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At high school we were told that a function has a domain and a range, the function maps from the domain to the range. Such that the domain contains all and only the possible inputs and the range contains all and only the possible outputs.

Now at University I'm told a function has a domain and a codomain, and that the codomain contains all the possible outputs but may also include other numbers. What is the point of having values in the codomain that can not be output by the function, how does that aid in describing the function? Does this also mean that the domain can include numbers that the are not inputs to the function?

Surely this means you could say the codomain of any function (that outputs numbers) is the complex set (so all numbers)?

EDIT: Wikipedia says the function $f : x \rightarrow x^2$ has codomain $\mathbb{R}$ but it's image (what I guess I knew as range in high school) is $\mathbb{R}^+_0$, so why not just say the codomain is $\mathbb{R}^+_0$.

EDIT2: And is it also then true that is a function is "onto" the codomain is the same as the image? So surely any function can be "onto" if you just change the what the codomain is?

What I'm really trying to ask I guess is the range/image of a function is defined by the function, what defines the codomain?

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One reason for identifying a “codomain” is that what the “range” is is often difficult to determine. The range of $y=x^4-x$? Hard to say. The codomain? Easy: it can be taken to be $\mathbb R$. What you say about ontoness is exactly right: the function is onto just when the range equals the codomain. As to your question about the domain, the answer is no, the domain is precisely the set on which the function is defined. Finally, when we write $f\colon A\to B$, we mean that the domain is $A$ and the codomain is $B$. –  Lubin May 1 at 21:53

3 Answers 3

First of all, not everything in mathematics is numbers. There are other objects, all of which are legitimate values for a function. Here is a simple example:

$f(x,n)=x^n$ can actually be seen as a function, $F(n)$ returns the function which maps $x$ to $x^n$. So $F$ takes in a natural number, and outputs a function from the real numbers to the real numbers.

Now we consider these for different $n$'s, when $n=0$ then $F(n)$ is the constant function $1$. When $n=1$ the range is $\Bbb R$. When $n=2$ the range is $\Bbb R^{\geq 0}$. And so on and so forth.

Each of these values gives us a different range. But if we think of all of them as functions from $\Bbb R$ to $\Bbb R$ then we can "meld" them together into one big function $f(x,n)=x^n$.

To some extent, you are right. The invention of "codomain" is a bit artificial. In parts of mathematics, a function comes with an associated codomain (and thus changing the codomain means changing the function), and in other parts of mathematics, the codomain is an external property assigned to the function and can be changed whenever we want to (as long as it includes the range of the function).

So this is a matter of how you intend to use "functions" as mathematical objects. Of course, a lot of the examples for when a codomain matters come from a slightly later stage. If you study mechanical engineering then it's possible that you'll never even have to worry about this; and if you study mathematics, then it's likely that you've already run into examples (without knowing that these are examples, perhaps).

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The codomain is a set which the function maps into. For example if $f:N \rightarrow R$ by $f(n) = n$ then R is the codomain.The range of the function is the subset of the codomain whose elements correspond to the mapping of some element from the domain. So with $f(n) = n$ the range in $R$ is the subset $N \subset R$.

If the range is equal to the codomain, then the function is called onto, or a surjection..

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What is the point of having values in the codomain that can not be output by the function, how does that aid in describing the function?

Here are a few reasons why we allow some functions to not be surjective.

  1. As Lubin mentioned, the range of a function can be difficult to determine. For example, determining the range of a polynomial of high even degree (such as $P(x)=x^6-3x^2+6x$) amounts to finding the zeroes of a high-degree polynomial (such as $P'(x)=6x^6-6x+6$, whose roots are not expressible as radicals), a difficult task in general. We could get around this by defining the codomain of every function $f$ to be $\operatorname{im} f$ (that is, $\{y\,:\,f(x)=y\text{ for some }x\in X \}$), but that doesn't really add any information.

  2. It's nice to separate surjective functions from other functions because surjective functions are dual to injective functions. When I say "dual" I'm referring to, for example, the following fact: a function $f:A\to B$ is injective if and only if there is a function $g:B\to A$ such that $g\circ f=1_A$ (by $1_A$ I mean the identity function on $A$); a function $f:A\to B$ is surjective if and only if there is a function $g:B\to A$ such that $f\circ g=1_B$. When you study the branch of mathematics known as category theory, you'll see that it's very natural to have dual properties like this.

Does this also mean that the domain can include numbers that the are not inputs to the function?

As others have remarked, the domain of a function can include other objects than numbers. For example, you could define a function which takes as input a person and returns his age. In any case, a function must be defined on all possible input values. The answer to your second question is no.

And is it also then true that is a function is "onto" the codomain is the same as the image? So surely any function can be "onto" if you just change the what the codomain is?

That's exactly right. You can make any function onto by changing the codomain. But as I remarked earlier, in general we don't know what the image of a function is and so it doesn't add any information to restrict the codomain.

What I'm really trying to ask I guess is the range/image of a function is defined by the function, what defines the codomain?

The codomain usually arises naturally in the definition of the function. For example, whenever you have a function which returns a number, the natural choice of codomain is $\mathbb R$. Of course, if by "number" you mean "complex number" then the codomain could be $\mathbb C$; if by "number" you mean "quaternion" then the codomain could be $\mathbb H$.

On the other hand, owing to the set-theoretic fact that "there is no set containing everything," it's not possible to pick a single universal codomain for functions.

When I wrote up this answer I realized that I used to ask the same questions as you, but I stopped once I had learned enough mathematics. I can't give you a single profound reason why we don't make all functions surjective besides a pragmatic one: surjectivity is a useful notion, and getting rid of it would be unprofitable.

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