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A basic description of a function is this: $$f: \text{dom}(f) \subseteq \mathbb{R}^n \to \text{ran}(f) \subseteq \mathbb{R}^m$$

Clearly, this function has FOUR things going on:

  • Domain: $\text{dom}(f)$, all inputs that gives valid output. For example, the real function $f(x) = \log(x)$ is defined on the domain $x > 0$. It is undefined elsewhere.

  • $\mathbb{R}^n$: the set for which domain belongs to.

  • Range: $\text{ran}(f) = f(\text{dom}(f))$

  • Codomain: the set which which contains the range.

Why is it that only "domain, range, codomain" are named, but not $\mathbb{R}^n$?

All my life I've seen $\mathbb{R}^n$ either being completely confused with the domain or sometimes is called the "ambient space", but this has strong physics undertone.

Why is there a loss of symmetry in the naming convention of these objects?

Your neighbor Todorovich
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    Not all functions are defined on (subsets of) $\mathbb R^n$. Indeed, the $n$ there isn't really specified uniquely...it might make sense in context, of course, but it's not really intrinsic to the function. – lulu Jan 19 '25 at 17:33
  • @lulu Usually a function can only take on a subset of value of all possible values. For example, all physical systems in the universe can be described as functions that take on a bounded set of input values, but not the entire possible values (at which point the physical system disintegrates/blows up). Why is it that we call the open set of feasible input value as the "domain" of the function but ignore giving a name to all possible values? – Your neighbor Todorovich Jan 19 '25 at 17:37
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    Well, you've distinguished range from co-domain. What more do you need? To be sure, the terminology has changed over time. I was taught that "range" meant what we know call codomain, while "image" was used to describe the set of values of the function. Still see some confusion from that shift in vocabulary. – lulu Jan 19 '25 at 17:38
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    You might be happy with the notion of a “partial function.” This removes the asymmetry between domain and codomain. Each need only contain the “relevant” piece. – Malady Jan 19 '25 at 17:55
  • @Malady Shame this idea is not more popular. – Your neighbor Todorovich Jan 19 '25 at 18:37
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    @Malady Partial functions are still "asymmetric" in the sense that each input-element is associated with at most one output-element but not conversely. The fully-symmetric notion is that of a (binary) relation: a relation between two sets $X,Y$ is any subset $R$ whatsoever of $X\times Y$. $Y$ is the codomain, the image is ${y\in Y:\exists x\in X((x,y)\in R)}$, and both these notions can be dualized to the "$X$-side." – Noah Schweber Jan 19 '25 at 20:52
  • A partial function between two finite sets of different cardinality can be bijective which is impossible for functions. This is extremely important for counting problems. – CyclotomicField Jan 19 '25 at 20:57
  • Functions can work on other things than numbers as input. I think a big stumbling block I had was when I came across functions whose domains were sets of functions. Somehow I could view numbers and n-tuples as "things" but not functions as things. But ANYTHING no matter how abstract can be an input of a function. As "anything" can be vague this idea of "but the anythings are all in some superset" isn't really useful nor clear. – fleablood Mar 03 '25 at 20:43
  • And what about when $dom(f)\subset X\to ran(f)\subset Y$ but $X$ and $Y$ are different things? And what's the difference between $f:$ Even natural numbers $\subset \mathbb R\to $ Natural Numbers $\subset \mathbb R $ and $f:$ Even natural numbers $\subset (\mathbb R\cup {$flavors of Oreo Cookies$}$? Why would the fact that the even numbers are also real numbers be relevant, but the fact that even number are either real number or a flavor of Oreo Cookie not be relevant? If an element is not in the domain does it matter why it's not in the domain? because it was odd, irrational, a cookie? – fleablood Mar 03 '25 at 20:53
  • In French, this set is named Ensemble de départ (start set) or Ensemble source (source set). I would naively use such terms if I had to name it. – Christophe Boilley Mar 04 '25 at 10:38

2 Answers2

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The set-theoretic definition of function is symmetric: it has only domain and codomain.

We start with the definition of relation:

$R \text { is a relation } \ ↔ ∀z(z \in R → ∃x \ ∃y \ (z = \langle x,y \rangle \ ))$,

that is a set of ordered couples. And we do not need to specify "from where" the elements of the pairs come from.

A function is simply a special relation.

Then we define:

$\mathcal {Dom}_R = \{ x : \exists y \ (x R y) \}$

and :

$\mathcal {Cdm}_R = \{ y : \exists x \ (x R y) \}$ (Converse domain or Image or Range).

See e.g. Suppes (1960) page 58, and see also the post Modern definition of domain and codomain of functions.

In calculus, where we deal with real functions, we often consider partial functions, i.e. function whose domain and range are subset of $\mathbb R$.

In this case we usually write $f : \mathbb R \to \mathbb R$, specifying in addition that the domain of definition of $f$ is some subset of $\mathbb R$, like e.g. $\mathbb R \setminus \{ 0 \}$ for the function $\dfrac 1 x $.

Going back to the set-theoretical viewpoint, we have that $f \subseteq \mathbb R \times \mathbb R$.

Mauro ALLEGRANZA
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    OK, I can see how this is supported by some texts, but it conflicts with the definitions in https://math.stackexchange.com/a/59435/139123, which are what I'm familiar with. Also, if the codomain can only contain values that are in the relation, what does it mean to say that a function is "onto"? "Onto" what? – David K Mar 06 '25 at 07:34
  • @DavidK - it was I've called above the "real function practice": we speak of real functions like $\dfrac 1 x : \mathbb R \to \mathbb R$ and $x^2 : \mathbb R \to \mathbb R$ and we perfectly understand that $\text {Dom}(\dfrac 1 x ) \ne \mathbb R$ and the same for... We want a specific name for the environment where Dom and Cdm "live"? for real fucntion it is $\mathbb R$, for complex one is... But I'm not the "legislator" of math jargon and it is a dream to legislate about languages in general. While $\times, \cdot, *$ for multiplication? why not a single symbol for a single concept? – Mauro ALLEGRANZA Mar 06 '25 at 07:45
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This distinction between the domain of the function and the (larger) space that contains this domain, and the importance to specify both, is extremely important in functional analysis. See for example Domain of an operator in functional analysis (or many, many textbooks): a (linear) operator is defined as an operator from some vector space $X$ to some other vector space $Y$; but it is not necessarily defined for every point of the initial space $X$, nor does it take every value in $Y$.

In that sense, the four objects you highlight are all equally important.

As for why the naming conventions don't reflect this, I have a couple guesses.

  • The first is that the naming convention gives a lesser importance to (your) $\mathbb R^n$ because it is often "obvious".

  • The second is that at the level where this vocabulary is introduced, mathematician don't like working with partial functions, and thus they will restrict the function so that (your) $\mathbb R^n$ is re-defined to be the domain of the function — which is why, for example, you would see the square root defined as $$ \begin{cases} [0, \infty) \to [0, \infty) \\ x \mapsto \sqrt{x} \end{cases} $$ even though you could define it as a partial function from $\mathbb R$ to $\mathbb R$ with domain $[0, \infty)$ and range $[0, \infty)$.

  • A third would be that some mathematicians don't really care about $X$ and $Y$ at all, because you can easily make them larger, so they just care about the range and domain; they would tell you that the codomain doesn't really matter either.

Thomas Lehéricy
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