Talk:Range of a function

Comments

Basically range is the largest number minus the smallest number of a set of data for example: the largest number is 9 and the smallest number is 3 so 9-3=6 —Preceding unsigned comment added by 213.249.237.201 (talk) 18:03, 10 November 2008 (UTC)

w how to define the range of a morphism in category theory. Wikipedia didn't know one, nor did MacLane nor that ACC online book. Therefore I made up a definition, possibly in violation of WP:NOR. Is this unacceptable? Is the definition even correct? Comments welcome. -lethe ^{talk +} 09:05, 25 March 2006 (UTC)

That is a violation of WP:NOR for sure. See maybe google books has any info on that, or mathworld or the Springer encyclopedia. Oleg Alexandrov (talk) 16:28, 25 March 2006 (UTC)

Comment for the above text about category theory

Morphisms and their composition are central concepts in category theory. Talking about morphisms instead of functions has the advantage to place more attention to their properties (one-one, onto, etc.) removing any ambiguity in this sense because explicit names, i.e. automorphism, isomorphisms, endomorphisms, etc. have precise meaning.

Given two morphisms $f:\gamma \to \beta$ and $g:\alpha \to \gamma$ may be composed resulting in a function $f\circ g:\alpha \to \beta$.

Facts like "if $f$ and $g$ are isomorphisms then $f\circ g$ is also an isomorphism", make unnecessary to mention the range.

In my humble opinion, as someone whose approach to category theory comes from functional programming, the category theory literature known to me use the more precise term codomain instead of range, but as exemplified above specific morphism names clearly states if a morphism is onto or not.

The book: Conceptual mathematics: a first approach to categories by F. William Lawvere and Stephen H. Schanuel (I am re-translating the title from the spanish edition) show different morphism and some laws of composition. —Preceding unsigned comment added by Elias (talk • contribs) 07:19, 14 September 2007 (UTC)

Codomain, range, image confusion

I was taught the definitions given in the article for codomain and range. In the years since and in innumerable and very modern (non-set theory) books I have seen again and again the word range referring to the so-called co-domain and the image being used for f(A). Therefore I am forced to reject the statement that "Older books sometimes call what is now called the codomain the range, and what is now called the range the image set." I think that even today this is the prevailing definition among working mathematicians. nadav 05:44, 23 October 2006 (UTC)

I agree, perhaps to help people who have been taught the im(A) notation we could put an entry on the image disamb. page. 128.211.223.73 14:21, 22 February 2007 (UTC)

May I chime in? The word codomain was absent from textbooks, lectures etc. during my grad school years -- early 50's. And I have since wondered why one needs it. The range of a function doesn't have to be specified. It's an intrinsic attribute of the function itself. Then \lightbulb I got it. When you say, for example, that the codomain of a function is the reals, you are, by implication, specifying that you can write f(x)+g(x) and mean by '+' addition among real numbers. And that's not an intrinsic attribute of the function. Maybe the article should point out that specifying the codomain of a function as an algebra of some sort allows the function to participate in the algebra's operations. If this is correct, "range" may not even be (at least in a literal sense) a subset of the codomain. Morseite 21:20, 20 August 2007 (UTC)

That seems to be a sensible definition (perhaps with some tweaking), but is it standard? I take it you are including that you could specify codomains without any algebraic properties; it seems restrictive that it has to be a field or something like it. Also, I think the image has to be a subset; at least the figure on the codomain page seems to imply that, and it would seem to add a lot of complication if it weren't true. --24.130.26.157 (talk) 05:20, 30 June 2008 (UTC)

There are various other reasons that codomains are important. For example, you can't even talk about whether a function is onto unless there's an implied codomain. For another example, you need to know the codomain of a linear operator if you want to be able to define its adjoint. Codomains are particularly important in category theory, where a category is defined as a collection of objects and a collection of morphisms, each of which has a domain and a codomain.

