Talk:Codomain

Errors in the Article

The first point I would like to make is that there is no sense in writing something like "let f be a function from R to R, where f(x) = x^(1/2)." This is because there is no such function. Suppose to the contrary that there is such a function f. Because it is a function, we know that for all a in R (its domain), there exists b in R (its codomain) such that b = f(a) = a^(1/2). However, it is easy to see that for a = -1, no b in R satisfies the statement b = a^(1/2) = (-1)^(1/2). This leads to a contradiction. Therefore, we conclude that there is no such function. Another interpretation would be that the "function" f is not well-defined, although this terminology is misleading as it suggests that f is in fact a function, which we have just disproved.

Of course, there is nothing wrong with supposing that such a function exists. It is just that doing so serves no purpose, since any statement follows from a false supposition. At any rate, the article should be changed to fix what is clearly a mistake.

The second point I would like to make is that even if the functions f and g were properly defined, it is trivial to show that f and g are in fact the same. By treating them as the sets they really are, proving that f = g is a simple task of proving set equality. This disproves the article's current claim that the functions are not the same.

-Your Friendly Anonymous Mathematician

Example

Copied from article

Let the function f be a function on the real numbers:

$f\colon \mathbb {R} \rightarrow \mathbb {R}$

defined by

$f\colon \,x\mapsto x^{2}$

The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R⁺—non-negative reals, i.e. the interval [0,∞):

$0\leq f(x)<\infty$

One could have defined the function g thus:

$g\colon \mathbb {R} \rightarrow \mathbb {R} ^{+}$ <== does not include zero!!

$g\colon \,x\mapsto x^{2}$

While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.

The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not.

Discussion

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.

It's just a case of mathematicians wanting to complicate things and make their profession seems advanced.

I too agree that it's stupid to consider a "mapped to" set to be anything different than the range of the function doing the mapping. But mathematicians allow for the mapped to set to have some useless elements to which the function will never map to (i.e., the codomain of f(x) = x^2 being R has no use, why not limit it to R+?).

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)

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)

CODOMAIN - "set of all possible values" ?

Assume f: A ->B, If the ran(f) is a proper subset of B, then how are the values in the set B-ran(f) "possible values" of the function f defined on A?

The set of all positive values is ran(f) not B. B is a set which contains the set of all positive values (a set being mapped "into" by f defined on A)

To describe ran(f) as the set of "actual values" implies that the function is actually used for all members of A. This is true if we're graphing y=f(x) or some other mechanism that actually feeds all members of A into f. But a function isn't like this, it's just a machine or rule that "can" take any value in A and map it to it's corresponding value in ran(f).

ran(f) is the set of all "possible" values!

From the point of view adopted in this context, the domain A and the codomain B are part of the data associated to the function f. Admittedly this doesn't make much sense when one considers one function in isolation. However it is very helpful when one is considering large collections of functions. One wants to say things like "consider all continuous functions with domain A and codomain B". Fiedorow 15:29, 19 December 2005 (UTC)

Yes but to explain the codomain as the set of possible values and range as the set of actual values is vague and makes even less sense in light of your comments above (and even less sense when going back and reading the explanation again).

The range is the "set of all possible output values from the function." The codomain is "the set that contains the set of possible output values of the function, and thus is a superset of the range."

A set, not the set. f could easily be defined to have a codomain of C instead of R or R₀⁺. (Correct me if I'm wrong.) Twifkak 21:06, 15 August 2007 (UTC)

On the composition example

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)

Suggestions from a non-mathematician

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)

168.209.97.34 09:10, 28 May 2007 (UTC)

A thousand words

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)

Something like this except cleaner

possible error ?

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)

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)

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)

Reverted?

Excuse me, please why have you reverted my changes if they're correct and i gave a source, which shows this page isn't correct?

