Talk:Range of a function

3054:

there are many inbound wikilinks pointed here (which it would IMO be disruptive to go change all at once). So what we want is sort of halfway between an article and a disambiguation page, more or less what we have currently, focused mostly on the term and its usage rather than the concepts of codomain or image (but it could be expanded and clarified a bit). The term 'range' is encountered frequently enough that there are going to be a significant number of readers looking it up in search engines / looking it up on Knowledge trying to figure out what the definition is and possibly confused about differences in definition from one source to another (this article typically gets about 100–150 page views per day). In my opinion leaving this as either a disambiguation page or a redirect with hatnote is going to poorly serve these readers, and is much more likely to leave them confused. –

95: 517:

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.

85: 64: 31: 1231:

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.

173: 3037:

solution would address that problem (if it were a problem), since a reader would still have to read the first sentence. You are welcome to try to refine incoming links, but I think in many cases it is not possible to unambiguously choose, because for many purposes it does not matter whether the codomain is the image or something larger, often in these cases one does not specify, and so the use of range could equally well mean either concept. --

2052: 1585:, 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 Knowledge to help perpetuate by making the initially wrong definitions -- the one that mathematicians don't use -- to be the main definition given here. 22: 2454:(who was opposed to renaming the article to "Range (function)", assuming that the comments by the anonymous user with this IP address were in fact made by the same person). However, it certainly is true that there were very few participants which may mean that the name of the article is good enough for most readers. Nevertheless, I think that renaming the article from 2236:: "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" ( 1400:- 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 1580:

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

608:

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

2774:

There is reference to Rudin for the line "To avoid any confusion, a number of modern books don't use the word "range" at all.", however I looked at page 99 in the book and found nothing to back this up. In fact, Rudin himself uses the word range to refer to the image (in the book, R(T)). Can someone

1462:

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

1353:

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

950:

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

592:

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

527:

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

497:

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

579:

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

516:

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

3053:

I think it's helpful to have an article about the term 'range', precisely because it's an ambiguous term. The way disambiguation pages are structured (by policy) on Knowledge is not sufficient to clear up potential reader confusion about the distinction between various definitions of 'range', and

1966:

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

1893:

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

1262:

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

1230:

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

1326:

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.

889:

Even the first sentence of this is throwing me off: "The codomain is a set containing the function's outputs" okay, so the codomain is the function's outputs. "whereas the image is the part of the codomain" And we're saying the image is the part of a functions outputs "which consists only of the

3036:

Your interpretation (that the existence of an article that begins "In mathematics, the range of a function may refer to either of two closely related concepts" might lead people to believe that it refers to three concepts) seems deeply implausible to me. Moreover it's not at all clear how your

1078:

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

284:

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,

1936:

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

741:

Aside from all the disagreement about the current and historical meaning of "codomain", "range" and "image", there's a more fundamental issue with the article - the "formal definition" section is confusing and actually incorrect. It gives the same formal definition for two things it claims are

1596:

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

462:

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.

1590:

Instead, Knowledge 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

1616:

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:

1796:

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.

1730:

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

970:

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

2637:

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 --

1504:

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

1004:

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

2288:

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. —

1472:

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.

1354:

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.

498:

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.

466:

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.

2611:

might not be ideal. But it doesn't change the undeniable fact that I point out above that the current title is highly misleading. I'm okay with whatever everyone else wants to see, but I'm not okay with the current title.

2380:, much more natural title that's also helpful to readers. I also think that the titles pointed out by Tobias Bergemann are very sweet titles and I think using a similar approach for other articles wouldn't be a bad idea. -- 2192:

2134:, 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. 853:" 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). 1404:

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

1327:

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.

1282:. 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. 1514:

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.

1533:

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 Knowledge has a duty not to perpetuate the

904:

Good point. I'll fix it. FYI: the codomain is some set (usually the real numbers) that includes all the outputs, while the image consists of those points in the codomain that actually are outputs.

1581:

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

583:

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.

2798:

So here we are in the domain! Strictly speaking, the context determines whether we use the noun or the verb, but the word is so confusing that I think that any good mathematician should avoid it!

2228:

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

1829:

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

1476:

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.

1575:

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.

1469:

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".

250:

w how to define the range of a morphism in category theory. Knowledge didn't know one, nor did MacLane nor that ACC online book. Therefore I made up a definition, possibly in violation of

151: 1058:

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.

388: 564:

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? --

3142: 3095: 2265: 2131: 350: 1097:

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.:

318: 688: 2523:. Closing as the nominator has withdrawn their nomination and there are no other arguments in support. Tobias - if you wish to contest the previous closure, you may list it at 1744:

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

2139:

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?

1466:(Please understand that I am *not* suggesting that we avoid intuitive definitions. But Knowledge should always include the rigorous one, since it is a reference work.) 457: 714:

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);

3132: 2657:

I propose that this second move request be closed now, without waiting another seven (well, five) days: it was opened just one day after the previous move request (

3147: 2994:

Do we really need an article about range of a function? It's just an ambiguous term, so a disambiguation page would be better? There are cons about this, though.

2846: 431: 411: 890:

function's outputs." So the codomain is the functions outputs, but that's very different from the image which is also the function's output. What am I reading?

1258:

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.

226:

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

2434:

While the proposal to move was closed with the result "No consensus", I humbly suggest that there is in fact some sort of consensus to move the article from

35: 3157: 141: 2013:

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.

696:

is an epimorphism (which concept is defined dually). He also proves that up to some isomorphisms and commutative diagrams this decomposition is unique (

3127: 593:

are standard references for undergraduates in their fields. I have a feeling working mathematicians might argue about which is the right convention.

580:

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.

2794:

I agree with some of the above discussions; there is more ambiguity about the word "range" in math. Considering the verb "range", wiktionary says:

3137: 3019:: there's no third concept, as might be misinterpreted by a casual reader. Furthermore, page is currently linked from a large number of articles: 2061: 3072: 2574:

of these articles are about "mathematical" uses of range, and it is clearly unclear to have one have the more general title and not the other.

117: 3152: 2589:

Wow, I can't believe this just got debated over the past two days! I know I must look like a sock for proposing this now, but I was just at

233: 1016:

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

955:

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.

1355: 1305: 1264: 609:

do have codomains, however, since they were defined as morphisms "between" specified objects, which you may call "domains" and "codomains".

