Talk:Function (mathematics)

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Ongoing discussion of the lead.

Isheden suggests introducing the idea of ordered pairs. Here is the current paragraph 4:

The set of inputs to a function is known as the domain, the set of paired inputs and outputs as the graph, and the set in which the outputs are constrained to fall as the codomain. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

How about something like this?

The input and output are often expressed as an ordered pair. In the example above, we have the ordered pair <–3, 9>. A complete definition of a particular function will give the set of inputs, called the domain, the set of paired input and outputs, called the graph, and a set in which the outputs are constrained to fall, called the codomain. For example, we could define the function f(x) = x² by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers <x, x²>.

Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

Rick Norwood (talk) 14:41, 2 April 2012 (UTC)

A newbie question: I was on-board until the example of the last sentece: Where did the "graph" go in f(x) = x². What physical object exactly is "the graph" in the case of a continuous (all points possible in the continuum) function as opposed to discrete pairs such as a lookup table? Bill Wvbailey (talk) 15:37, 2 April 2012 (UTC)

It's an infinitely long (notional) lookup table. --Cybercobra (talk) 15:52, 2 April 2012 (UTC)

Worse than that, almost every entry is infinitely long, so the table is an infinitely long listing of infinitely long entries that cannot be specified (see quote below0. This is not an object in the newbie intuitive sense of the word. I'd be more inclined to describe the graph as an analog machine, say a really big slide-rule, say, or a plotting device such that, fractal-like, no matter how close you squint there's always another digit. Again it's not surprising that newbies cannot grab the idea: "Where's the graph"? What is "the graph"? Here's a quote to back it up:

From the algorithm article:

"No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13] cf Boolos & Jeffrey (1974, 1999)

BillWvbailey (talk) 17:35, 2 April 2012 (UTC)

Intuitively, the graph of f(x) = x² is the curve in the Cartesian plane (at least for a domain such as -5 ≤ x ≤ 5). Formally, it is the set of all ordered pairs (x, x²). If the domain is a countable set such as the integers, you could write a possibly infinitely long lookup-table. For an uncountable set this might not be possible. Isheden (talk) 09:24, 3 April 2012 (UTC)

I think the example is good. However, at the beginning of paragraph 4 I would like to propose "In modern mathematics, a function is defined by its set of inputs, a set containing its outputs, and a set of input-output pairs describing the relation between inputs and outputs." Possibly "describing the relation ..." could be skipped. Isheden (talk) 15:53, 2 April 2012 (UTC)

One more remark: The set of ordered pairs is just all pairs of the form <x, x²>; that x and x² are real numbers is determined by the domain and codomain definitions, respectively. Isheden (talk) 16:22, 2 April 2012 (UTC)

Maybe we should explain the difference between a graph in the sense the word is used in elementary mathematics and a graph in the sense it is used in abstract mathematics. The article graph (mathematics) which this article links to also needs work, but I hesitate to add yet another page to my watch list. How about something like the following:

The input and output are often expressed as an ordered pair. In the example above, we have the ordered pair <–3, 9>. This ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. But no picture can exactly define every point in an infinite set. In modern mathematics, a function is defined by its set of inputs, called the domain, a set containing the outputs, called its codomain, and the set of all paired input and outputs, called the graph. For example, we could define the function f(x) = x² by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers <x, x²>. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis.

I don't think it hurts to say "real numbers" twice, since that avoids having to say that the graph is a subset of the cross product.

Let me know what you think. Rick Norwood (talk) 12:07, 3 April 2012 (UTC)

It's probably good enough to insert in the lead and to let other editors work on it. Isheden (talk) 13:09, 3 April 2012 (UTC)

I made the change. Rick Norwood (talk) 14:21, 3 April 2012 (UTC)

Gandalf61's edit

Gandalf61 changed f(x) = x² to x → x². I kind of like this, but I reverted it, because I think we need to discuss it here, first. There are (at least) three ways we could introduce the example, f(x) = x², x → x², and y = x², and we could use any one, any two, or all three. Any thoughts about how to be clear without overloading the lead? Rick Norwood (talk) 14:20, 3 April 2012 (UTC)

I think x → x² has some merit because it indicates a direction from input to output that is not visible when using the other two ways and it does not introduce any extra notation. Isheden (talk) 14:28, 3 April 2012 (UTC)

Shouldn't it be $x\mapsto x^{2}$ with a mapsto? — Carl (CBM · talk) 15:03, 3 April 2012 (UTC)

--

Edit conflict: Is this logical implication? The introduction of a new sign to replace the familiar "arithmetic equivalence" (the unfortunate choice → being the familiar logical sign for implication), is a rabbit-hole we do not want to descend into (at least in the lead). The theories I'm familiar with (e.g. used by Goedel) either add arithmetic equivalence as an axiom outright, or regard it as a generalization (over all variables of a specified class, cf PM and Goedel 1931) in other words, the sign = presupposes the notion of "function" (e.g. classes of individuals of type 2 here):

