Talk:Codomain

message: Why should f and g be considered different functions? No doubt this is very confusing for those who are meeting functions for the first time; also I know of no use for the surjection concept (although I still have to read that article). Brianjd
:I have a math degree and I was never taught any such thing. I was taught that the range is the set of all outputs, and that if someone says x->x^2 has a range of R, then they were just wrong or imprecise.
:Secondly the use of the words possible and actualized is totally confusing
:Thirdly I can understand how x->x^2 is different if you are talking about R and C, but when both map R->R+ I don't see how misstating the proper range changes the function at all. This seems to be some kind of semantic nuance that is only important when abusive language is used with classes of functions.
:I just don't get it. I'm gonna check my analysis books when I get home.
:The concept of codomain makes sense when the image is either complicated or unknown, in which it is possible to give an imprecise set as the codomain.  — Preceding unsigned comment added by 68.32.100.93 (talk) 19:45, 23 April 2022 (UTC)
replies: text:  It is important in many areas of advanced mathematics, particularly algebraic topology to distinguish between the range of the function and the codomain. The reason is that when you apply a functor like homology or fundamental group to a function, the resulting homorphisms can be entirely different. For example proofs of the fundamental theorem of algebra and its generalizations to quaternions and octonions using algebraic topology techniques rely on clever use of this distinction. Fiedorow 20:09, 7 December 2005 (UTC)

text: I do agree that in the case discussed, it seems quite useless to claim that f and g are different functions (and that range and codomain are two separate, different concepts). However, in a more general case, it is IMO worthwhile. For example, let there be a function p: R^3 -> R^3 that to each vector in R^3 maps its orthogonal projection on a plane L. It makes sense to claim that p is not the same as q, another R^3 -> R^3 function that to each vector in R^3 maps its orthogonal projection on a plane W, right? And a function s: R^3 -> R^2 is another case. While both L and W are "two dimensional" in a sense, it is quite clear that they are not equal to R^2. So I guess my point is that I agree with the current article, but recognize the need for an example that more clearly illustrates the point. 85.224.198.251 17:01, 3 May 2007 (UTC)
thread_title: Discussion

message: The article gives an example where two functions with the same graph are given, but with different stated codomains, and claims that in one caase a certain composition is possible and in another case it is not.   This could be expanded into a longer explanation of why it is sometimes necessary to track the codomain explicitly; it is not about the ability of the functions qua functions to be composed, but about the ability of the functions qua morphisms in a category to be composed.  In a noncategorical context, all that is needed in order to compose f and g into f \circ g is that the range of g is contained in the domain of f, and this is a property of the graphs alone. CMummert 13:06, 14 October 2006 (UTC)
thread_title:  On the composition example 

message: I don't have any mathematical training other than a high school diploma, so I don't have the confidence to make the changes to this article myself, but I have a few questions/comments. I think I already know the answers, but I'm probably wrong. Either way, I think these are questions that a lot of untrained readers will be asking, and they're worth explaining in the article.
1. A function has to be defined for every number in its domain, right? The principal difference between range and codomain seems to be that this distinction does not apply to codomains: just because a number is within the codomain of a function doesn't mean that that function has to be capable of producing that number as output. 
:Correct
2. How, then, do we determine what the codomain of a function is? When the domain of a function isn't explicitly limited, we assume it to be the set for which that function is defined. (At least, that's what I think I did in high school.) The codomain seems to be quite different. It seems like it's not determinable (and probably not relevant) unless it's explicitly defined. 
:It usually is, I.E. all reals or complex numbers
3. I assume that a big, easy-to-understand instance where the codomain and the range would differ is if the domain is limited. For example, if we define f(x) = x^2, f has a codomain of the real numbers that are greater than or equal to zero. However, if we define its domain as [3:infinity), then its codomain remains the same, but its range is now [9:infinity) instead of [0:infinity). Right? 
:Yes--Cronholm144 11:39, 29 May 2007 (UTC)
thread_title:  Suggestions from a non-mathematician 