2830:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

2510:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

2087:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

2799: 2733: 2711: 2687: 2662: 2639: 2613: 2594: 2575: 2478: 975:

of the function. Other times the range is defined as a larger set that describes the possible output values, which is often called the

891: 827: 565: 529: 1602:

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

2796:

Of a variable, to be able to take any of the values in a specified range. Ex: The variable x ranges over all real values from 0 to 10.

2715: 2699: 2695: 2451: 2398: 2198: 1650: 875: 108: 69: 1204:

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?

560:, where a category is defined as a collection of objects and a collection of morphisms, each of which has a domain and a codomain. 2152:) 22:08, 11 January 2013 (UTC) Tobias Bergemann made some very good points. Based on his comment, I am changing my target name to 178:

This article has previously been nominated to be moved. Please review the prior discussions if you are considering re-nomination.

1392:

where it has two possible meanings relating to the output values of function. I just had a look at the disambiguation page for

528:

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. --

264:

That is a violation of WP:NOR for sure. See maybe google books has any info on that, or mathworld or the Springer encyclopedia.

3122: 2930: 1543:

As for what the "sentence about range sometimes being restricted" means, it would be best if it stated what it means clearly/

544:

There are various other reasons that codomains are important. For example, you can't even talk about whether a function is

1859:

I'm planning to write a brief article about this, but have never seen it in print, although it's actually quite elementary.

2983:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

2385: 507:

I agree, perhaps to help people who have been taught the im(A) notation we could put an entry on the image disamb. page.

2462:

would be an improvement, and I would rename the article in a day or three if nobody complains, no disrespect intended. —

3162: 44: 3096:

Wikipedia_talk:WikiProject_Mathematics/Archive/2023/Dec#h-Disambiguation_of_links_to_Range_of_a_function-20231205081900

1304:

is false (because of the "thus"). Since the point being made is incorrect, I think both sentences should be removed.

984: 482: 2963: 2892:

can be redirected to the dab page. The proposed title has the advantages of 1) using a natural disambiguator (this

2723: 2467: 2336: 2294: 2821: 2501: 2078: 1699:

for the independent variable. We shall not make this distinction." Calculus and Analytic Geometry, 2nd edition.

269: 2264:("the difference between the highest and the lowest values in a set") as the very first terms. And yet, reading 789:. Is this okay to mention in this article? Or what mathematical terminology (in words) is appropriate for this? 2706:

does not fully remove the ambiguity or may even introduce more ambiguity. My proposal to rename the article to

2683: 2658: 2443: 2381: 2237: 1359: 1309: 1268: 1017: 983:

then its image is equal to its codomain, and their is not confusion. In a representation of a function in a xy

800: 355: 237: 1609:

Different mathematicians use words differently, but range is still more common than image, hense this article.

2803: 2345:

Very good points. I hadn't considered that my proposal might cause that kind of confusion. Request edited. --

1455: 1038:

In fact I think I'll go and add that to the article and copy over the nice picture explaining range from the

1001:

The lead should try and avoid too much indecision, an extra paragraph is better for less common alternatives.

895: 831: 569: 533: 2737: 2643: 2617: 2598: 2579: 2482: 879: 2719: 2463: 2332: 2290: 1489:

This makes as much sense as saying "The value of a number N may depend on which number N is defined to be."

2901: 2881: 2402: 2316: 2281: 2273: 2253: 2245: 2202: 1713: 1565: 1520: 1406: 1397: 927: 909: 846: 508: 323: 3080: 3027: 2955: 2831: 2760: 2524: 2511: 2425: 2354: 2165: 2149: 2088: 1079:

science which used it in the second sense. If you have some evidence otherwise then I'd like to see it.

858: 747: 291: 50: 94: 1008:

The range is the image of the domain of the function, you can have an image of a subset of the domain.

645: 2951: 2938: 2897: 2893: 2540: 2532: 2312: 1630: 871: 470: 265: 229: 951:

make this change, the examples section will need to be re-worked. (And some edits at articles like

21: 3103: 3058: 3042: 2889: 2869: 2865: 2834:

after discussing it on the closer's talk page. No further edits should be made to this discussion.

2780: 2707: 2691: 2675: 2671: 2561: 2551: 2459: 2455: 2439: 2435: 2371: 2324: 2320: 2308: 2285: 2277: 2269: 2249: 2241: 2233: 2229: 2225: 2217: 2153: 2127: 2123: 2113: 2105: 1634: 1441:

reason this article should exist is to explain the unfortunate history of the word "range" -- with

1389: 1236: 1187: 1063: 960: 598: 545: 116:

on Knowledge. If you would like to participate, please visit the project page, where you can join

2926: 2877: 2590: 2565: 2328: 2261: 1506: 1385: 1279: 920:

The last line says "thus the range is [0, ∞)." I actually came to this page to find out what the

731: 100: 84: 63: 2873: 1560:

You seem to think that it is possible for a name to be an error. Names are arbitrary symbols.

722:

as the inclusion of the range into the codomain. However, I do not think that this merits more

2929:

exists, "mathematics" isn't a good enough disambiguator. The proposed title seems like a good

2913: 2350: 2161: 2145: 2010: 1826: 1709: 1561: 1516: 905: 613: 553: 548:

unless there's an implied codomain. For another example, you need to know the codomain of a

436: 3076: 3023: 2703: 2608: 2260:, a set of real numbers that includes all numbers between any two numbers in the set"), and 2221: 2136: 854: 621: 3003: 2934: 2536: 2528: 557: 549: 478: 2564:

is a very general topic, and one, which unfortunately, could also refer to this article:

3099: 3055: 3038: 2896:

to parenthetical disambiguators), 2) following the naming pattern of related articles (

2855: 2776: 2109: 2101: 2018: 1972: 1903: 1895: 1864: 1834: 1802: 1775: 1662: 1548: 1494: 1414: 1332: 1287: 1232: 1209: 1183: 1127:

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

1084: 1059: 1047: 1025: 956: 935: 594: 255: 1449:-- mentioned immediately. This article should then refer the reader to the article on 416: 396: 3116: 3073:

Wikipedia_talk:WikiProject_Mathematics#Disambiguation_of_links_to_Range_of_a_function

2593:

the other day and that's what brought me here. I still think this should get moved.

1638: 1459:-- which does not yet exist -- depending on which meaning of "range" is of interest. 1182:

I can only point to personal experience as to the common usage among mathematicians.

