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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)

subtraction and division convention — PEMDAS, anyone?[edit]

The article has this text:

The subtraction operation, , is not associative, for instance. However, the notation is well-defined under the convention that the operation is understood as addition of the opposite, thus is the same as . Division is also non-associative. However, does not have an unambiguous conventional interpretation, so this expression is ill-defined.

As much as subtraction is understood as the addition of the opposite, division does have an unambiguous conventional interpretation as outlined in order of operations: top to bottom, left to right. In this application, if , then ; likewise, if , then . If there really is evidence to the contrary, that evidence should be presented, because every scientific calculator that I've ever used would respond to an entry of 36/6/6 with 1, not 36. D. F. Schmidt (talk) 21:24, 7 August 2014 (UTC)

The meaning of a/b/c and of a/bc has been extensively discussed, and the outcome has been inconclusive. People have strong views on both sides. Rick Norwood (talk) 00:39, 8 August 2014 (UTC)

I find it fairly problematic that this topic is discussed in the same article as what constitutes a well-defined map on, say, a quotient space (has to not depend on choice of representative). To my mind this is more evidence that this article does not have a clear rationale. That's not to say any of the material is bad, just that we probably should not have an article under this title. The subject matter can be treated in other articles. --Trovatore (talk) 01:05, 8 August 2014 (UTC)

I agree. I think most mathematicians would reserve "well-defined" for functions on quotient spaces and the like, and group a/b/c and similar problems under "order of operations". Rick Norwood (talk) 14:38, 8 August 2014 (UTC)
Bearing in mind the text appearing in the first paragraph—"A well-defined function gives the same output for 0.5 that it gives for 1/2", added by user:Rick Norwood earlier today—and understanding that the onus is on the function and how the input is processed, I propose changing the text of #Well-defined notation to the following, or a better paraphrase:
For real numbers, the product is unambiguous because . [1] In this case this notation is said to be well-defined. However, if the operation (here ) did not have this property, which is known as associativity, then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined.
The subtraction operation, , is not associative, for instance. However, the notation is well-defined under the convention that the operation may be understood as addition of the opposite, thus is the same as . However, if the operation is upon a variable a and an input u, if the given input is processed prior to subtraction from a, then .
Division is also non-associative and depends on whether the input is processed before further operations or inline with further operations, so the expression is ill-defined. See Order of operations, especially § Exceptions to the standard.
D. F. Schmidt (talk) 15:25, 8 August 2014 (UTC)
OK, so first of all, never ever ever rely on MathWorld on terminological issues. NEVER EVER EVER!!!! It's utterly unreliable on such things.
But my complaint is not really with the wording. I don't think that these two things (well-definedness on equivalence classes, and non-ambiguity of mathematical notation) should be treated in the same article at all. --Trovatore (talk) 19:54, 8 August 2014 (UTC)
Tell us how you really feel, Trovatore. lol. That source preceded my edit, and since I had no justification to remove it nor indeed any interest in touching that paragraph as it appears here, I simply left it alone. Since we're discussing article scope, I guess it sounds like you want to see well-defined#Well-defined notation go away altogether, right? I have no objection to this, which is my primary objection to the article as it exists now, and in fact any very simple overview--if any at all--should refer to order of operations as the main article to see on the topic. However, I feel that this would negate user:Rick Norwood's edits earlier today. D. F. Schmidt (talk) 22:30, 8 August 2014 (UTC)

My edit was just to try to make the lead clearer to someone without any knowledge of mathematics. I have no problem with the section on well-defined notation either going away, or being shortened to a sentence with a reference to order of operations. Maybe something to the effect that mathematics has many conventions, such as order of operations, to make sure that all mathematical expressions are unambiguous.Rick Norwood (talk) 23:49, 8 August 2014 (UTC)

not very good[edit]

I think the current header is not very good, so I searched for better ones. Wolfram starts like this: An expression is called "well-defined" (or "unambiguous") if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to not be well-defined, or ambiguous.

I would also try to avoid "well-defined function" because a function is by definition well-defined. Admittedly, this does make the wording longer. Instead, we can say that in math, any function f must satisfy the requirement that a = b implies f(a) = f(b), and that "well-defined" is the name of this requirement. A common example is that f(a/b)=a+b does not define a function over the rational numbers because f is not well-defined, e.g. 1/2 = 2/4 but f(1/2)=3 is not f(2/4)=6. MvH (talk) 21:42, 29 April 2015 (UTC)MvH

I like all of these suggestions. Rick Norwood (talk) 12:05, 30 April 2015 (UTC)
Thanks. MvH (talk) 14:32, 30 April 2015 (UTC)MvH

Several algebra textbooks explain the issue using sets mod an equivalence relation. Dummit and Foote give this example: A = A1 union A2 define f: A --> {1,2} as f(a)=1if a in A1 and f(a)=2 if a in A2. What I like about Dummit/Foote's example is (a) that it is more elementary than most other examples, and (b) that it gives an example where f is well-defined, and an example where f is ambiguous. If A1 intersect A2 is empty, then f is well-defined, but if A1 intersect A2 is not empty, then f is not well-defined since f(a) is ambiguous for a is in A1 intersect A2. MvH (talk) 14:32, 30 April 2015 (UTC)MvH

I made a number of changes. I propose to delete this part: "In set theory, functions are special cases of binary relations. For such a relation to give a well-defined function, if there exist two ordered pairs in the function with the same first coordinate ... An equivalent way of expressing the definition above ...". MvH (talk) 15:08, 30 April 2015 (UTC)MvH.

I deleted the "In set theory, functions are special cases of binary relations...." because I think people that can read this probably already know what well-defined is. I inserted the example from Dummit/Foote because it is the easiest example I could find (other than the f(1/2) = f(0.5) example that was already in the page). MvH (talk) 02:49, 2 May 2015 (UTC)MvH

I guess we have to tell people that well-defined is not the same as defined. If a mathematician uses the term "well-defined" he knows
  1. that he is going to define sth, but
  2. that it's not done as usual by writing down the definition, and that he has to prove sth, namely that the new object is a very specific type of relation.
And only after that proof the object is really defined (e.g. as a function).
@MvH:Nobody says: "As binary relation, f is always well-defined." (Nor did I!) Instead: "The binary relation f is defined ... ." And that's it, and there is no need for the term "well-defined". (What I want to say: we as mathematicians should use the term "well-defined" only when a "definition" contains also an assertion the proof of which is required.)
And sorry, I do not see how the breaking down defeats the purpose of the example:
The breaking down of the "definition" of a function into 2 sound steps, is IMHO mathematically correct, and does not require apostrophs around "define". It could help people to understand what is going on mathematically. (See also some discussion above) --Nomen4Omen (talk) 21:55, 30 June 2015 (UTC)
Sorry for the slow response. The usefulness of the Dummit/Foote example comes from its simplicity, it can explain the concept of "well-defined" to beginning undergrad students. Adding additional concepts to the example defeats that purpose. MvH (talk) 19:06, 10 July 2015 (UTC)MvH

Too technical in the first paragraph after the lead.[edit]

I appreciate the work of editors to make this article mathematically accurate, but the first paragraph after the lead should be written in language that it is at least possible for a non-mathematician (someone who does not already know what well-defined means) to understand. Rick Norwood (talk) 13:03, 1 July 2015 (UTC)

An effort has been made to address this problem. Thank you. However, nobody who is not a mathematician an read even the first sentence of the simple example. Rick Norwood (talk) 11:44, 2 July 2015 (UTC)

  1. ^ Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.