For the first example, couldn't you can simply say e.g. "sin(x) is not onto \mathbb{R}", just as "tan(x) is not onto \mathbb{C}"? i.e., no implication / codomain definition necessary? --24.130.26.157 (talk) 05:25, 30 June 2008 (UTC)

In any case, my experience is that most mathematicians try to avoid using the word "range" unless they are dealing with real-valued functions, or the codomain is otherwise clear. "Codomain" and "image" are both unambiguous, but "range" has two possible meanings. On the other hand, calculus books all seem to use the word "range" to mean "image" Also, I've sometimes heard the phrase "range space" or "range set" used to refer to the codomain.

As for the article, I think the current text is somewhat misleading. My suggestion would be to add a section entitled "Range vs. codomain" that discusses the difference and mentions the ambiguity. Jim 05:24, 22 August 2007 (UTC)

This might not be a bad idea. The difficulty is different authors take different points of view on the terms. For example in Dummit and Foote's book on Abstract Algebra they define the range and image to be the same. In Munkres' topology he defines range to be the same as co-domain. These books are standard references for undergraduates in their fields. I have a feeling working mathematicians might argue about which is the right convention. Thenub314 (talk) 14:14, 27 April 2008 (UTC)

I'm not going to comment on the relative merit of "range" versus "image"; I just give a brief comment on codomains, images, and ranges from a category theoretical point of view. In a general category, morphisms are not nevcessarily functions. Hence, they do not automatically have defined ranges. They do have codomains, however, since they were defined as morphisms "between" specified objects, which you may call "domains" and "codomains".

In order to define "images" without reference to function properties, e.g. Saunders Mac Lane (Categories for the working mathematician) approaches the concept in a fairly roundabout manner. He proves, that in any abelian category any morphism (or "arrow") f has a factorisation f = me, where m is a monomorphism, i.e., satisfies

$mg=mh\implies g=h$

and e is an epimorphism (which concept is defined dually). He also proves that up to some isomorphisms and commutative diagrams this decomposition is unique (op. cit. VIII.3, Proposition 1), and then defines the image of f as the morphism m.

Now, in e.g. the category of all left modules over a fixed (unitary) ring, the decomposition precisely corresponds to identifying the range (or set-theoretical image); e may be chosen as the surjective homomorphism onto the range, and m as the inclusion of the range into the codomain. However, I do not think that this merits more in this article than a brief reference to the category theoretical concept. JoergenB (talk) 19:20, 3 October 2008 (UTC)

Sequence of integer numbers

In for example computer science and numerical computing, the range from a to b refers to a, a+1, … b, i.e. the sequence or series of integer numbers from a to b. Is this okay to mention in this article? Or what mathematical terminology (in words) is appropriate for this? Mange01 23:49, 3 December 2006 (UTC)

I don't know if this was the case when you posted this, but your subject is found in Range (computer science) which can be reached from the disambiguation page for Range. Maghnus 23:28, 25 September 2007 (UTC)

Wordings

If it is possible could someone add to the article a simple explanation of what a range is. Thanks 59.100.252.71 (talk) 09:50, 20 May 2008 (UTC)

The first line says "the range of a function is the set of all "output" values produced by that function". I think it's hard to get simpler than that, assuming the reader knows what a set and a function is (and that should not be explained in every article mentioning the two concepts). PrimeHunter (talk) 13:52, 20 May 2008 (UTC)

what is a range how would you explain it to an 6th grader —Preceding unsigned comment added by 72.208.9.191 (talk) 02:39, 20 August 2008 (UTC)

Syntax

The last line says "thus the range is [0, ∞)." I actually came to this page to find out what the [ and ( represent. I know it has to do with whether the numbers are inclusive but couldn't remember which was which. the article uses them without giving a definition. It should explain what the [ and ( syntax represent. Mloren (talk) 04:12, 4 January 2009 (UTC)

I've put a reference and example of interval (mathematics) in the introductory section along with the statistics meaning. Dmcq (talk) 11:38, 4 January 2009 (UTC)