799:

I don't know if this was the case when you posted this, but your subject is found in

743: 727: 584: 518: 251: 2876:, the current title is ambiguous as there are other uses of "range" in mathematics: 2248:

already does a good job at leading the layman reader to the right page: it mentions

2750: 2679: 2447: 2415: 2346: 2157: 2141: 1708:

Moral: Writers at M.I.T. et al use words to mean whatever they want them to mean.

808: 790: 2194: 1675:

Here is George Thomas writing at M.I.T. in 1951, "The set of numbers over which

499: 113: 1300:

I'm not saying that any other meaning should be covered here. I'm saying that

620:) approaches the concept in a fairly roundabout manner. He proves, that in any 3107: 3084: 3061: 3046: 3031: 3010: 2997: 2970: 2942: 2916: 2859: 2807: 2784: 2759:

Subsequent comments should be made in a new section on this talk page or in a

2741: 2727: 2647: 2621: 2602: 2583: 2544: 2486: 2471: 2424:

Subsequent comments should be made in a new section on this talk page or in a

2406: 2389: 2358: 2340: 2298: 2232:

should be renamed to a somewhat more specific title, within the limits set by

2206: 2169: 2117: 2022: 1976: 1907: 1868: 1838: 1806: 1779: 1717: 1666: 1569: 1552: 1524: 1498: 1418: 1409:

to refer to range and then come back here again with that as another meaning.

1363: 1336: 1313: 1291: 1272: 1240: 1213: 1191: 1088: 1067: 1051: 1029: 980: 964: 939: 899: 883: 862: 835: 811: 793: 751: 735: 602: 587: 573: 537: 521: 511: 502: 486: 474: 273: 258: 254:. Is this unacceptable? Is the definition even correct? Comments welcome. - 241: 90: 2732:

I hereby withdraw my move proposal and await Tobias' superior move proposal.

2851: 2694:

is ambiguous and that the article should be renamed. The same editors (plus

2224:

to me suggests an article about a function named "Range". Perhaps rename to

2014: 1968: 1899: 1860: 1830: 1798: 1771: 1687:. In most of our applications, the ranges of our variables will consist of 1658: 1544: 1490: 1410: 1328: 1283: 1205: 1080: 1043: 1021: 931: 612:

In order to define "images" without reference to function properties, e.g.

2659:

Talk:Range (mathematics)#Proposal to move - current title is too ambiguous

3094:

for discarding this confusing article in favor of a disambiguation page.

2909: 2908:

the single oppose ¬vote in the previous RM, none of the things known as "

2634: 2189: 1646: 1450: 1039: 988: 976: 952: 761:

In for example computer science and numerical computing, the range from

1654: 1642: 1618: 190: 2912:" outside of mathematics are commonly talked of as having ranges.) – 1622: 850: 2665:

has commented that he or she had not seen the previous move request.

2284:

would help to solve that problem, but this would make the header of

2885: 1393: 972: 804: 718:

may be chosen as the surjective homomorphism onto the range, and

1626: 459:

is also an isomorphism", make unnecessary to mention the range.

1484:

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

1463:

X, there is exactly one y in Y such that (x,y) belongs to f.

930:

in the introductory section along with the statistics meaning.

2252:("a set containing the output values produced by a function", 2244:

fails here. I would have thought that the disambiguation page

2046: 167: 15: 3022:. They should link to either of the two more precise terms. 726:

than a brief reference to the category theoretical concept.

868:

what is a range how would you explain it to an 6th grader

1691:

of numbers..." Footnote: "Some writers reserve the word

2710:

appears to be acceptable to all participants, including

2500:

The following discussion is an archived discussion of a

2077:

The following discussion is an archived discussion of a

1165:

Tools of the Trade: Introduction to Advanced Mathematics

3020: 2556: 2178: 2064:

on 11 February 2013. The result of the move review was

998:

There are a couple of problems with what you say there

803:

which can be reached from the disambiguation page for

781:, i.e. the sequence or series of integer numbers from 700:

VIII.3, Proposition 1), and then defines the image of

2749:

The above discussion is preserved as an archive of a

2414:

The above discussion is preserved as an archive of a

822:

If it is possible could someone add to the article a

648: 439: 419: 399: 358: 326: 294: 1013:

