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WikiProject Mathematics (Rated Start-class, Mid-importance)
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 Field: Foundations, logic, and set theory

Example: definitions based on a representative of an equivalence class[edit]

a good example from group theory is when you define something on an equivalence class in terms of one of its members. Naturally, you need to get the same result no matter which member was chosen. -- Tarquin 21:01 26 Jun 2003 (UTC)

Yes, this is a "canonical" meaning of "well-defined", when certain operations or functions, more generally, don't depend on choice of representatives. This precise meaning isn't really articulated in the article here. Revolver

It would be nice to have a specific example of what you need to verify in order for a function to be "well defined".

Example added, although possibly you may want more detail. Geometry guy 11:06, 14 February 2007 (UTC)

Talk: Well defined.[edit]

Is it meaningful to ask " Is the definition of "well defined" itself well defined, under its own definition ?" is this a paradox ? (talk) 16:47, 12 November 2009 (UTC)

on "well-defined-ness"[edit]

something is well defined when what the definition claims to exist exists. thus, to show something is well-defined you show it exists. that's all.

if you always show something exists before you name it, you never need to show your definition is well-defined, because you've already did. (see for example "the structure of the real number system" by cohen and ehrlich; they never use the term well-defined, because they carefully show everything exists before they even think about naming it for good.


let f be a function bla bla bla; you need to show f exists. some functions clearly exist; but that doesn't mean you can't show it exists. for instance, let f be a function that multiply 2 to its argument x; that's clearly a function; but you can show it is exists; if exists, then its name --- f --- is well-defined.

here's something ill-defined: let joe be a kangaroo that's not a kangaroo. every kangaroo is a kangaroo. such joe doesn't exist.

here's something ill-defined: let f be a function that maps its rational argument p/q to the number p + q. try to map 0.5 = 1/2 = 4/8. such f is not a function; hence, such function doesn't exist; hence, this definition isn't good.

more examples:

how would one show that f x = 2*x is well-defined? look at what it rests on; it rests on *; and * is a function; a binary function; we're just fixing one of its arguments; hence, f is just a restriction of *'s domain; therefore, f is a function; hence well-defined.

on psychology:

why do people always talk in terms of "ambiguity"? because a function is never ambiguous; it always maps each domain-guy to only one in the co-domain; if you map twice, then ambiguity arises; usually though, people go straight to the ambiguity talk because they haven't realized what well-defined-ness really means and they're usually talking about functions.

but relations and sets can also be ill-defined; even kangaroos can be ill-defined. -- danulus bastilis :) —Preceding unsigned comment added by (talk) 19:59, 8 February 2010 (UTC)

Merge with Defined and undefined[edit]

According to WP:ADJECTIVE, Titles should be nouns or noun phrases. Adjective and verb forms (e.g. democratic, integrate) should redirect to articles titled with the corresponding noun (Democracy, Integration), although sometimes they will be disambiguation pages, as at Organic. For this reason, "Defined and undefined" is not an appropriate article title. Anything which could be discussed under such a title would be better discussed on the Well-definition article, therefore the Defined and undefined article should be merged here. Neelix (talk) 22:54, 11 February 2010 (UTC)

I agree to merge the “defined and undefined” and “well-definition” to the “well-definition” article – this is the same. But with one condition: there should be a distinct article undefined (or, say, undefined value if you are so concerned about grammatical issues), because this is an important topic itself, not only in mathematics, but in some programming languages also. “Undefined” is an important case regardless of its cause, such as function's argument out of domain, uninitialized variable or an argument which has not be passed (as in MediaWiki templates). There should be two articles: “well-definition” about the concept, and “undefined value” about the case when something is not defined and possible consequences. So, a completely new article should be written, with the title undefined or with a redirect from. Incnis Mrsi (talk) 07:48, 24 March 2010 (UTC)

I don't see why "Undefined" couldn't be a section in the article on "Well-definition"... By the way, how about merging both articles into the general article on Definition? FilipeS (talk) 12:22, 22 April 2010 (UTC)

