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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
Let the function f be a function on the real numbers:
defined by
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,∞):
One could have defined the function g thus:
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.
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
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+?).
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!
=== g(x) = √x ? The example for g(x) states the codomain is R. Now I understand that you can define the codomain to be any set that the range is a subset of, which *seems* ok. The implication of this is g(9) = {3, -3}. This is a one-to-many mapping and I thought violated the basic definition of a function? Usually f(x) = √x is defined to be f(x) = +√x.... doesn't this definition *require* the codomain to be limited to [0,∞) ??? Perhaps the problem is the unqualified use of the radical sign to indicate the operation of taking a positive square root allowing people like me to think -3 *is* in the *range* of g(9) when you meant it not to be? 71.31.152.112 ( talk) 19:24, 17 October 2010 (UTC)
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 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)
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.
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.
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?
168.209.97.34 09:10, 28 May 2007 (UTC)
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
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)
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)
ser talk:123unoduetre|talk]]) 00:46, 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)
123unoduetre, this type of condescending edit comment is certainly not helping your case, especially when you are simultaneously proving that you believe that concepts such as "function" have unique, immutable, definitions throughout mathematics. (Said he, going on to prove that Wikipedia has other editors who are more skilled in condescension.) Your definition is indeed the traditional one and still the more common one in set theory. But there is a general shift from the old style to the new style. Whenever a mathematician talks about a function that is "onto" without explaining "onto what", or about a surjective function, they are using the modern definition that includes the codomain in the official definition of the function. I made a little survey of set theory books:
Now this was just set theory. Perhaps the most natural context of the present article is category theory, and I find it hard to believe there is a single serious book on category theory that uses the old definition of functions.
The reason I felt it worth to do so much research is that the treatment at function (mathematics) is not entirely satisfactory. (Do have a look; it gives the modern definition as the standard and explains the old one as an alternative.) -- Hans Adler ( talk) 11:59, 23 May 2009 (UTC)
That's why i proposed to show both definitions and explain differences between them in one of my previous posts. 123unoduetre ( talk) 12:26, 23 May 2009 (UTC)
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)
here's a related source:
Does anyone know a primer on proofs that uses the 'mathematician's' definition of function? It's not surprising that a book about 'how to prove things' takes a more logical approach. Adam.a.a.golding ( talk) —Preceding undated comment added 11:22, 29 December 2009 (UTC).
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)
Yes. I've written short note in my version of article about category theory, that codomains are primitives there. I suppose my version wasn't as good as i thought. I would like to work with all of you to create better unbiased one. Thank you. 123unoduetre ( talk) 16:43, 23 May 2009 (UTC)
I am not sure that function composition should require the codomain of the function on the right side of the composition to be same as the domain of the function on the left side and that the image is "indeterminate at the level of the composition". I would say both codomain and image are consequences of a function's definition, so that as soon as we can prove that the image of f is a subset of the domain h (for which it suffices but is not necessary to prove that the codomain of f is a subset of the domain of f), we can define the composition .
The question is actually, whether X and Y in the triple (X, Y, F) are mere sets or some kind of type variables. The statement that image is "indeterminate at the level of the composition" would only make sense if X and Y in the triple were not just sets but some metamathematical type variables so that they were more "determinate" than the image of F. In some programming languages such a type system might indeed be beneficial because it helps to assure that the code is correct. E.g. in Java one might use Java classes to represent sets and then using generics one might create an interface
interface Function<S,T>
representing a function that converts an instance of class S to an instance of class T, and a class
class CompositeFunction<S,T> implements Function<S,T>
with a constructor
<U> CompositeFunction(Function<? super S,? extends U> first, Function<U,? extends T> second)
But in a mathematical proof one has got other means of proof than a type system, so I think the type system is not absolutely necessary.
Anyhow, I think the more important thing about including codomain in the definition of function is whether we can make statements like "Function f is a surjection." and "Function f is a bijection." or we have to (or at least formally would have to) explicitly mention the target set in order to be able to talk about surjectivity and bijectivity, i.e. we could only make sentences like "Function f is a surjection to Y.". So I think it should be said before the composition example that the "new" definition of function (the one that defines function as a triple (X, Y, F)) makes surjectivity and bijectivity properties of a function. After that, one might give the composition example with the explanation that there are different possible formalisms and in one formalism, where one regards codomain as a type variable, one might say that the image of a function is "indeterminate at the level of the composition". In fact, without such an explanation I would rather cut out the composition example because it is a bit confusing.
Well, and then the domain and codomain are also commonly used to make statements like "Function f is continuous." or "Function f is open." or "Function f is monotone." or "Function f is a group isomorphism.", which are actually formally incorrect if one considers the domain and codomain as just sets. In practice, of course, one has the justification that the algebraic/topological/order structure of domain and codomain are often obvious from the context. The justification might indeed be a bit more formally sound if one considers the domain and codomain as metamathematical variables.
But I still think that the simple consequence of incorporating codomain in function definition - being able to talk about a function being surjection - should be mentioned first and after that, if at all (i.e. if someone takes the effort to elaborate on it), the more complex issues that depend on whether we consider the domain and codomain as mere sets (the composition example and being able to talk whether a function is continuous, for example).
I don't quite understand what the linear transformation example is trying to say but it also seems to be something more complex than merely a consequence of codomain being a set and should also (if at all) be more clearly worded. —Preceding unsigned comment added by Jaan Vajakas ( talk • contribs) 14:14, 1 June 2009 (UTC)
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)
Range? 68.173.113.106 ( talk) 03:23, 6 March 2012 (UTC)
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)
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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
Let the function f be a function on the real numbers:
defined by
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,∞):
One could have defined the function g thus:
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.
