplete) definition. Some texts I have reviewed have a very simple statement that defines a function, such as, "a function is a mathematical rule between two sets which assigns each member of the first, exactly one member of the second" [McGraw-Hill Dictionary of Scientific and Technical Terms: Fifth Edition, 1993, McGraw-Hill, page 816], or "a function is a rule that assigns to each element from one set exactly one element from another set" [Lial, Greenwell, and Ritchey (2002) Calculus With Applications: Brief Version: Seventh Edition, Pearson Education, page 50], or "a function is an association between two or more variables, in which to every value of each of the independent variables, or arguments, corresponds exactly one value of the dependent variable in a specified set" [Gullberg (1996) Mathematics From the Birth of Numbers, W.W. Norton & Company, NY, page 336]

The second and third paragraphs can carry the details that make the definition come alive and inviting to the reader, while serving to make the definition more formally correct.

The notion of 'function' is so fundamental to mathematics that it screams out for perfection in its explanation. That perfection, however, needn't be completely captured in the first paragraph.

The typical person that I imagine so enjoys Wikipedia that she might type "function" into the search bar is looking first for a simple, comprehensible, first-order definition with which she might gain a little-toehold in her quest for true enlightenment, and should not be scared away by a profusion of confusing adjectival modifiers that overwhelms the notion being explained. Also, a reader needs to be encouraged (invited by dint of the way the notion is expressed in language) to read the entire article.

That wasn't my experience when I typed in 'function', and I have a reasonable, yet clearly dated, background in mathematics. When I began reading that first paragraph something smacked me in the face. I had always thought of a function as having a domain and range; I found the term "codomain" stunning! After urgently reading the commentary on this discussion page and realizing what was meant by the term 'codomain', indeed, it seems to have a place in the definition of a function, but I shouldn't have had to go through all of that "what the heck is going on; am I crazy?" reading adventure just because the term 'range' is so familiar to me.

In my opinion, that first paragraph simply tries to do to much work.

Without a mention of the term 'range' in the text of the lead, notice that just to the right is an illustration with the caption:

Graph of example function, \begin{align}&\scriptstyle \\ &\textstyle f(x) = \frac{(4x^3-6x^2+1)\sqrt{x+1}}{3-x}\end{align} Both the domain and the range in the picture are the set of real numbers between −1 and 1.5.

So, my beloved 'range' was there all along!

In his book, "The Road To Reality: A Complete Guide to the Laws of the Universe," [Vintage Books, Copyright by Roger Penrose, 2004] Roger Penrose introduced the notion of 'function' in the following way; I found it suitably engaging, and quoted him:

"To Euler, and the other mathematicians of the 17th and 18th centuries, a 'function' would have meant something that one could write down explicitly, like x^2 or sin x or log(3-x+e^x), or perhaps something defined by some formula involving an integration or maybe by an explicitly given power series."

"Nowadays, one prefers to think in terms of 'mappings', whereby some array A of numbers (or of more general entities) called the domain of the function is 'mapped' to some other array B, called the target of the function. The essential point of this is that the function would assign a member of the target B to each member of the domain A. (Think of the function as 'examining' a number that belongs to A and then, depending solely upon which number it finds, it would produce a definite number belonging to B) This kind of function can be just a 'look-up table'. There would be no requirement that there be a reasonable-looking 'formula' which expresses the action of the function in a manifestly explicit way."

While we cannot steal Penrose's work, we must have, amongst the mathematicians who contributed to this discussion page, at least one who, like Penrose, can write better than the way the lead paragraphs of this topic are currently written.Langing (talk) 00:59, 18 May 2011 (UTC)

I've rewritten the lede with the above ideas in mind. Comments? Rick Norwood (talk) 13:02, 19 May 2011 (UTC)

Well I'd have much preferred you'd not stuck in all that stuff about range and accepted that it is an ambiguous and confused word. The article itself points this out, I don't know why you had to then just one meaning in the lead. There's other sources mentioned here besides a dictionary and a popular science work. Dmcq (talk) 13:43, 19 May 2011 (UTC)

I have no problems with what Langing said. However I hope you can see that 'range' was assumed by Langing to mean something because of previous experience so there is no real reason for preferring it to codomain if you are talking about someone new looking up this article, and there is a lot against sticking it in so often in the lead as a main thing when it has no agreed definition. Dmcq (talk) 13:53, 19 May 2011 (UTC)

The word "range", like the word "ring", has different definitions in different, equally authoritative, books. We need to mention both definitions of "range" or neither. My reason for mentioning the definition that defines range as image is that it avoids the "ordered triple" definition. If a function is defined by a formula, there is no way to know what the codomain is unless it is stated explicitly. Rick Norwood (talk) 16:02, 19 May 2011 (UTC)

Well image is the word if you really mean just the values that actually occur or codomain if you are talking about functions in general and the possible target values for instance real valued functions. The ordered triple is the usual definition nowadays as it allows people to deal with the concept of for instance real valued functions properly as it allows dealing with things like composition easily, the other way of going about it is only useful for particular special functions or in things like set theory when you want everything pared down to the minimum rather than being useful in itself. Dmcq (talk) 16:12, 19 May 2011 (UTC)

The point is that all mathematicians already know what a function is, and that anyone who turns to this article for information is almost certainly trying to understand the meaning of function when it is understood that the function is a real valued function of a real variable. Therefore, function as ordered triple belongs further down in the article. A Wikipedia article must not say anything that is wrong, but should not try to say everything that is right. The rule is: address the lede to the layperson. Rick Norwood (talk) 17:59, 19 May 2011 (UTC)

I think the current third paragraph needs to go. The lead should summarize the article and provide an accessible introduction. The lead is not the place to discuss fine terminological distinctions such as the one between range, image, and codomain. Nor does it seem helpful to discuss how these are often "understood" in context at this point. These are relatively minor considerations that should be addressed elsewhere. I do agree that referring to the range rather than the codomain improves the overall accessibility, though. Sławomir Biały (talk) 19:23, 19 May 2011 (UTC)

I propose the following as a replacement for the first several paragraphs:

A function, in mathematics, takes as argument a set of quantities, and assigns to each and every quantity one value. The set of all quantities input to a function is called its domain; the set of all quantities output by a function is called its range.