"[21] x₁ = y₁ is to be regarded as defined by x₂∀(x₂(x1)⊃ x₂(y₁) as in PM (I, *13) similarly for higher types." (Goedel 1931 in van H:600, with ∀ replacing Goedel's ∏ "for all" but retaining his sign for logical equivalence "⊃")

Maybe a different sign, e.g. =>, or one of these from unicode: ⇛ ⇒ ⇝ ↣. The idea that a function is indeed a one-way "process" is interesting, and it is "correct", e.g. we know that the function {(3, 5), (4, 5)} is not reversible, given output sign 5 we cannot deduce the exact sign that fell into the hopper. But I'd discourage introducing something as subtle as this in the lead. BillWvbailey (talk) 15:31, 3 April 2012 (UTC)

Should we then start a new paragraph to introduce f(x) = x² and y = x²? Or maybe put the entire example into a separate paragraph, and add the ordered pair notation. Maybe something like this:

In mathematics, a function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output.

An example is the function that squares any real number. This concept may be expressed in any one of several different ways. We may name the input $x$ and write $x\mapsto x^{2}$. We may name the function $f$ and write $f(x)=x^{2}$ (read "f of x equals x squared"). Or we may name the output $y$ and write $y=x^{2}$. Another notation sometimes used is to write the input and output as an ordered pair, $<x,x^{2}>$. Suppose the input is –3. Then the output is 9 and using each of the notations above we could write $-3\mapsto 9$, $f(-3)=9$, $9=(-3)^{2}$ or we could write the ordered pair $<-3,9>$.

If we go with this or something like it, we'll need to do a little rewriting in the last paragraph.

Rick Norwood (talk) 15:06, 3 April 2012 (UTC)

This is OK with me if the \mapsto is corrected. — Carl (CBM · talk) 15:08, 3 April 2012 (UTC)

This notation is even more attractive if it is called a rule. Otherwise one might wonder why it is not called an equation or formula instead. Isheden (talk) 15:14, 3 April 2012 (UTC)

Let's postpone discussing "rule" further until we get mapsto sorted out. I'm having trouble with a parsing error and have not sorted out what I'm doing wrong yet. Can either of you see what I'm doing wrong? Rick Norwood (talk) 15:16, 3 April 2012 (UTC)

I think the commas may be problematic inside the math environment. Isheden (talk) 15:17, 3 April 2012 (UTC)

Thanks. But it doesn't mind the comma in $<x,x^{2}>$.

Note to CBM, please join the ongoing discussion. We've talked at length above about whether or not $f(x)=x^{2}$ is or is not a function, and I cited three books, and could easily cite thirty books, that call expressions like this functions. To go to great lengths to avoid calling this a function is a little bit like going to great lengths to avoid calling 2 a number. Yes, 2 is a numeral rather than a number, but everybody calls 2 a number anyway. Rick Norwood (talk) 15:22, 3 April 2012 (UTC)

I think I am in the ongoing discussion (unfortunately the volume is so large that it's very hard to follow everything). Of course there are some books that call those things functions - probably the calculus book I teach from says it. But those books are being sloppy in a way that we cannot afford to be sloppy in this article. In a calculus class we don't really expect students to know what functions are, and there is a lot of context because pretty much all the objects they work with are real numbers. But in general math we care much more what a function is, and we no longer have the fixed context of the reals. So we have to be more careful about things than a calculus textbook might be. We can't say that a function is a rule, or that "x^2" is a function. — Carl (CBM · talk) 15:31, 3 April 2012 (UTC)

At the same time I do not think we need to mention domain or codomain at all in the first paragraph - we don't have to say what a function is in that paragraph, but we should avoid making claims that are either false or would be embarrassing to make in a room full of mathematicians. — Carl (CBM · talk) 15:36, 3 April 2012 (UTC)

Thanks to whoever fixed the parsing error. Please tell me how you did it.

Actually, the examples I gave were well above the calculus level. They were from Keryszig, Advanced Engineering Mathematics, Munkres, Topology, and Lomen & Lovelock, Differential Equations. That said, I have no problem with the changes you made. And, yes, the length of these discussions can be daunting.

Rick Norwood (talk) 15:42, 3 April 2012 (UTC)

The problem with the math is that it used a minus symbol instead of a hyphen, but latex only accepts a hyphen and displays it as a minus symbol. This is one reason that it's better to type − as "&minus;"

The Munkres quote I see above is 'In calculus, a function is often given by a simple formula such as f(x) = 3x^2 + 2." Note that Munkres is distinguishing between the formula and the function just as I have tried to do in the article. I would tend to lump diff eq. books in with calculus because they are aimed at engineers as much as mathematicians. Many physics books still use infinitesimals(!).