message: I think that many problems with the understanding of this concept could be alleviated by the addition of the set theory blob mapping to the codomain blob. If you have read texts on algebra I think you know what I mean. Unfortunately I think that any picture I make will look haphazard at best. I will give it a shot though... don't hesitate to stop me.--Cronholm144 11:31, 29 May 2007 (UTC)
thread_title:  A thousand words 

message: Do any of you think that the sentence at the start of this article:
"Unlike the range, which is a consequence of the definition of a function, the codomain is part of the definition of a function. "
contains a minor error ?
Shouldn't the sentence be something like:
"Unlike the range, which is a consequence of the definition of a function, the codomain is NOT part of the definition of a function. "
On an unrelated note. I was reading your comments pertaining to comparison of the "codomain" and "range" concepts. Do any of you think that the computer science concept of "data type" might be useful here? A codomain, could be regarded as the set of all possible values having a specific data type, or some arbitrary subset of such a set, for example the set of all complex numbers, the set of all real numbers, the set of all groups, etc..., and the range, of a function could be thought of as that subset of a codomain, to which that function maps values.  Do any of you know if there exists a term that is regarded as acceptable in mathematics, whose meaning is roughly equivalent to the meaning of the computer science "data type" concept ?  —Preceding unsigned comment added by 76.178.75.237 (talk) 03:14, 11 June 2008 (UTC)
replies: text: No I think the current wording is perfectly correct. For instance the square function on the reals would normally be defined with codomain the reals but its range is just the positive reals. Computer science tends to refer to the codomain as the range which can cause trouble and also has partial functions. Otherwise the word 'type' seems to cover both maths and computyer science pretty well I'd have thought. Dmcq (talk) 13:06, 25 April 2009 (UTC)

text: Some people working in logic like to define functions as just by the mapping of values without explicitly defining a domain or codomain, this can be easier as sometimes the 'function' is defined going from one class to another rather than between sets. Leaving out the typing of the function avoids having to cope with that. For most of mathematics though the codomain is part of the definition of a function. Dmcq (talk) 17:26, 18 May 2009 (UTC)
thread_title:  possible error ? 

message: Instead of arguing about whether the codomain is or should be part of the definition of a function, how about adding some reliable secondary sources, and basing the article on these references? That would be a tad more encyclopedic than unsourced analysis as to why the codomain is or isn't part of the definition, no? Geometry guy 12:54, 23 May 2009 (UTC)
replies: text: And I don't mind if it is in Polish, I would like to see the word codomain in it though please. Dmcq (talk) 12:59, 23 May 2009 (UTC)

text: Of course it won't be word codomain. But it'll be polish translation of it. 123unoduetre (talk) 13:01, 23 May 2009 (UTC)
replies: text: Fine, I'll struggle through, my guess is it comes from the German zielmenge but my wife knows Russian. Dmcq (talk) 13:20, 23 May 2009 (UTC)

text: Przeciwdziedzina is codomain: http://wazniak.mimuw.edu.pl/index.php?title=Matematyka_dyskretna_1/Wyk%C5%82ad_3:_Zliczanie_zbior%C3%B3w_i_funkcji but there is nothing about function equivalence. But i gave you example above where author discuss codomains (RCRH7.pdf) 123unoduetre (talk) 13:41, 23 May 2009 (UTC)

text: I suppose i've shown that definition i'd like to use is really used by mathematicians. It seems for me, that it's reasonable to create articles which explain both definitions and differences between them. I'd be thankful to people who could help me create them well. Thank you. And of course thank you for interesting discussion. 123unoduetre (talk) 14:27, 23 May 2009 (UTC)

text: And big thanks for Hans Adler for making research on this topic. 123unoduetre (talk) 14:31, 23 May 2009 (UTC)
replies: text: Hans and I have added a bunch of sources to the article. Additionally I've linked to a bunch of quotes from Google books which demonstrate the diversity of views on the notion of codomain in the literature. Try to write the article based on what its sources say. If you don't think the sources are a representative sample, add to them or change them for others. Geometry guy 15:11, 23 May 2009 (UTC)