A particular function's argument, also called input, and its value, also called output, could both be the set of real numbers. But a great many functions exist in mathematics, so a function's argument and value can be elements from any possible sets of mathematical entities.

A simple example of a function is f(x) = 2x, where x is any real number. This function associates every real number with a real number twice as large. So, for example, 5 is associated with 10, written f(5) = 10. Notice that for this function the domain is the set of real numbers, and the range is also a set of real numbers; the two sets are not identical. Langing (talk) 19:41, 19 May 2011 (UTC)

Agree with Bialy--- Current third paragraph, that starts describing the codomain, must go.Langing (talk) 19:55, 19 May 2011 (UTC)

I have no objection to moving codomain further down in the article. On the other hand, I see serious problems with Langing's proposed lede. I do not think it is standard usage to have "argument" and "domain" be synonyms, but Langing defines both as the set of inputs. I think standard usage is for the argument to be an element of the domain, not the domain itself. Langing's first paragraph says functions take as argument a set of "quantities", the second paragraph says they may take anything as an input, not necessarily only quantities. And I'm not sure what the reader is supposed to understand by the assertion that the set of real numbers is not identical to the set of doubles of real numbers. Two sets are in the interest identical if they have the same elements, and domain and range of f(x) = 2x have the same elements. Rick Norwood (talk) 20:19, 19 May 2011 (UTC)

Referring to range the way the lead does now is a really bad idea. Range c an mean either image or codomain in mathematics and in statistics it can mean an interval. It may be a more common name for a layman but that doesn';t mean it is a better word to use because it doesn't necessarily mean what they think it means. You're far better off with words like image and codomain which only have one well defined meaning in maths. Cutting the codomain out of the lead entirely is to destroy the article as any sort of description of the concept in modern maths. Could you please not try turning this back to the time of Euler. There is no point in describing something that bears little relation to modern maths just because you wanted something more accessible. Dmcq (talk) 23:10, 19 May 2011 (UTC)

Have a look at Range (mathematics) and seer the problems. It has got to be mentioned because so many older texts mention it and computer science uses it in the sense of codomain but it just causes confusion. Dmcq (talk) 23:16, 19 May 2011 (UTC)

The exact same problem occurs with "ring". Are the even integers a ring or not? But mathematicians all understand the ambiguity and have no choice but to live with it. The lay reader needs to be told that "range" has two meanings, or else the lede should not use the word at all. Many people, not mathematicians, remember the word "range". They have a right to be told what it used to mean, and what it means today. What you have written is fine, except for a minor typo which I fixed. Rick Norwood (talk) 00:17, 20 May 2011 (UTC)

Personally I can't think of any ring without unity that can't be treated as an ideal just as well or better but yes it's good example - but at least people are very aware there is that problem. The range business is just chaos though with people assuming what it means. In computing too they have the same confusion, normally it means the codomain but it also means an interval so they refer to an argument being out of range and mean the image when they ask what values the output ranges over. Dmcq (talk) 09:56, 20 May 2011 (UTC)

Note to TheDinParis

I'll let you finish your edit, and then continue mine, but please note that I had not finished removing the repeated definitions, so there is still a lot of repetition. Rick Norwood (talk) 13:39, 21 May 2011 (UTC)

Ordered triple of sets?

There is something that doesn't make sense to me on the definition of function as an ordered triple of sets (domain, codomain, graph), because a triple is an Tuple of 3 elements, and citing the wiki of Tuple: An n-tuple can also be regarded as a function whose domain is the tuple's set of element indices, and whose codomain is the tuple's set of elements.

So a function is a tuple, and a tuple is a function. Isn't this a circular definition? Wich of the definitions should be dropped in that case? — Preceding unsigned comment added by 186.58.68.193 (talk) 19:09, 27 May 2011 (UTC)

"So a function is a tuple, and a tuple is a function." A tuple is regarded as a function and a tuple is not defined as a function. In set theory a tuple is a set and in Martin-Lof's type theory a tuple is primitive. Actually I am unaware of a theory which defines a tuple as a function. --Beroal (talk) 11:46, 18 July 2011 (UTC)

Didn't notice this query, but the definition using a triple is no more strange than the alternative definition of a function in terms of just the graph which is also a set. That doesn't mean that sets are defined as functions. Dmcq (talk) 12:05, 18 July 2011 (UTC)

Image of a set (notation)

I admit I'd never seen f`x and f``A; however, Set Theory for the Mathematician by Jean E. Rubin uses f "A. (at least, I think that's the character used; it could be f’’A or f''A.) — Arthur Rubin (talk) 09:34, 2 August 2011 (UTC)

I just noticed in the Image (mathematics) article a mention of f "A as a notation used in set theory, and it didn't have a citation. So I'll copy your citation over to that article. It also featured more prominently which is why I missed that before some other notations so I'm not sure about its due weight here. Dmcq (talk) 17:09, 2 August 2011 (UTC)