Talk:Range (mathematics)

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. expand, example or two, maybe a picture, etc.. --Cronholm144 00:49, 21 May 2007 (UTC)

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)

Contents 1 Comment for the above text about category theory 2 Codomain, range, image confusion 3 Sequence of integer numbers 4 Wordings 5 Syntax 6 Lead paragraph

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 and may be composed resulting in a function .

Facts like "if f and g are isomorphisms then 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)

[edit] 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

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)