I think the situation with the number/numeral distinction is a little different because, even though the numeral "2" is not the number 2, the numeral generally only defines one number. But the expression x² can define many different functions, and the issue that the domain has to be specified is exactly the issue that an article like this has to be very careful to keep straight. — Carl (CBM · talk) 15:47, 3 April 2012 (UTC)

Thanks. Are we good to go with the paragraph above? Is there any way to stop the first $x$ from being higher than the second? Rick Norwood (talk) 15:59, 3 April 2012 (UTC)

Does this symbol show up for you? ↦ — Carl (CBM · talk) 16:36, 3 April 2012 (UTC)

Just barely. If I wasn't looking for the arrow head, I'd mistake it for a dash. Rick Norwood (talk) 16:39, 3 April 2012 (UTC)

That's out, then. If we are using <math> then there is no easy way to make things line up, unfortunately. Maybe we could do a hack like this: x $\mapsto$ x². The rest could be in regular wikitext. — Carl (CBM · talk) 16:46, 3 April 2012 (UTC)

I think, unless someone objects, I'm going to insert the paragraph above exactly as it appears. Thanks. Rick Norwood (talk) 19:54, 3 April 2012 (UTC)

First, I think the section is excessive in notation. There is no need to introduce 3-4 various ways of denoting the same thing in the lead. Second, I think it's better to wait with the lead paragraph. We should not move on without consensus this time. Isheden (talk) 20:10, 3 April 2012 (UTC)

Ok, I'll wait and see if others voice an opinion. Rick Norwood (talk) 20:45, 3 April 2012 (UTC)

No comments so far, so I'll mention another option. If the paragraph above is too long for the lead, it could serve as the first paragraph of the "Notation" section. Rick Norwood (talk) 12:12, 4 April 2012 (UTC)

Unless someone objects, I'm going to go ahead and put the paragraph in the Notation section. Rick Norwood (talk) 13:36, 5 April 2012 (UTC)

A goes to 1 or triangle goes to 3.

In the lead, the example of a function whose domain was not a set of "numbers" was the familiar map that sends A to 1, B to 2, C to 3, and so on. At some point a few weeks ago, this was first supplimented and then replaced by a map that sends a triangle to 3 and a square to 4. This isn't wrong, but there is a problem with it, in that it is not clear what the domain is. All polygons? All equivalence classes of polygons? All polygons in the Euclidean plane? My inclination is to restore the more familiar A maps to 1, B maps to 2, and so on. Comments? Rick Norwood (talk) 12:19, 4 April 2012 (UTC)

Unless you specify the domain and the range and not leave it to "intuition" or "a priori knowledge", either image is insufficient. To specify a function you have to specify a triple, correct?: ( "source", "destination", "graph"), i.e. ( S, D, G ) like the Bourbaki folks label it (my preference). And it would seem (more tacit, a priori knowledge) that you need to specify how the "triple" is to be read, i.e. ( (S, D), G), as opposed to ( S, (D, G)). Correct? Bill Wvbailey (talk) 16:41, 4 April 2012 (UTC)

I put that in because the previous sentence said the sets could contain anything for instance geometric figures. If geometric figures is not a sufficiently good domain for you then it should be removed from the previous sentence as well. Its seemed silly to just stick in geometric figures and the set of letters was not talked about and seems very limiting. Dmcq (talk) 18:07, 4 April 2012 (UTC)

The reference to "geometric figures" bothered me, too. What does a circle map to? How about a fractal? I think clearly "polygon" is intended. You say the set of letters is "not talked about". That's why I'm talking about it here before making any changes. Rick Norwood (talk) 18:56, 4 April 2012 (UTC)

Well the function isn't fully defined by just a couple of examples, in fact one could have hexagon go to 5 as far as the definition of a function is concerned. Personally by geometric figure I think of something drawn by a straightedge and compass, but from the origami standpoint I'd be quite happy with just lines. Dmcq (talk) 19:43, 4 April 2012 (UTC)

So, how about my suggestion to go back to a function that sends A to 1, B to 2, C to 3, etc. Well defined domain and codomain, familiar example, no hassle. Rick Norwood (talk) 20:06, 4 April 2012 (UTC)

Edit conflict: :Isn't there a risk of confusion if the mapping is per any kind of rule, such as for a "geometric figure" (e.g. a 2-dimensional n-sided symbol with n-exterior sides)? Don't you want an example that contains some elements very abstract and seemingly arbitrary together with something familiar, e.g. ( { ✻, ✣, ✩, ✦, ✶}, { suits of a deck of playing cards }, { ( ✻, ♣),(✦, ♦),(✣, ♡),(✶, ♥), (✩, ♣) } ) Or maybe the musical notations { sharp ♯, flat♭, natural ♮ } mapped to { the suits of a deck of cards: ♣ ♠ ♡ ♢ ♧ ♤ ♥ ♦ }. ( These symbols appeared okay on my browser.) BillWvbailey (talk) 20:24, 4 April 2012 (UTC)