Lead paragraph

I would like to work on the lead to make clear that books vary about what precisely the range is. What do people think about this? Above I gave to examples (Munkres;Dummit and Foote) of standard texts that take different definitions, so I hope people agree it is worth mentioning. Of course if we make this change, the examples section will need to be re-worked. (And some edits at articles like codomain should also take this into account.) Here is my suggestion about what a lead paragraph may look like, feel free to edit it or critique it. Thenub314 (talk) 13:05, 4 January 2009 (UTC)

In mathematics, the range of a function describes the set "output" values produced by that function. The precise definition varies from author to author. In some cases the range is defined as the set of all output values produced by that function, this is also called the image of the function. Other times the range is defined as a larger set that describes the possible output values, which is often called the codomain. If a function is a surjection then its image is equal to its codomain, and their is not confusion. In a representation of a function in a xy Cartesian coordinate system, the range is represented on the ordinate (on the y axis).

There are a couple of problems with what you say there

• The lead should try and avoid too much indecision, an extra paragraph is better for less common alternatives.
• The possible output values is the codomain not the range in common mathematical parlance nowadays. In computing and some maths books the range is the possible values and an extra bit could be added saying something like that and referring to range in computing.
• The range is the image of the domain of the function, you can have an image of a subset of the domain.

I'd go for an extra paragraph between the two like:

In some books, especially older ones, the range means the set of all possible values, i.e. the codomain. This is also the current usage for range in computer science. Dmcq (talk) 13:42, 4 January 2009 (UTC)

In fact I think I'll go and add that to the article and copy over the nice picture explaining range from the codomain article. Dmcq (talk) 13:46, 4 January 2009 (UTC)

I disagree that one is a less common alternative. I also disagree about the common parlance of mathematics these days. I see it really as a "six of 1 half dozen of the other" situation. Thenub314 (talk) 15:00, 4 January 2009 (UTC)

Well I just had a look through google books with "function domain range" and I didn't find a single example in mathematics in the first 6 pages I looked at where it meant anything different from the principal meaning as given in the first paragraph. There was a philosophy book and some computer science which used it in the second sense. If you have some evidence otherwise then I'd like to see it. Dmcq (talk) 18:53, 4 January 2009 (UTC)

Well, doing a quick check, I can point to these books (I know of at least one of which that is in active use in an undergraduate curriculum) All of which I think count as recent.:

Real Analysis and Foundations

By Steven George Krantz

Published by CRC Press, 1991

Real Analysis and Applications: Including Fourier Series and the Calculus of Variations

By Frank Morgan

Published by AMS Bookstore, 2005

Mathematical Analysis: A Concise Introduction

By Bernd S. Schroder

Published by John Wiley & Sons, 2008

Tools of the Trade: Introduction to Advanced Mathematics

By Paul J. Sally, Jr.

Published by AMS, 2008

I can only point to personal experience as to the common usage among mathematicians. Thenub314 (talk) 20:35, 4 January 2009 (UTC)

That's very surprising, okay you've proved your point. I wonder why I didn't find anything with my search. How did you find these? Dmcq (talk) 22:17, 4 January 2009 (UTC)

Well I specifically was looking at books for a "first rigorous class" in an undergrad curriculum. After some thought as how to search that category of books I decided to start with introductory analysis books. (Looking back you'll notice a slant in the style and nature of the book). Now to play devil's advocate for your side of the argument. All elementary calculus books like to talk about the range as the image, because they love to ask silly questions about "finding the range" (I should mention I dislike this type of question, and feel it got too much focus in my education). Calculus is a much more common course for people to take, so if we are going to start with one definition we should start with the range as the image, but should quickly mention other alternatives. Thenub314 (talk) 07:42, 5 January 2009 (UTC)

Ranges and intervals

Many ranges are of the form of an interval, e.g. [0,1) for the numbers from 0 to 1 including 0 and excluding 1. Thus intervals are often referred to as ranges.

