Talk:Well-defined expression

message: 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)
thread_title:  Example: definitions based on a representative of an equivalence class 

message: Is it meaningful to ask " Is the definition of "well defined" itself well defined, under its own definition ?" is this a paradox ? 
http://www.dpmms.cam.ac.uk/~wtg10/welldefined.html 81.141.30.210 (talk) 16:47, 12 November 2009 (UTC)
thread_title:  Talk: Well defined. 

message: 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.
examples:
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 77.250.91.80 (talk) 19:59, 8 February 2010 (UTC)
thread_title:  on "well-defined-ness" 

message: 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)
replies: text: 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)
replies: text: "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)
thread_title:  Merge with [[Defined and undefined]] 

message: 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)
thread_title:  Neologism? 

message: "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)
thread_title:  Undefined versus well-defined 

message: 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)
replies: text: 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)
thread_title:  Ill-defined 

message: 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)
thread_title:  Too technical in the first paragraph after the lead. 

message: Isn't the following sentence "For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined" wrong? If f is a function on the real numbers, and I can evaluate f(1/2) at all, then that has to mean that the argument has first to be somehow convert to a real number. And presumably this would happen via division on the real numbers, with result 0.5, or not? Put another way, if you read the sentence on a degree of sloppiness so that the statement "If f takes real numbers as input and is well-defined, then f(0.5) does equal f(1/2)" is correct, then f(0.5) has to equal f(1/2) (on this given level of rigour) no matter whether it is well-defined or not.Seattle Jörg (talk) 14:40, 28 April 2017 (UTC)
replies: text: You're right: The sentence does not catch the point, although it has some similarity to Eric Weisstein's. --Nomen4Omen (talk) 15:15, 28 April 2017 (UTC)
thread_title:  Lead paragraph 

message: Currently this section only contains the sentence "A solution to a partial differential equation is said to be well-defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed." There is reference to MathWorld, where not further references exist. I find the whole claim very questionable. Perhaps the author (in MathWorld) was thinking of "well-posedness in the sense of Hadamard", which is stronger than the existence and uniqueness of solutions in some given (distribution, classical, viscocity etc.) sense, which would be the correct meaning for "well-defined" in the context of solutions of PDE's. Should the whole section be removed? Lapasotka (talk) 07:40, 1 August 2018 (UTC)
thread_title:  Other uses of the term