text: Thank you. I'll check it after answering to the point Ryan Reich made. 123unoduetre (talk) 15:21, 23 May 2009 (UTC)

text: I think I can can see the problem from the Polish article. Mostly it is fine but it says things like (x,y) \in f rather than in the graph of f. I think they were just being sloppy but I'm not absolutely sure. I wasn't able to spot an equivalent of range (mathematics) which is totally confused between image and codomain in the English but they do have image (mathematics). As to changing this article I haven't seen evidence of a diversity of views where codomain is mentioned, range certainly but not codomain. Dmcq (talk) 15:53, 23 May 2009 (UTC)
replies: text: Just had a good look at those references inserted into the article. The Ian Stewart et al one is pretty unambiguous about accepting the set-theoretic version and calling it codomain as being an acceptable practice though he also says the triple is more formal. Okay that pretty much clinches it, the additions that have just been added deserve their place in I guess. Dmcq (talk) 16:06, 23 May 2009 (UTC)

text: Thank you for incorporating changes. 123unoduetre (talk) 17:27, 23 May 2009 (UTC)

text: The book you mentioned seems to be okay about functions as far as modern mathematics is concerned, why did you give a search highlighting 'relation'? Dmcq (talk) 12:21, 29 December 2009 (UTC)
thread_title:  Radical suggestion 

message: While we're at it, this article also needs to contain the definition of the codomain of a morphism as used in category theory. Paul August ☎ 16:27, 23 May 2009 (UTC)
thread_title:  Codomain of a morphism 

message: Since the purpose of this example seems to be to demonstrate the importance of the codomain to people who don't generally deal with maps at that level of abstraction, it might make sense to make the example more concrete.  For example instead of talking about linear transformations from R^n to R^m generally, specific spaces could be used, (R^2 to itself for instance, with the example of a non-surjective map being some simple rank 1 transformation, like (x,y) -> (x,0))  This is an idea that would be familiar to anyone with high school math.  I would change it, but I'm sort of new to this, and also there seems to be some controversy surrounding this example and I'm scared.  Anyway, just a thought.
On second though, I made the change, but obviously feel free to edit or revert it if this made things less clear rather than more.Rckrone (talk) 20:17, 21 June 2009 (UTC)
thread_title:  linear transformation example 

message: Range? 68.173.113.106 (talk) 03:23, 6 March 2012 (UTC)
replies: text: See the second sentence in the lead. Also range can be clicked on as it is blue and that takes you to an article about it. Dmcq (talk) 09:12, 6 March 2012 (UTC)
thread_title:  Alternative name 

message: So apparently, based on this article, and the lengthy discussion on the talk page, a codomain is some set that contains the set of actual output values from the function. It might even be the exact same set as the actual output values, or it could be the entire universe.
Given the vagueness of that specification, perhaps this article could improve in the direction of answering some these questions:
1. what is the point or virtue of the codomain idea?
2. If there is a virtue, how is that virtue used to select a particular set as the codomain?
3. If two functions differs only by codomain, are they different functions?
4. Do similar considerations pertain to the domain -- that is, can the domain of a function be larger than the actual set of values for which the function is defined?
To me, "f:R-->R" looks an awful lot like a declaration of a function in a programming language, in which the programmer specifies general data types for the input and output arguments. This constrains what data types the compiler will permit into the function, and gives a guarantee of the data types coming out of the function. That latter guarantee is less strict than the precise set of values that actually can come out.
Regarding "f:R-->R" in a math context: does the codomain actually impose any constraints, or imply any guarantees? Or is it merely informational, setting expectations, or just documenting the author's intent?
If I write "f:R-->R" , f:x |--> sqrt(x), 
then is this a function that is prohibited from outputting imaginary results? Or is it a function that produces imaginary results, and the "f:R-->R" is a documentation error?  68.7.23.65 (talk) 02:02, 24 February 2021 (UTC)
thread_title:  Codomain: what's the point?