Really? We call an interval a "range" because the image of a function is sometimes an interval? And usages such as "the range of data", a "salary range", the "range" of a singer, and "the range of possibilities" come from the range of a function? I seriously doubt that. If anything, I imagine it's the other way around. 68.239.116.212 (talk) 02:02, 13 January 2010 (UTC)

Those meanings are covered in Range (statistics). This is about the meaning of range in maths, in particular analysis. If you can phrase it better please do but consider that 'range' can also refer to all fair haired people or all the odd numbers for instance. Dmcq (talk) 08:09, 13 January 2010 (UTC)

I'm not saying that any other meaning should be covered here. I'm saying that Thus intervals are often referred to as ranges is false (because of the "thus"). Since the point being made is incorrect, I think both sentences should be removed. 68.239.116.212 (talk) 13:23, 13 January 2010 (UTC)

I was not saying any other meaning should be covered. I was simply pointing out that you were picking particular types of range that suited your point rather than using the meaning of range in general. Someones removed the whole business which I also believe is wrong but you said you agree with. Would you agree with a neutral statement that an interval is sometimes referred to as a range or do you blieve that should be confined to the statistics view? I thought it occurred in other areas besides statistics. Dmcq (talk) 13:51, 13 January 2010 (UTC)

I'm failing to convey my meaning. I'm not sure what you mean about "the statistics view". Let me rephrase my original point. The sentences in question made, in part, an etymological claim. That claim is, I'm fairly certain, incorrect, so it should be removed. The only remaining content is that intervals are often referred to as ranges. This is of course true, but not particularly helpful here (indeed, I assumed that the whole point was to make the etymological connection). Thus, removing the whole thing makes sense to me. 68.239.116.212 (talk) 14:26, 13 January 2010 (UTC)

By the statistics view I meant Range (statistics), this is Range (mathematics) where it has two possible meanings relating to the output values of function. I just had a look at the disambiguation page for Range and it gives as the first maths meaning an Interval (mathematics) - but that article never mentions the word range at all. I don't think anyone wanted to make an etymological claim here and I'm happy for the first part of that paragraph to go. I have just looked up google books and it is clear that intervals are occasionally referred to as ranges in mathematics besides statistics, statements like where x is in the range 0 to 1 where more properly they should say something like the domain is the closed interval 0 to 1, here they weren't even referring to output values of a function. I think I should see about getting Interval (mathematics) to refer to range and then come back here again with that as another meaning. Dmcq (talk) 17:46, 13 January 2010 (UTC)

Article is misguided

The only reason this article should exist is to explain the unfortunate history of the word "range" -- with both of its meanings -- not just one -- mentioned immediately. This article should then refer the reader to the article on codomain and the article on ''image (mathematics)'' -- which does not yet exist -- depending on which meaning of "range" is of interest.

Here is why: As discussed in a section above, the word "range" is ambiguous. During my mathematical training from the mid 1960s through the early 1970s, a function f was defined intuitively in numerous ways, but rigorously in only one way: a subset of X x Y (for sets X and Y) such that for each x in X, there is exactly one y in Y such that (x,y) belongs to f.

(Please understand that I am *not* suggesting that we avoid intuitive definitions. But Wikipedia should always include the rigorous one, since it is a reference work.)

In this case X was called the "domain of f", and Y was called the "range of f". The name for the set of values that f takes (i.e., the set {f(x) such that x is in X}) was the "image of f".

At some point, some calculus books erroneously defined range as what was properly called only the image, and this terminology caught on and was repeated in many later textbooks written by people without rigorous mathematical training.

We cannot undo this error, but it helps no one to perpetuate it! As the article states -- a little too far down for my taste -- it is preferable to avoid the word "range" due to the ambiguity.

P.S. One part of the article begins:

"Range may also be restricted by the definition of the function."

This makes as much sense as saying "The value of a number N may depend on which number N is defined to be."Daqu (talk) 16:38, 3 February 2010 (UTC)

Different mathematicians use words differently, but range is still more common than image, hense this article. Wishing mathematical usage made perfect sense is natural, but we have to live in the real world. If you think this article is bad, take a look at ring (mathematics).