I merged the more sensible parts of defined and undefined with this page, as was suggested by several people. I included a bit more about exponentiation and some links to the corresponding notions in complex analysis. Tilmanbauer (talk) 12:30, 17 October 2010 (UTC)

"Defined and undefined" is an entirely different subject from "well-defined". In Euclidean geometry, "line" and "point" are undefined terms, "triangle" and "circle" are defined terms. In abstract algebra, f(a/b) = 2a/b is well-defined, f(a/b) = a + b is not well-defined. Rick Norwood (talk) 13:23, 3 January 2012 (UTC)


The new title is better than defined and undefined in one way; it is a noun, and in most cases article titles should be nouns. But it's worse in a possibly more important way, which is that it appears to be a neologism. Oh, I imagine that the term well-definition is probably attested somewhere, but it is certainly not in common use. I don't think the article can stay here. --Trovatore (talk) 00:45, 18 October 2010 (UTC)

I agree, I'm not sure about that word, either, although I've read it here and there (e.g. Davis-Kirk, Lecture Notes in Algebraic Topology). A short, unrepresentative search seems to indicate that "well-definedness" is used more often. I personally find that awkward, but who am I to judge. Unrelated to that, the page did exist before I merged "defined and undefined" into it, so would you agree that the content of the old page fits into this page, whatever its name? I don't think we need a separate page for "undefined values". Tilmanbauer (talk) 07:21, 18 October 2010 (UTC)

New title for this article.[edit]

"Well-definition" is a phrase you almost never hear. What you hear is "well-defined". Also, the noun form of a two-word adjective is not hyphenated. I would like to so some work on this article, providing the required references, but before I begin I would like to move it to "Well-defined". Any comments? Rick Norwood (talk) 13:20, 3 January 2012 (UTC)

It's commonly used in mathematics, it is common in Europe and UK education. --Jorgen W (talk) 05:19, 20 January 2012 (UTC)

If that is the case, it should be easy to provide references, which the article lacks. Rick Norwood (talk) 15:54, 20 January 2012 (UTC)

Hearing no response, my inclination is to make the change. In a google search for "well-definition" all the front page hits (except the one for this article) are about a "well" meaning a hole in the ground. On the other hand a search for "well-defined" gives many appropriate mathematical hits. I also note that this article itself uses "well-defined" eight times, and only uses "well-definition" twice in the body of the text. However, I will wait 24 hours before making the change, to give Jorgen W a chance to respond. Rick Norwood (talk) 13:22, 21 January 2012 (UTC)

Hearing no objection, I'm going ahead with the move. My aim is to 1) have the article title reflect the most common form of the idea, 2) separate out the difference between well-defined and undefined, and 3) add references. Rick Norwood (talk) 23:05, 22 January 2012 (UTC)

I've made the move, added a reference, an started working on links -- there are a lot of them and most link the phrase "well-defined" first to "well-definition" and then back here. Help would be appreciated, especially reference to a bot that will do the job. Rick Norwood (talk) 00:39, 23 January 2012 (UTC)

I certainly agree that you hear well-defined a lot, and well-definition almost never, and this common use is probably enough to override WP:NOUN, assuming the article should exist at all.
But frankly I don't think the article should exist at all. The principle that titles should be nouns is not really just about titles; it's about the kind of thing that should have articles. Articles should be about a thing, and "well-defined" is not a thing. Articles should not exist merely to document jargon. I think we should look for another solution. --Trovatore (talk) 05:29, 23 January 2012 (UTC)

I respectfully disagree. The concept is much more basic than many esoteric mathematical terms that have Wikipedia articles. It really isn't possible to understand why f(a/b) = 2a/b is well-defined and f(a/b) = a + b is not well-defined unless you have the definition of well-defined under your belt. I've been teaching abstract algebra for a long time, and while mathematicians take understanding well-defined for granted, students struggle with the concept. In class on Friday I discovered that none of my students in Modern Algebra II knew what well-defined meant. And these are students who habitually turn to Wikipedia for answers. I'd appreciate your help in making this a better article. Note that there are more than a hundred articles which have links to this page. Rick Norwood (talk) 13:15, 23 January 2012 (UTC)