Didn't work in my browser, sad to say. I don't think we want something exotic, just something that is not a number. Rick Norwood (talk) 21:11, 4 April 2012 (UTC)

I suppose we could pick selected symbols from: {Roman alphabet} to { Greek alphabet } or to { Cyrillic }. But not in alphabetical order. Or more abstractly, from { Greek alphabet } map to selected symbols from { Cyrillic alphabet }: { Π, Φ, Λ, Ψ }, { Д, Л, Ж, И }, { (Π, Ж), (Φ, Л), (Λ, И), (Ψ, Л) }. Plus the symbols are pretty and had better work on all browsers. Bill Wvbailey (talk) 22:54, 4 April 2012 (UTC)

I really think we want as simple and short an example as possible in the lead. Rick Norwood (talk) 12:09, 5 April 2012 (UTC)

I also didn't like letters because once you make them into mathematical objects they are practically numbers anyway. That's why I'm not so keen on things like date to day of the week either though I guess it could work from the familiarity point of view. I'd prefer something that started off a bit mathematical. How about something like triangle to area? Dmcq (talk) 13:05, 5 April 2012 (UTC)

If we are going to use triangles at all, I think polygon to number of sides is simpler than triangle to area. But there is still the problem of exactly what the domain is. I'm not sure how you would precisely define the set of all triangles or the set of all polygons, the problem being the larger space in which such object exist. You can't use Rⁿ for any fixed n, because there will be a triangle in Rⁿ⁺¹ that is not in Rⁿ, though of course it will lie in a two-dimensional subspace. Rick Norwood (talk) 13:34, 5 April 2012 (UTC)

I think you're just finding problems where there aren't any. After all you wouldn't even have an area in projective geometry, I think most people would be pretty happy with something like that anyway. It is just an introduction and we don't need things like n-dimensional space in it. Dmcq (talk) 13:45, 5 April 2012 (UTC)

---

Here's Halmos 1970:31 Section 8 Functions:

"It is easy to find examples of functions in the precise set-theoretic sense of the word in both mathematics and everyday life; all we have to look for is information, not necessarily numerical, in tabulated form. One example is a city directory; the arguments of the function are, in this case, the inhabitants of the city, and the value are their addresses."

[There may be a slight argument with this in the case of some families with two domiciles or two phones but the phone book takes care of that. There's an example in our book where a couple has a residence and a barn, but the 2nd entry is listed as "barn" under their name:

John and Mary Doe [address, phone-number}

John and Mary Doe barn [phone-number]

The domain is clear-cut: names of people in the city. BillWvbailey (talk) 14:15, 5 April 2012 (UTC)

Halmos is certainly a good reference, but in his day fewer people had two phones. I still don't see the problem with the function that sends A to 1, but since several people don't like that, I guess we keep looking. Meanwhile, the way the lead is now is not bad. Rick Norwood (talk) 14:25, 5 April 2012 (UTC)

Shouldn't that be Joseph and Mary Doe stable rather than John and Mary Doe barn? ;-) If you want things that close to real world I'd have thought date to day of week was quicker and easier. Dmcq (talk) 14:34, 5 April 2012 (UTC)

The only problem with date to day of the week is that it has a number input, but really any of these would probably get the point across. Rick Norwood (talk) 14:52, 5 April 2012 (UTC)

---

Chess pieces (32 total) to name/shape/ID of the pieces: { pawn, knight, rook, bishop, king, queen }. Or to number of pieces { 16, 4, 2 }, or to color "K" { K, ~K }. Or type of piece { pawn, knight, rook, bishop, king, queen } to number of pieces in the set. Bill Wvbailey (talk) 15:50, 5 April 2012 (UTC)

Help at Simple Wikipedia

There are only a few (none?) math editors at SW so I have started work on "function" over there with the idea of understanding just what "simple" might mean in this context. Anyone not too busy here would like to help out, SW Function Selfstudier (talk) 17:46, 4 April 2012 (UTC)

Moving right along

It seems as if we've done about as much on the lead as we can get a consensus on, and I think we've done a good job. Turning to the rest of the article, I've made a few minor changes, mostly matters of style. I'll pause for comments before I do any more. Rick Norwood (talk) 12:26, 6 April 2012 (UTC)

Formal Definition

From the section "Formal Definition":

"we say f is a real valued function of a real variable, and the study of such functions is called real variables."

I think it should be real analysis.

--ConferAll (talk) 13:19, 5 June 2012 (UTC)