As I understand it, the sentence about range being sometimes restricted by definition means that, for example, the range of the square root function is, by definition, restricted to the non-negative numbers. Rick Norwood (talk) 21:01, 3 February 2010 (UTC)

Range is not used by mathematicians much at all, precisely because of the ambiguity. It is mainly found in calculus books, where this error began. That is why Wikipedia has a duty not to perpetuate the confusion.

As for what the "sentence about range sometimes being restricted" means, it would be best if it stated what it means clearly/Daqu (talk) 19:23, 8 February 2010 (UTC)

You seem to think that it is possible for a name to be an error. Names are arbitrary symbols. Rick Norwood (talk) 23:59, 8 February 2010 (UTC)

Names for concepts -- often referred to as terminology -- become problematic when they acquire more than one meaning in the same area of discussion, especially when each meaning is widely used.

As I said, the use of "range" to mean the image of a function was originally an error, since range has long had a different (but closely related) meaning among mathematicians. Now that it has widely caught on, I don't know if it can still be called an error . . . but the situation with the same word having two distinct meanings, both concerned with the values that a function can take, has caused much confusion. E.g., when a student has just taken a year of calculus with the word range meaning the set of values a given function actually takes, and then goes to a more advanced math course where the word has the standard mathematical meaning of the set that a function (by definition) takes values *in*, this leads to confusion. It's a situation that one would not want Wikipedia to help perpetuate by making the initially wrong definitions -- the one that mathematicians don't use -- to be the main definition given here.

Instead, Wikipedia should describe the situation (of two meanings for the word) as it exists, and mention that the definition used by mathematicians (who, after all, should know) is the set Y, such that a function f having domain the set X, is a subset of XxY (where X is the domain of f).

The reason, by the way, that the codomain concept is so important is that for many if not most functions, there is no convenient way to determine the exact set of values it takes. But it's always necessary and possible to describe a set Y such that all the values of a given function f lie in Y.

Also, there's no way to tell whether a function is surjective (aka "onto"), or has an inverse function -- unless its codomain is known.

Furthermore: Someone above wrote: "Different mathematicians use words differently, but range is still more common than image, hense this article."

Hmm, I wonder what the source of this data is. I'd be extremely surprised if anyone can find a use of the word "range" to mean the set of all values a function actually takes, in any book by a mathematics department faculty member since, say, 1950 at any of these 10 universities: Harvard, M.I.T., Yale, Princeton, Columbia, Johns Hopkins, Cornell, Stanford, Berkeley, or Caltech.Daqu (talk) 10:11, 16 February 2010 (UTC)

Here is George Thomas writing at M.I.T. in 1951, "The set of numbers over which x may vary is called the range of x. In most of our applications, the ranges of our variables will consist of intervals of numbers..." Footnote: "Some writers reserve the word range for the dependent variable, and use the word domain for the independent variable. We shall not make this distinction." Calculus and Analytic Geometry, 2nd edition.

Moral: Writers at M.I.T. et al use words to mean whatever they want them to mean. Rick Norwood (talk) 14:31, 16 February 2010 (UTC)

Touché! I had a feeling my challenge was so broad it might be easy to fulfill. In fact, I did write above that the main place that misuse of "range" occurred was in calculus books . . . and Thomas was not known around M.I.T. for doing much mathematics beyond that calculus book.

Calculus books are known for getting subtle things wrong. For example, tens of millions of calculus students have been told that (where x is non-zero) any antiderivative of

                g(x) = 1/x

is of form

                G(x) = loge|x| + C.