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

112:, a collaborative effort to improve the coverage of 2514:. No further edits should be made to this section. 2091:. No further edits should be made to this section. 2763:. No further edits should be made to this section. 2702:who may or may not be the same person) agree that 2428:. No further edits should be made to this section. 1894:idea of what one should say about it. I'll ask at 682: 451: 425: 405: 382: 344: 312: 2633:"function" is more ambiguous than "mathematics". 2220:can be misleading to the layman reader. However, 2188:"function" is more ambiguous than "mathematics". 2043:Proposal to move - current title is too ambiguous 1583:the set of values a given function actually takes 210: 280:Comment for the above text about category theory 3143:Knowledge level-5 vital articles in Mathematics 2477:I belatedly support Tobias's statement above. 1302:Thus intervals are often referred to as ranges 1695:for the dependent variable, and use the word 1146:Mathematical Analysis: A Concise Introduction 8: 2268:suggests to me that many readers coming to 1396:and it gives as the first maths meaning an 556:. Codomains are particularly important in 2820:The following is a closed discussion of a 383:{\displaystyle f\circ g:\alpha \to \beta } 285:endomorphisms, etc. have precise meaning. 58: 647: 438: 418: 398: 357: 325: 293: 2690:and myself agree that the current title 2442:, with one explicit supporting !vote by 2272:are actually looking for something like 1152:Published by John Wiley & Sons, 2008 669: 664: 618:Categories for the working mathematician 352:may be composed resulting in a function 3133:Knowledge vital articles in Mathematics 2450:and myself, and one "Neutral" !vote by 826:explanation of what a range is. Thanks 60: 19: 2670:I repeat my proposal that the article 3148:C-Class vital articles in Mathematics 1898:and see if anyone has an idea there. 1479:P.S. One part of the article begins: 552:if you want to be able to define its 200: 7: 2839:The result of the move request was: 2519:The result of the move request was: 2446:, two implicit supporting !votes by 2096:The result of the move request was: 2011:List of integrals#Rational functions 926:I've put a reference and example of 345:{\displaystyle g:\alpha \to \gamma } 106:This article is within the scope of 2009:I have warned about the problem at 1607:Furthermore: Someone above wrote: " 313:{\displaystyle f:\gamma \to \beta } 199:, 2 February 2013, from to ], see 189:, 13 January 2013, from to ], see 49:It is of interest to the following 3158:High-priority mathematics articles 2775:confirm or refute this reference? 2266:the feedback this article's gotten 2132:the feedback this article's gotten 987:, the range is represented on the 640:is a monomorphism, i.e., satisfies 14: 2979:The discussion above is closed. 2197:is not a disambiguation page. -- 683:{\displaystyle mg=mh\implies g=h} 209:, 18 April 2020, from to ], see 126:Knowledge:WikiProject Mathematics 3128:Knowledge level-5 vital articles 2884:and two more articles listed in 2535:) 01:45, 10 February 2013 (UTC) 2050: 1133:Published by AMS Bookstore, 2005 493:Codomain, range, image confusion 171: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 2904:, etc.), 3) being unambiguous ( 2661:) was closed, and the proposer 2216:I agree that the current title 2108:) 08:22, 1 February 2013 (UTC) 2062:listed at Knowledge:Move review 1384:By the statistics view I meant 146:This article has been rated as 3138:C-Class level-5 vital articles 3108:22:58, 24 September 2024 (UTC) 3071:: The discussion continues at 2847:closed by non-admin page mover 1278:Those meanings are covered in 812:23:28, 25 September 2007 (UTC) 666: 487:07:19, 14 September 2007 (UTC) 374: 336: 304: 1: 2545:01:45, 10 February 2013 (UTC) 2023:19:21, 23 February 2010 (UTC) 1977:19:14, 23 February 2010 (UTC) 1908:22:32, 17 February 2010 (UTC) 1869:20:31, 17 February 2010 (UTC) 1839:14:34, 17 February 2010 (UTC) 1807:10:16, 17 February 2010 (UTC) 1780:09:17, 17 February 2010 (UTC) 1718:14:31, 16 February 2010 (UTC) 1667:10:11, 16 February 2010 (UTC) 1108:Real Analysis and Foundations 512:14:21, 22 February 2007 (UTC) 242:18:03, 10 November 2008 (UTC) 120:and see a list of open tasks. 