123unoduetre (talk) 21:29, 22 May 2009 (UTC)

There are several problems with your edits and overall I don't think they constituted an improvement to the article. I do admit that this article could be improved, and that some of the ideas you were trying to incorporate into the article might have merit. First though you mention that this page is incorrect, can you please state which statements in the article are incorrect? Thanks. Paul August ☎ 21:40, 22 May 2009 (UTC)

While f and g map a given x to the same number, they are not, in the modern view, the same function because they have different codomains. This sentence is false for example. There are other things which i wouldn't call mathematics, but hand-waving. Should i list them all here? 123unoduetre (talk) 21:57, 22 May 2009 (UTC)

If it's only about style and some wiki standards i would be grateful if some more experienced user help me with doing it correctly. I could write new, correct article from scratch too. 123unoduetre (talk) 22:30, 22 May 2009 (UTC)

As I said above logicians don't seem to be up to speed with the concept of domain and codomain as used in most of maths. I had a quick look in Google Books and the very fist book my search on "codomain image" turned up was page 16 of Discrete algorithmic mathematics By Stephen B. Maurer, Anthony Ralston where it says that the square function from the reals to the reals is different from the square function from the reals to the positive reals. The definition you gave is not useful and completely misses the point of having a codomain. It stops one talking about onto or one to one functions functions or the composition of functions. Dmcq (talk) 23:02, 22 May 2009 (UTC)

You might notice that in this book author doesn't give any mathematical definition on function. He only gaves intuitions about it. Mathematical definition would be set theoretical definition in some axiomatic set theory. I assume you agree with me that mathematics is science which is based on formal systems. Citation from book you gave in chapter about functions: "A function is a rule or process that associates each element of one set with a unique element of another (the codomain or output space). If one emphasizes the dynamic process of getting from inputs to outputs one often call the function the mapping." As you might notice it isn't correct definition of function and he never gives any set theoretical definition. Now i'll show you why it isn't correct definition. 1. it defines function by process or rule, but rule and process are leaved undefined. Let's assume rule is some algorithm. So this definitions says only about computable functions. But there exist also non-computable functions. Of course they're not interesting from point of view of computer science, but we speak about mathematics. Do you have some other sources i could show are incorrect? (Books are wrong sometimes, sorry about that). 123unoduetre (talk) 23:42, 22 May 2009 (UTC)

First book on google books about mathematical analysis: http://books.google.pl/books?id=Pjk60RP-IeUC&printsec=frontcover&dq=mathematical+analysis#PPA4,M1 page 3-4 123unoduetre (talk) 23:47, 22 May 2009 (UTC)

This book clearly states a function is a type of relation from X to Y. A relation from X to Y is defined as a subset of X x Y. Thus the codomain is specified, contrary to your assertion. Indeed, the book also then defines surjectivity, making it even clearer that this is a property of the function (and thus codomain is part of the function). --C S (talk) 11:18, 23 May 2009 (UTC)

If you really want i could give you many examples. But it doesn't making sense to search google books. But if you really, really want i could do it, and count books which use correct and incorrect definitions of functions. Then i could show you some numbers. Of course i could accept that some authors (but not the one you gave (he doesn't give any definition of function)) use different definition. But wikipedia is encyclopedia, and should give most common understanding of terms. 123unoduetre (talk) 23:51, 22 May 2009 (UTC)

http://books.google.pl/books?id=qA5FTMT7HE4C&pg=PP1&dq=mathematical+analysis#PPA12,M1 page 12 123unoduetre (talk) 23:54, 22 May 2009 (UTC)

http://books.google.pl/books?id=jn_h9eIWIzUC&printsec=frontcover&dq=discrete+mathematics&lr=#PPA78,M1 page 78 this is discrete mathematics book 123unoduetre (talk) 00:11, 23 May 2009 (UTC)

Again, this book defines a function as specifying a domain and codomain. See several pages earlier in the book for discussion which makes this clear. Not to mention the very page you cite states that two functions must have the same "type" to be equal. A type is a specification of domain and codomain, written A --> B. --C S (talk) 11:18, 23 May 2009 (UTC)

http://books.google.pl/books?id=Er1r0n7VoSEC&pg=PP1&dq=set+theory+jech#PPA24,M1 page 24 This book is written by Karel Hrbacek, Thomas J. Jech. I hope you know who Jech is. I suppose it should be enough for you. It took too much of my time. I'm tired. 123unoduetre (talk) 00:41, 23 May 2009 (UTC)

I'm waiting for help of somebody who could explain to me how could i edit this page to comply to wikipedia standards and style. Thank you. 123unoduetre ([[U

ser talk:123unoduetre|talk]]) 00:46, 23 May 2009 (UTC)

