Talk:Range (mathematics)

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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

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)

[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)


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.

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)

[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)