3153:C-Class mathematics articles 3085:03:17, 7 December 2023 (UTC) 3062:02:10, 6 December 2023 (UTC) 3047:17:13, 6 December 2023 (UTC) 3032:01:57, 6 December 2023 (UTC) 2874:in the previous move request 2813:Requested move 11 April 2020 2785:05:42, 28 January 2015 (UTC) 2742:18:13, 5 February 2013 (UTC) 2728:09:39, 4 February 2013 (UTC) 2648:00:27, 3 February 2013 (UTC) 2622:20:42, 2 February 2013 (UTC) 2607:Okay, I guess I can see why 2603:20:40, 2 February 2013 (UTC) 2584:20:38, 2 February 2013 (UTC) 2487:20:54, 2 February 2013 (UTC) 2472:10:09, 1 February 2013 (UTC) 2407:05:15, 14 January 2013 (UTC) 2390:21:05, 13 January 2013 (UTC) 2359:20:02, 13 January 2013 (UTC) 2341:19:13, 13 January 2013 (UTC) 2299:09:39, 13 January 2013 (UTC) 2240:). I think it is clear that 2207:05:59, 13 January 2013 (UTC) 2170:20:02, 13 January 2013 (UTC) 2118:08:22, 1 February 2013 (UTC) 1570:23:59, 8 February 2010 (UTC) 1553:19:23, 8 February 2010 (UTC) 1525:21:01, 3 February 2010 (UTC) 1499:16:38, 3 February 2010 (UTC) 1419:17:46, 13 January 2010 (UTC) 1364:14:26, 13 January 2010 (UTC) 1337:13:51, 13 January 2010 (UTC) 1314:13:23, 13 January 2010 (UTC) 1292:08:09, 13 January 2010 (UTC) 1273:02:02, 13 January 2010 (UTC) 1114:Published by CRC Press, 1991 794:23:49, 3 December 2006 (UTC) 503:05:44, 23 October 2006 (UTC) 2397:on the revised request. -- 1263:it's the other way around. 1241:07:42, 5 January 2009 (UTC) 1214:22:17, 4 January 2009 (UTC) 1192:20:35, 4 January 2009 (UTC) 1089:18:53, 4 January 2009 (UTC) 1068:15:00, 4 January 2009 (UTC) 1052:13:46, 4 January 2009 (UTC) 1030:13:42, 4 January 2009 (UTC) 985:Cartesian coordinate system 965:13:05, 4 January 2009 (UTC) 940:11:38, 4 January 2009 (UTC) 884:02:39, 20 August 2008 (UTC) 757:Sequence of integer numbers 736:19:20, 3 October 2008 (UTC) 588:05:24, 22 August 2007 (UTC) 522:21:20, 20 August 2007 (UTC) 3179: 3011:18:32, 3 August 2020 (UTC) 2971:21:50, 11 April 2020 (UTC) 2943:16:12, 11 April 2020 (UTC) 2917:12:16, 11 April 2020 (UTC) 2860:14:55, 18 April 2020 (UTC) 752:18:43, 25 April 2015 (UTC) 624:any morphism (or "arrow") 603:14:14, 27 April 2008 (UTC) 274:16:28, 25 March 2006 (UTC) 259:09:05, 25 March 2006 (UTC) 2808:22:13, 21 June 2017 (UTC) 1018:range in computer science 900:19:55, 19 June 2014 (UTC) 841:The first line says "the 574:05:25, 30 June 2008 (UTC) 538:05:20, 30 June 2008 (UTC) 145: 78: 57: 2981:Please do not modify it. 2931:WP:NaturalDisambiguation 2827:Please do not modify it. 2756:Please do not modify it. 2507:Please do not modify it. 2421:Please do not modify it. 2238:Knowledge:Disambiguation 2084:Please do not modify it. 863:13:52, 20 May 2008 (UTC) 836:09:50, 20 May 2008 (UTC) 801:Range (computer science) 452:{\displaystyle f\circ g} 152:project's priority scale 1679:may vary is called the 1456:''image (mathematics)'' 1111:By Steven George Krantz 979:. If a function is a 109:WikiProject Mathematics 3123:C-Class vital articles 2902:Argument of a function 2882:Interval (mathematics) 2317:argument of a function 2282:Interval (mathematics) 2274:Interval (mathematics) 2254:Interval (mathematics) 1407:Interval (mathematics) 1398:Interval (mathematics) 1171:Published by AMS, 2008 928:interval (mathematics) 684: 453: 433:are isomorphisms then 427: 407: 384: 346: 314: 2894:is usually preferable 2276:. Another hatnote at 1967:doesn't make it true. 1168:By Paul J. Sally, Jr. 685: 454: 428: 408: 385: 347: 315: 36:level-5 vital article 2898:Domain of a function 2313:domain of a function 1433:Article is misguided 1253:Ranges and intervals 1149:By Bernd S. Schroder 646: 628:has a factorisation 437: 417: 397: 356: 324: 292: 288:Given two morphisms 132:mathematics articles 3163:Closed move reviews 2890:Range (mathematics) 2870:Range of a function 2866:Range (mathematics) 2770:Rudin 1991 Citation 2708:range of a function 2692:Range (mathematics) 2684:The Evil IP address 2676:Range of a function 2672:Range (mathematics) 2562:Range (mathematics) 2552:Range (mathematics) 2460:Range of a function 2456:Range (mathematics) 2444:The Evil IP address 2440:Range of a function 2436:Range (mathematics) 2382:The Evil IP address 2372:Range of a function 2325:limit of a function 2321:graph of a function 2311:would nicely match 2309:range of a function 2286:range (mathematics) 2278:range (mathematics) 2270:Range (mathematics) 2250:Range (mathematics) 2242:range (mathematics) 2230:range (mathematics) 2226:range of a function 2218:Range (mathematics) 2154:Range of a function 2128:Range of a function 2124:Range (mathematics) 2100:and 14+ days stale 1453:and the article on 1445:of its meanings -- 1390:Range (mathematics) 849:is the set of all " 2927:Range (statistics) 2888:. After the move, 2878:Range (statistics) 2591:Range (statistics) 2566:Range (statistics) 2329:zero of a function 2262:Range (statistics) 1770:But this is wrong. 1507:ring (mathematics) 1386:Range (statistics) 1280:Range (statistics) 680: 670: 665: 449: 423: 403: 380: 342: 310: 101:Mathematics portal 45:content assessment 3006: 2967: 2886:Range#Mathematics 2850: 2072: 2071: 1827:list of integrals 886: 874:comment added by 614:Saunders Mac Lane 489: 473:comment added by 426:{\displaystyle g} 406:{\displaystyle f} 232:comment added by 219: 218: 166: 165: 162: 161: 158: 157: 3170: 3009: 3004: 3000: 2968: 2965: 2962: 2958: 2844: 2829: 2758: 2720:Tobias Bergemann 2704:Range (function) 2609:Range (function) 2559: 2557:Range (function) 2509: 2493:Requested move 2 2464:Tobias Bergemann 2423: 2333:Tobias Bergemann 2291:Tobias Bergemann 2256:("also called a 2222:Range (function) 2181: 2179:Range (function) 2137:Range (function) 2086: 2066:Closure endorsed 2054: 2053: 2047: 991:(on the y axis). 