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
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+?).
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!
=== g(x) = √x ? The example for g(x) states the codomain is R. Now I understand that you can define the codomain to be any set that the range is a subset of, which *seems* ok. The implication of this is g(9) = {3, -3}. This is a one-to-many mapping and I thought violated the basic definition of a function? Usually f(x) = √x is defined to be f(x) = +√x.... doesn't this definition *require* the codomain to be limited to [0,∞) ??? Perhaps the problem is the unqualified use of the radical sign to indicate the operation of taking a positive square root allowing people like me to think -3 *is* in the *range* of g(9) when you meant it not to be? 71.31.152.112 ( talk) 19:24, 17 October 2010 (UTC)
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 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)
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.
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.
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?
168.209.97.34 09:10, 28 May 2007 (UTC)
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
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)
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)
ser talk:123unoduetre|talk]]) 00:46, 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)
123unoduetre, this type of condescending edit comment is certainly not helping your case, especially when you are simultaneously proving that you believe that concepts such as "function" have unique, immutable, definitions throughout mathematics. (Said he, going on to prove that Wikipedia has other editors who are more skilled in condescension.) Your definition is indeed the traditional one and still the more common one in set theory. But there is a general shift from the old style to the new style. Whenever a mathematician talks about a function that is "onto" without explaining "onto what", or about a surjective function, they are using the modern definition that includes the codomain in the official definition of the function. I made a little survey of set theory books:
Now this was just set theory. Perhaps the most natural context of the present article is category theory, and I find it hard to believe there is a single serious book on category theory that uses the old definition of functions.
The reason I felt it worth to do so much research is that the treatment at function (mathematics) is not entirely satisfactory. (Do have a look; it gives the modern definition as the standard and explains the old one as an alternative.) -- Hans Adler ( talk) 11:59, 23 May 2009 (UTC)
That's why i proposed to show both definitions and explain differences between them in one of my previous posts. 123unoduetre ( talk) 12:26, 23 May 2009 (UTC)
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)
here's a related source:
Does anyone know a primer on proofs that uses the 'mathematician's' definition of function? It's not surprising that a book about 'how to prove things' takes a more logical approach. Adam.a.a.golding ( talk) —Preceding undated comment added 11:22, 29 December 2009 (UTC).
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)
Yes. I've written short note in my version of article about category theory, that codomains are primitives there. I suppose my version wasn't as good as i thought. I would like to work with all of you to create better unbiased one. Thank you. 123unoduetre ( talk) 16:43, 23 May 2009 (UTC)
I am not sure that function composition should require the codomain of the function on the right side of the composition to be same as the domain of the function on the left side and that the image is "indeterminate at the level of the composition". I would say both codomain and image are consequences of a function's definition, so that as soon as we can prove that the image of f is a subset of the domain h (for which it suffices but is not necessary to prove that the codomain of f is a subset of the domain of f), we can define the composition .
The question is actually, whether X and Y in the triple (X, Y, F) are mere sets or some kind of type variables. The statement that image is "indeterminate at the level of the composition" would only make sense if X and Y in the triple were not just sets but some metamathematical type variables so that they were more "determinate" than the image of F. In some programming languages such a type system might indeed be beneficial because it helps to assure that the code is correct. E.g. in Java one might use Java classes to represent sets and then using generics one might create an interface
interface Function<S,T>
representing a function that converts an instance of class S to an instance of class T, and a class
class CompositeFunction<S,T> implements Function<S,T>
with a constructor
<U> CompositeFunction(Function<? super S,? extends U> first, Function<U,? extends T> second)
But in a mathematical proof one has got other means of proof than a type system, so I think the type system is not absolutely necessary.
Anyhow, I think the more important thing about including codomain in the definition of function is whether we can make statements like "Function f is a surjection." and "Function f is a bijection." or we have to (or at least formally would have to) explicitly mention the target set in order to be able to talk about surjectivity and bijectivity, i.e. we could only make sentences like "Function f is a surjection to Y.". So I think it should be said before the composition example that the "new" definition of function (the one that defines function as a triple (X, Y, F)) makes surjectivity and bijectivity properties of a function. After that, one might give the composition example with the explanation that there are different possible formalisms and in one formalism, where one regards codomain as a type variable, one might say that the image of a function is "indeterminate at the level of the composition". In fact, without such an explanation I would rather cut out the composition example because it is a bit confusing.
Well, and then the domain and codomain are also commonly used to make statements like "Function f is continuous." or "Function f is open." or "Function f is monotone." or "Function f is a group isomorphism.", which are actually formally incorrect if one considers the domain and codomain as just sets. In practice, of course, one has the justification that the algebraic/topological/order structure of domain and codomain are often obvious from the context. The justification might indeed be a bit more formally sound if one considers the domain and codomain as metamathematical variables.
But I still think that the simple consequence of incorporating codomain in function definition - being able to talk about a function being surjection - should be mentioned first and after that, if at all (i.e. if someone takes the effort to elaborate on it), the more complex issues that depend on whether we consider the domain and codomain as mere sets (the composition example and being able to talk whether a function is continuous, for example).
I don't quite understand what the linear transformation example is trying to say but it also seems to be something more complex than merely a consequence of codomain being a set and should also (if at all) be more clearly worded. —Preceding unsigned comment added by Jaan Vajakas ( talk • contribs) 14:14, 1 June 2009 (UTC)
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)
Range? 68.173.113.106 ( talk) 03:23, 6 March 2012 (UTC)
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)