But this is wrong.Daqu (talk) 09:17, 17 February 2010 (UTC)

Yeah I had this one going on for months on Wkipedia that I was wrong in an article because of some silly thing in one of the Schaum's Outline series. Having other people coming along and giving their opinion and giving proofs and examples where they were wrong were ineffective. Dmcq (talk) 10:16, 17 February 2010 (UTC)

I see list of integrals does exactly what you say for the integral of 1/x. Have you got a cite for a book with a better expression in?, the text has the real number version and just sticking in arg looks wrong. Thanks Dmcq (talk) 14:34, 17 February 2010 (UTC)

I'm planning to write a brief article about this, but have never seen it in print, although it's actually quite elementary.Daqu (talk) 20:31, 17 February 2010 (UTC)

You really need some sort of source if all the books give that expression above for the integral. And how can one actually say it is wrong if working with real analysis anyway? The best I could think of was saying a definite integral including 0 had an indeterminate value, I haven't a really good idea of what one should say about it. I'll ask at WP:RDMA and see if anyone has an idea there. Dmcq (talk) 22:32, 17 February 2010 (UTC)

I can say it's wrong because I understand why it's wrong. The reason it's wrong is so simple that anyone who got an A or B in a rigorous first course in calculus that they haven't forgotten can figure it out. (I finished my first calculus course over 45 years ago and have taught it many times since.)

It is also true that I have not explained here why it's wrong, but the reason for that is simply that I'm writing a brief article on this at the moment. And just to be clear, the only reason I mentioned this example is to show that just because a claim is widely published in calculus books, that doesn't make it true.Daqu (talk) 19:14, 23 February 2010 (UTC)

I have warned about the problem at List of integrals#Rational functions based on what the reference desk said but I haven't put in a citation. Waiting for some comment there or some citation befor I extend it anywhere else. Dmcq (talk) 19:21, 23 February 2010 (UTC)

Proposal to move - current title is too ambiguous

Range (mathematics) → Range (function) – Looking at the feedback this article's gotten, it seems clear that the concept we're discussing here is not the concept they're looking for. I feel that the title is deceptive. We're discussing something too specific here to warrant having such a general title. Range (function) already exists and redirects here. I think that's backwards. This content should be at that page, and this page should redirect to the disambiguation page. Any thoughts? Aurochs (Talk | Block) 22:08, 11 January 2013 (UTC)

• Strong Oppose "function" is more ambiguous than "mathematics". function is a disambiguation page. This isn't about some function commonly found in a programming library for writing computer programs. Nor is it some sort of cooking device at a social function. Or any other form of function. mathematics is not a disambiguation page. -- 76.65.128.43 (talk) 05:59, 13 January 2013 (UTC)

• Comment I agree that the current title Range (mathematics) can be misleading to the layman reader. However, Range (function) to me suggests an article about a function named "Range". Perhaps rename to range of a function instead? Within mathematics, I believe that "range of a function" is the primary meaning of "range" but this wouldn't be obvious to the layman reader, and so I agree that the page currently at range (mathematics) should be renamed to a somewhat more specific title, within the limits set by WP:PRECISION: "Usually, titles should be precise enough to unambiguously define the topical scope of the article, but no more precise than that." The goal is to find a title that ensures that "a reader who searches for a topic using a particular term can get to the information on that topic quickly and easily, whichever of the possible topics it might be" (Wikipedia:Disambiguation). I think it is clear that range (mathematics) fails here. I would have thought that the disambiguation page range already does a good job at leading the layman reader to the right page: it mentions Range (mathematics) ("a set containing the output values produced by a function", Interval (mathematics) ("also called a range, a set of real numbers that includes all numbers between any two numbers in the set"), and Range (statistics) ("the difference between the highest and the lowest values in a set") as the very first terms. And yet, reading the feedback this article's gotten suggests to me that many readers coming to Range (mathematics) are actually looking for something like Interval (mathematics). Another hatnote at range (mathematics) leading to Interval (mathematics) would help to solve that problem, but this would make the header of range (mathematics) function as another disambiguation page, and I think we should avoid having more than one level of disambiguation. I would prefer a solution were the article about the range of a function does not need to carry any disambiguating hatnotes at all. — Tobias Bergemann (talk) 09:39, 13 January 2013 (UTC)