922: 921: 869: 689: 687: 686: 681: 622:abelian category 468: 458: 456: 455: 450: 432: 430: 429: 424: 412: 410: 409: 404: 393:Facts like "if 389: 387: 386: 381: 351: 349: 348: 343: 319: 317: 316: 311: 244: 175: 174: 168: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 3178: 3177: 3173: 3172: 3171: 3169: 3168: 3167: 3113: 3112: 3002: 2995: 2992: 2987: 2964: 2960: 2956: 2872:– As discussed 2825: 2815: 2792: 2772: 2767: 2754: 2555: 2505: 2495: 2432: 2419: 2367: 2177: 2082: 2058:This discussion 2051: 2045: 1763: 1761: 1753: 1435: 1255: 1130:By Frank Morgan 948: 918: 820: 759: 724:in this article 644: 643: 558:category theory 550:linear operator 495: 435: 434: 415: 414: 395: 394: 354: 353: 322: 321: 290: 289: 282: 266:Oleg Alexandrov 234:213.249.237.201 227: 224: 172: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 3176: 3174: 3166: 3165: 3160: 3155: 3150: 3145: 3140: 3135: 3130: 3125: 3115: 3114: 3111: 3110: 3092:Strong support 3088: 3087: 3065: 3064: 3051: 3050: 3049: 2991: 2990:Disambiguation 2988: 2986: 2985: 2975: 2974: 2973: 2957:cookie monster 2945: 2914:Uanfala (talk) 2863: 2837: 2836: 2822:requested move 2816: 2814: 2811: 2791: 2788: 2771: 2768: 2766: 2765: 2751:requested move 2746: 2745: 2744: 2678:. The editors 2674:be renamed to 2667: 2666: 2651: 2650: 2627: 2626: 2625: 2624: 2549: 2548: 2521:page not moved 2517: 2516: 2502:requested move 2496: 2494: 2491: 2490: 2489: 2431: 2430: 2416:requested move 2410: 2409: 2392: 2368: 2366: 2365: 2364: 2363: 2362: 2361: 2302: 2301: 2210: 2209: 2174: 2173: 2121: 2094: 2093: 2079:requested move 2073: 2070: 2069: 2055: 2044: 2041: 2040: 2039: 2038: 2037: 2036: 2035: 2034: 2033: 2032: 2031: 2030: 2029: 2028: 2027: 2026: 2025: 1992: 1991: 1990: 1989: 1988: 1987: 1986: 1985: 1984: 1983: 1982: 1981: 1980: 1979: 1951: 1950: 1949: 1948: 1947: 1946: 1945: 1944: 1943: 1942: 1941: 1940: 1939: 1938: 1921: 1920: 1919: 1918: 1917: 1916: 1915: 1914: 1913: 1912: 1911: 1910: 1880: 1879: 1878: 1877: 1876: 1875: 1874: 1873: 1872: 1871: 1848: 1847: 1846: 1845: 1844: 1843: 1842: 1841: 1816: 1815: 1814: 1813: 1812: 1811: 1810: 1809: 1787: 1786: 1785: 1784: 1783: 1782: 1759: 1757: 1751: 1750: 1749: 1748: 1747: 1746: 1745: 1737: 1736: 1735: 1734: 1733: 1732: 1723: 1722: 1721: 1720: 1703: 1702: 1701: 1700: 1670: 1669: 1613: 1612: 1604: 1603: 1599: 1598: 1593: 1592: 1587: 1586: 1577: 1576: 1558: 1557: 1556: 1555: 1538: 1537: 1536: 1535: 1528: 1527: 1511: 1510: 1434: 1431: 1430: 1429: 1428: 1427: 1426: 1425: 1424: 1423: 1422: 1421: 1373: 1372: 1371: 1370: 1369: 1368: 1367: 1366: 1356:68.239.116.212 1344: 1343: 1342: 1341: 1340: 1339: 1319: 1318: 1317: 1316: 1306:68.239.116.212 1295: 1294: 1265:68.239.116.212 1254: 1251: 1250: 1249: 1248: 1247: 1246: 1245: 1244: 1243: 1221: 1220: 1219: 1218: 1217: 1216: 1197: 1196: 1195: 1194: 1177: 1176: 1175: 1174: 1173: 1172: 1169: 1166: 1158: 1157: 1156: 1155: 1154: 1153: 1150: 1147: 1139: 1138: 1137: 1136: 1135: 1134: 1131: 1128: 1120: 1119: 1118: 1117: 1116: 1115: 1112: 1109: 1101: 1100: 1099: 1098: 1092: 1091: 1075: 1074: 1073: 1072: 1071: 1070: 1036: 1035: 1034: 1033: 1032: 1011: 1010: 1009: 1006: 1002: 993: 992: 947: 946:Lead paragraph 944: 943: 942: 917: 914: 866: 865: 819: 816: 815: 814: 758: 755: 739: 738: 712: 690: 679: 676: 673: 668: 663: 660: 657: 654: 651: 641: 610: 577: 576: 543: 541: 540: 509:128.211.223.73 494: 491: 448: 445: 442: 422: 402: 379: 376: 373: 370: 367: 364: 361: 341: 338: 335: 332: 329: 309: 306: 303: 300: 297: 281: 278: 277: 276: 247: 223: 220: 217: 216: 215: 214: 204: 197:Page not moved 194: 176: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 3175: 3164: 3161: 3159: 3156: 3154: 3151: 3149: 3146: 3144: 3141: 3139: 3136: 3134: 3131: 3129: 3126: 3124: 3121: 3120: 3118: 3109: 3105: 3101: 3097: 3093: 3090: 3089: 3086: 3082: 3078: 3074: 3070: 3067: 3066: 3063: 3060: 3057: 3052: 3048: 3044: 3040: 3035: 3034: 3033: 3029: 3025: 3021: 3018: 3015: 3014: 3013: 3012: 3007: 2999: 2989: 2984: 2982: 2977: 2976: 2972: 2969: 2959: 2953: 2952:WP:NATURALDIS 2949: 2946: 2944: 2940: 2936: 2932: 2928: 2924: 2921: 2920: 2919: 2918: 2915: 2911: 2907: 2903: 2899: 2895: 2891: 2887: 2883: 2879: 2875: 2871: 2867: 2862: 2861: 2857: 2853: 2848: 2842: 2835: 2833: 2828: 2823: 2818: 2817: 2812: 2810: 2809: 2805: 2801: 2800:78.15.201.195 2797: 2789: 2787: 2786: 2782: 2778: 2769: 2764: 2762: 2757: 2752: 2747: 2743: 2739: 2735: 2734:50.193.171.69 2731: 2730: 2729: 2725: 2721: 2717: 2713: 2712:50.193.171.69 2709: 2705: 2701: 2697: 2693: 2689: 2688:50.193.171.69 2685: 2681: 2677: 2673: 2669: 2668: 2664: 2663:50.193.171.69 2660: 2656: 2653: 2652: 2649: 2645: 2641: 2640:65.92.180.137 2636: 2632: 2631:Strong Oppose 2629: 2628: 2623: 2619: 2615: 2614:50.193.171.69 2610: 2606: 2605: 2604: 2600: 2596: 2595:50.193.171.69 2592: 2588: 2587: 2586: 2585: 2581: 2577: 2576:50.193.171.69 2573: 2572: 2567: 2563: 2558: 2553: 2547: 2546: 2542: 2538: 