See, I just don't think that's part of the encyclopedic mission. You have to remember that we are not teachers, in our role as encyclopedists. Wikipedia should be, and is, an excellent resource for self-teaching, but it absolutely MUST NOT NOT NOT attempt to teach in and of itself. I can't over-emphasize how fundamental this point is. -Trovatore (talk) 18:30, 23 January 2012 (UTC)

Is this article solely about well-defined functions? I agree that "well-defined" is an important description, but I don't see why it needs its own page when distinguishing "function" from "well-defined function" from "functions with holes in their domains" could done through example on a page about functions. I do hear the term used fairly often to describe problems, as in a "well-defined problem," by physicists, cognitive scientists, and mathematicians (see this wikibook for an example) and within mathematics I've also heard it used to describe operations or expressions (eg, MathWorld and PlanetMath). I've never in my life "well-definition." Scoresomecake (talk) 04:23, 24 January 2012 (UTC)

The concept of "well-defined" has nothing to do with "functions with holes in their domains" nor has it anything to do with what physicists mean by a "well-defined problem". It is an essential concept in abstract algebra. The most important elementary example has to do with fractions. A function defined on the rational numbers must take on the same value for 2/4 that it takes on for 1/2. The large number of pages that link to this page shows its importance. I, too, never heard to "well-definition" until I saw it in the old name for this article. Rick Norwood (talk) 14:53, 24 January 2012 (UTC)
Thank you for taking on the page -- it looks much nicer. I asked about functions (and other things that "well-defined" can describe) because this page offers examples of well-defined functions and there are other legitimate uses of the term. Indeed, the definition is given as "a function is well-defined if..." It seems that other uses of the adjective should not be directed to here if this page is only about "well-defined functions, foundational concept in abstract algebra." Frankly, I got to this page while wondering about criteria for "well-defined problems." I do not find the term indexed in Artin. It is indexed in Dummit & Foote (1999). On page 1, "If the function f is not specified on elements it is important in general to check that f is well defined [note -- no hyphen!], i.e., is unambiguously determined." So, this has to do with functions (and it comes up again in the context of cosets and homomorphisms, in D&F), but the indexed pages do not offer a stand-alone definition of the term. Examples following on page 2 use the phrase ``unambiguously defines." Given this usage and the usage on PlanetMath, I think "well-defined" has to do, in general, with lack of ambiguity, not just whether or not the same input might yield different outputs (as with the f(.5) vs. f(1/2) example -- not the only kind of ambiguity) or the example with the function that excludes 0 from its domain and so is well-defined on its domain. For example, whether a*b*c is unambiguous for some operation *. The "ambiguity" view would be consistent with other objects that "well-defined" can describe, as well. Scoresomecake (talk) 02:05, 25 January 2012 (UTC)

Thanks for your comment. Can you give me an example of a function that is "ambiguous" but does not fail the "If x = y, f(x) = f(y)" test? Rick Norwood (talk) 15:43, 25 January 2012 (UTC)

Undefined versus well-defined[edit]

"A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, the f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is. But 0 is not in the domain of the function." This sounds like POV to me. References, or it goes... FilipeS (talk) 14:24, 6 February 2012 (UTC)


I wonder if ill-defined should redirect here. I was looking for the definition, and comparing other sources like online dictionaries I found out that the meaning is perhaps the opposite. I suggest then that ill-defined should have its own page, or at least one section under well-defined explaining the distinction, instead of being simply redirected and presented as the same thing.

Luiscarlosrubino (talk) 19:42, 3 April 2013 (UTC)

Often it is best to define something in terms of its opposite. The phrase ill-defined is not often used, rather people say "not well-defined". Rick Norwood (talk) 12:52, 4 April 2013 (UTC)