The book you originally quoted did not specifically say functions were the same if the codomain was different, and the book you quoted in answer to me didn't mention codomain at all. What you are trying to put in is not right, that is why it shouldn't be put in. I'm quite willing to accept some other work you've quoted will say what you said but it isn't a very sensible thing to say as it destroys the whole point of a codomain. As I said before logicians really are happier without the domain and codomain part of the modern definition of a function and only using the set of pairs, however in the rest of mathematics that is not how things are done. I'll post a bit on the Mathematics project talk page at WT:WPM#Codomain definition to get other people's opinions on it. Dmcq (talk) 10:30, 23 May 2009 (UTC)

I had a better look at the definition in the next book as well after that one I said didn't mention codomain and it didn't mention codomain either. However they did say that a function was equal to another function if its type was the same and the mapping was the same and it gave the type as A->B where A and B are sets. It then had an explanation which didn't mention B, so basically their definition was okay but the explanation was incomplete. Dmcq (talk) 10:37, 23 May 2009 (UTC)

I don't know about logicians, but amongst mathematicians, I wouldn't be surprised if analysis books were split either way, all topology/geometry and algebra books say a function has a specified codomain. --C S (talk) 11:18, 23 May 2009 (UTC)

Wrong, for example this book defines function as set of pairs. It's about topology: http://books.google.com/books?id=-goleb9Ov3oC&printsec=frontcover&dq=topology&hl=pl 123unoduetre (talk) 11:25, 23 May 2009 (UTC)

This one too: http://books.google.com/books?id=-o8xJQ7Ag2cC&printsec=frontcover&dq=topology&hl=pl#PPA4,M1 123unoduetre (talk) 11:27, 23 May 2009 (UTC)

This is about abstract alebra: http://books.google.com/books?id=LE4mPB-1RFQC&printsec=frontcover&dq=abstract+algebra&hl=pl#PPA16,M1 123unoduetre (talk) 11:30, 23 May 2009 (UTC)

In your previous post I don't know which book are you refering. Could you please use title? I'll try to make it more clean to all parts involved in discussion: we are arguing about definition of function. I gave some formal definition as set of pairs. You didn't gave any formal definition of function. I gave in my text some non-standard (in my opinion) definition of function as <F,B>, where F i set of pairs and B is codomain, and told it's nonstandard. You are saying that in common mathematics functions definitions include codomains, so i suppose either you're proponent of non formal treatment of mathematics, or would accept some definition similar to mine. So the whole point is to decide which definition of function is standard and which isn't. I gave you some books which use my definition of function as set of pairs. You gave me one book which doesn't give any formal definition of function, but only guide intuitions about it. I suppose there is one nice solution of this problem. We could write in text 3 things: 1. what intuition is about functions 2. two possible definition of function, one iny my way (set of pairs), and one in your way (maybe similar to my "nonstandard" definition, you have to choose). 3, We should show difference between these definitions when comparing functions. For first definition it's enough to have domains and values of function for domain members to be the same. For second definition they also have to have the same codomain. Do you accept such solution? 123unoduetre (talk) 11:04, 23 May 2009 (UTC)

Michael Artin's Algebra contains the following note in its Appendix, pp. 585-6:

A map φ from a set A to a set T is any function whose domain of definition is S and whose range is T....We also take the domain and range of a function as part of its definition. If we restrict the domain to a subset, or if we extend the range, then the function obtained is considered to be different.

His terminology is a bit wrong (he uses "range" for what we call "codomain"; he calls what is properly the range, the "set of values") but the idea is clear. Ryan Reich (talk) 11:14, 23 May 2009 (UTC)

I wouldn't call it wrong, or even a "bit wrong". The use of "range" for codomain is very common. --C S (talk) 11:19, 23 May 2009 (UTC)

Not in this article. It would be aggravating if 123unoduentre were to argue that the words were ambiguous (I mean, two functions definitely are different if their ranges-as-sets-of-values are different). After all, there are four other sources on my bookshelf defining functions as triples but not explicitly outlining the point in question here, and I wouldn't use those as evidence like I have Artin because although they support me, they don't unambiguously rule out the alternative. But next thing you'll know we'll be hearing that he defines "map" using the word "function" so the definition is circular and vacuous. Ryan Reich (talk) 11:32, 23 May 2009 (UTC)