2534: 2530: 2526: 2522: 2515: 2513: 2508: 2503: 2498: 2497: 2492: 2488: 2484: 2480: 2479:50.193.171.69 2476: 2475: 2474: 2473: 2469: 2465: 2461: 2457: 2453: 2449: 2445: 2441: 2437: 2429: 2427: 2422: 2417: 2412: 2411: 2408: 2404: 2400: 2396: 2393: 2391: 2387: 2383: 2379: 2376: 2375: 2374: 2373: 2360: 2356: 2352: 2348: 2344: 2343: 2342: 2338: 2334: 2330: 2326: 2322: 2318: 2314: 2310: 2306: 2305: 2304: 2303: 2300: 2296: 2292: 2287: 2283: 2279: 2275: 2271: 2267: 2263: 2259: 2255: 2251: 2247: 2243: 2239: 2235: 2231: 2227: 2223: 2219: 2215: 2212: 2211: 2208: 2204: 2200: 2196: 2191: 2187: 2186:Strong Oppose 2184: 2183: 2182: 2180: 2172: 2171: 2167: 2163: 2159: 2155: 2151: 2147: 2143: 2140: 2138: 2133: 2130:– Looking at 2129: 2125: 2120: 2119: 2115: 2111: 2107: 2103: 2099: 2092: 2090: 2085: 2080: 2075: 2074: 2067: 2063: 2059: 2056: 2049: 2048: 2042: 2024: 2020: 2016: 2012: 2008: 2007: 2006: 2005: 2004: 2003: 2002: 2001: 2000: 1999: 1998: 1997: 1996: 1995: 1994: 1993: 1978: 1974: 1970: 1965: 1964: 1963: 1962: 1961: 1960: 1959: 1958: 1957: 1956: 1955: 1954: 1953: 1952: 1935: 1934: 1933: 1932: 1931: 1930: 1929: 1928: 1927: 1926: 1925: 1924: 1923: 1922: 1909: 1905: 1901: 1897: 1892: 1891: 1890: 1889: 1888: 1887: 1886: 1885: 1884: 1883: 1882: 1881: 1870: 1866: 1862: 1858: 1857: 1856: 1855: 1854: 1853: 1852: 1851: 1850: 1849: 1840: 1836: 1832: 1828: 1824: 1823: 1822: 1821: 1820: 1819: 1818: 1817: 1808: 1804: 1800: 1795: 1794: 1793: 1792: 1791: 1790: 1789: 1788: 1781: 1777: 1773: 1769: 1768: 1767: 1766: 1765: 1764: 1756: 1743: 1742: 1741: 1740: 1739: 1738: 1729: 1728: 1727: 1726: 1725: 1724: 1719: 1715: 1711: 1707: 1706: 1705: 1704: 1698: 1694: 1690: 1686: 1682: 1678: 1674: 1673: 1672: 1671: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1640: 1639:Johns Hopkins 1636: 1632: 1628: 1624: 1620: 1615: 1614: 1610: 1606: 1605: 1601: 1600: 1595: 1594: 1589: 1588: 1584: 1579: 1578: 1574: 1573: 1572: 1571: 1567: 1563: 1554: 1550: 1546: 1542: 1541: 1540: 1539: 1532: 1531: 1530: 1529: 1526: 1522: 1518: 1513: 1512: 1508: 1503: 1502: 1501: 1500: 1496: 1492: 1487: 1485: 1480: 1477: 1474: 1470: 1467: 1464: 1460: 1458: 1457: 1452: 1448: 1444: 1440: 1432: 1420: 1416: 1412: 1408: 1403: 1399: 1395: 1391: 1387: 1383: 1382: 1381: 1380: 1379: 1378: 1377: 1376: 1375: 1374: 1365: 1361: 1357: 1352: 1351: 1350: 1349: 1348: 1347: 1346: 1345: 1338: 1334: 1330: 1325: 1324: 1323: 1322: 1321: 1320: 1315: 1311: 1307: 1303: 1299: 1298: 1297: 1296: 1293: 1289: 1285: 1281: 1277: 1276: 1275: 1274: 1270: 1266: 1260: 1259: 1252: 1242: 1238: 1234: 1229: 1228: 1227: 1226: 1225: 1224: 1223: 1222: 1215: 1211: 1207: 1203: 1202: 1201: 1200: 1199: 1198: 1193: 1189: 1185: 1181: 1180: 1179: 1178: 1170: 1167: 1164: 1163: 1162: 1161: 1160: 1159: 1151: 1148: 1145: 1144: 1143: 1142: 1141: 1140: 1132: 1129: 1126: 1125: 1124: 1123: 1122: 1121: 1113: 1110: 1107: 1106: 1105: 1104: 1103: 1102: 1096: 1095: 1094: 1093: 1090: 1086: 1082: 1077: 1076: 1069: 1065: 1061: 1057: 1056: 1055: 1054: 1053: 1049: 1045: 1041: 1037: 1031: 1027: 1023: 1019: 1015: 1014: 1012: 1007: 1003: 1000: 999: 997: 996: 995: 994: 990: 986: 982: 978: 974: 969: 968: 967: 966: 962: 958: 954: 945: 941: 937: 933: 929: 925: 924: 923: 915: 913: 911: 907: 902: 901: 897: 893: 892:184.20.43.154 887: 885: 881: 877: 873: 864: 860: 856: 852: 848: 844: 840: 839: 838: 837: 833: 829: 828:59.100.252.71 825: 817: 813: 810: 806: 802: 798: 797: 796: 795: 792: 788: 784: 780: 776: 772: 768: 764: 756: 754: 753: 749: 745: 737: 733: 729: 725: 721: 717: 713: 710: 707: 703: 699: 695: 691: 677: 674: 671: 661: 658: 655: 652: 649: 642: 639: 635: 631: 627: 623: 619: 615: 611: 607: 606: 605: 604: 600: 596: 590: 589: 586: 581: 575: 571: 567: 566:24.130.26.157 563: 562: 561: 559: 555: 551: 547: 539: 535: 531: 530:24.130.26.157 526: 525: 524: 523: 520: 514: 513: 510: 505: 504: 501: 492: 490: 488: 484: 480: 476: 472: 464: 460: 446: 443: 440: 420: 400: 391: 377: 371: 368: 365: 362: 359: 339: 333: 330: 327: 307: 301: 298: 295: 286: 279: 275: 271: 267: 263: 262: 261: 260: 257: 253: 248: 245: 243: 239: 235: 231: 221: 212: 208: 205: 202: 198: 195: 192: 188: 185: 184: 183: 182: 181:Discussions: 177: 170: 169: 153: 149: 148:High-priority 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 73:High‑priority 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 3091: 3068: 3016: 2993: 2980: 2978: 2947: 2922: 2905: 2864: 2840: 2838: 2826: 2819: 2795: 2793: 2773: 2755: 2748: 2716:76.65.128.43 2700:76.65.128.43 2696:76.65.128.43 2654: 2630: 2570: 2569: 2550: 2520: 2518: 2506: 2499: 2452:76.65.128.43 2433: 2420: 2413: 2399:76.65.128.43 2394: 2377: 2369: 2307:By the way, 2257: 2234:WP:PRECISION 2213: 2199:76.65.128.43 2185: 2175: 2135: 2122: 2098:no consensus 2097: 2095: 2083: 2076: 2065: 2057: 1755:is of form 1754: 1710:Rick Norwood 1696: 1692: 1688: 1684: 1680: 1676: 1608: 1582: 1562:Rick Norwood 1559: 1517:Rick Norwood 1488: 1483: 1481: 1478: 1475: 1471: 1468: 1465: 1461: 1454: 1447:not just one 1446: 1442: 1438: 1436: 1401: 1301: 1261: 1257: 1256: 949: 919: 906:Rick Norwood 903: 888: 876:72.208.9.191 867: 842: 823: 821: 786: 782: 778: 774: 770: 766: 762: 760: 740: 723: 719: 715: 708: 705: 701: 697: 693: 637: 633: 629: 625: 617: 591: 582: 578: 542: 515: 506: 496: 465: 461: 392: 287: 283: 249: 246: 225: 206: 196: 187:No consensus 186: 180: 179: 147: 107: 51:WikiProjects 34: 3077:fgnievinski 3024:fgnievinski 2832:move review 2761:move review 2525:move review 2512:move review 2426:move review 2280:leading to 2195:mathematics 2089:move review 1752:g(x) = 1/x 870:—Preceding 855:PrimeHunter 742:different. 469:—Preceding 228:—Preceding 123:Mathematics 114:mathematics 70:Mathematics 3117:Categories 2935:Rreagan007 2841:Page moved 2537:Malcolmxl5 2529:Malcolmxl5 1758:G(x) = log 1534:confusion. 1388:, this is 1005:computing. 981:surjection 769:refers to 211:discussion 201:discussion 191:discussion 3100:Taylor 49 3056:jacobolus 2910:functions 2790:Ambiguity 2777:J.Dong820 2655:Proposal: 2110:Tiggerjay 2102:Tiggerjay 1762:|x| + C. 1689:intervals 1631:Princeton 1233:Thenub314 1184:Thenub314 1060:Thenub314 1042:article. 957:Thenub314 595:Thenub314 39:is rated 2925:. Since 2635:function 2190:function 1651:Berkeley 1647:Stanford 1635:Columbia 1451:codomain 1040:codomain 989:ordinate 977:codomain 953:codomain 872:unsigned 847:function 818:Wordings 744:Srfahmy1 728:JoergenB 706:morphism 698:op. cit. 636:, where 519:Morseite 483:contribs 471:unsigned 230:unsigned 222:Comments 2948:Support 2933:to me. 2923:Support 2680:Aurochs 2448:Aurochs 2395:Neutral 2378:Support 2347:Aurochs 2214:Comment 2158:Aurochs 2142:Aurochs 1937:since.) 1896:WP:RDMA 1655:Caltech 1643:Cornell 1619:Harvard 809:Maghnus 791:Mange01 704:as the 554:adjoint 150:on the 41:C-class 3017:Agreed 2961:(2020) 2327:, and 1825:I see 1697:domain 1623:M.I.T. 916:Syntax 851:output 824:simple 777:+1, … 252:WP:NOR 47:scale. 2998:3125A 2355:Block 2258:range 2246:range 2166:Block 2150:Block 1731:book. 1693:range 1681:range 1653:, or 1394:Range 973:image 845:of a 843:range 805:Range 500:nadav 475:Elias 256:lethe 207:Moved 28:This 3104:talk 3081:talk 3069:Note 3043:talk 3028:talk 3005:talk 2950:per 2939:talk 2906:pace 2856:talk 2852:Jerm 2804:talk 2781:talk 2738:talk 2724:talk 2718:. — 2714:and 2698:and 2644:talk 2618:talk 2599:talk 2580:talk 2571:Both 2541:talk 2533:talk 2527:. -- 2483:talk 2468:talk 2403:talk 2386:talk 2351:Talk 2337:talk 2331:. — 2295:talk 2203:talk 2162:Talk 2156:. -- 2146:Talk 2114:talk 2106:talk 2060:was 2019:talk 2015:Dmcq 1973:talk 1969:Daqu 1904:talk 1900:Dmcq 1865:talk 1861:Daqu 1835:talk 1831:Dmcq 1803:talk 1799:Dmcq 1776:talk 1772:Daqu 1714:talk 1663:talk 1659:Daqu 1627:Yale 1566:talk 1549:talk 1545:Daqu 1521:talk 1495:talk 1491:Daqu 1443:both 1439:only 1437:The 1415:talk 1411:Dmcq 1360:talk 1333:talk 1329:Dmcq 1310:talk 1288:talk 1284:Dmcq 1269:talk 1237:talk 1210:talk 1206:Dmcq 1188:talk 1085:talk 1081:Dmcq 1064:talk 1048:talk 1044:Dmcq 1026:talk 1022:Dmcq 961:talk 936:talk 932:Dmcq 910:talk 896:talk 880:talk 859:talk 832:talk 748:talk 732:talk 692:and 599:talk 570:talk 546:onto 534:talk 479:talk 413:and 320:and 270:talk 238:talk 142:High 3059:(t) 3039:JBL 2966:755 2753:. 2568:. 2458:to 2438:to 2418:. 1683:of 1591:f). 785:to 765:to 585:Jim 3119:: 3106:) 3098:. 3083:) 3075:. 3045:) 3030:) 3008:}} 2996:{{ 2954:. 2941:) 2900:, 2880:, 2868:→ 2858:) 2843:. 2824:. 2806:) 2783:) 2740:) 2726:) 2686:, 2682:, 2646:) 2620:) 2601:) 2582:) 2560:– 2554:→ 2543:) 2504:. 2485:) 2470:) 2405:) 2388:) 2370:→ 2357:) 2353:| 2339:) 2323:, 2319:, 2315:, 2297:) 2205:) 2176:→ 2168:) 2164:| 2148:| 2126:→ 2116:) 2081:. 2021:) 1975:) 1906:) 1867:) 1837:) 1805:) 1778:) 1716:) 1665:) 1649:, 1645:, 1641:, 1637:, 1633:, 1629:, 1625:, 1621:, 1597:Y. 1568:) 1551:) 1523:) 1497:) 1486:" 1417:) 1362:) 1335:) 1312:) 1290:) 1271:) 1239:) 1212:) 1190:) 1087:) 1066:) 1050:) 1028:) 1020:. 963:) 938:) 912:) 898:) 882:) 861:) 834:) 807:. 773:, 750:) 734:) 667:⟹ 634:me 632:= 601:) 572:) 536:) 485:) 481:• 444:∘ 390:. 378:β 375:→ 372:α 363:∘ 340:γ 337:→ 334:α 308:β 305:→ 302:γ 272:) 240:) 3102:( 3079:( 3041:( 3026:( 3001:| 2937:( 2854:( 2849:) 2845:( 2802:( 2779:( 2736:( 2722:( 2642:( 2616:( 2597:( 2578:( 2539:( 2531:( 2481:( 2466:( 2401:( 2384:( 2349:( 2335:( 2293:( 2201:( 2160:( 2144:( 2112:( 2104:( 2068:. 2017:( 1971:( 1902:( 1863:( 1833:( 1801:( 1774:( 1760:e 1712:( 1685:x 1677:x 1661:( 1657:. 1611:" 1564:( 1547:( 1519:( 1509:. 1493:( 1482:" 1413:( 1402:x 1358:( 1331:( 1308:( 1286:( 1267:( 1235:( 1208:( 1186:( 1083:( 1062:( 1046:( 1024:( 959:( 934:( 908:( 894:( 878:( 857:( 830:( 787:b 783:a 779:b 775:a 771:a 767:b 763:a 746:( 730:( 720:m 716:e 711:. 709:m 702:f 694:e 678:h 675:= 672:g 662:h 659:m 656:= 653:g 650:m 638:m 630:f 626:f 616:( 597:( 568:( 532:( 477:( 447:g 441:f 421:g 401:f 369:: 366:g 360:f 331:: 328:g 299:: 296:f 268:( 236:( 213:. 203:. 193:. 154:. 53::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index