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Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology.

  1. $X$ is the source of $f$
  2. $Y$ is the target
  3. $\{x \in X \mid \exists y \in Y : xy \in f\}$ is the domain
  4. and $\{y \in Y \mid \exists x \in X : xy \in f\}$ is the range.

However, it has come to my attention that some people use 'domain' to mean $X$, rather than the subset of $X$ described above. So, is there any standard, unambiguous terminology for these concepts that does not have this problem?

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  • $\begingroup$ My impression is that $f:X\rightarrow Y$ more often means $f$ is defined on all of $X$. Now, there are areas in which it is convenient not to make that assumption: Birrational geometry, functional analysis, PDE. $\endgroup$ – OR. Jul 18 '13 at 12:33
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    $\begingroup$ When you say correspondence, do you just mean any relation, or some specific type of relation? $\endgroup$ – Tobias Kildetoft Jul 18 '13 at 12:42
  • $\begingroup$ @TobiasKildetoft, I just mean a relation that is more than its graph, its an ordered triple $f=(F,X,Y)$ with $F \subseteq X \times Y$. $\endgroup$ – goblin Jul 18 '13 at 12:43
  • $\begingroup$ In that case, I would not use the notation $f: X\to Y$ as pretty much everyone will think you mean a function when you write that. $\endgroup$ – Tobias Kildetoft Jul 18 '13 at 12:46
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    $\begingroup$ In that case, your last two definitions need not make sense at all. Not all arrows in category theory have anything to do with relations. But if $X$ and $Y$ are sets and you write $f: X\to Y$ you will need to point out very clearly that $f$ is not assumed to be a function, and even then, many people would still be confused. $\endgroup$ – Tobias Kildetoft Jul 18 '13 at 12:51
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No, there is no standard unambiguous terminology. The standard terms are "domain" and "range" but they are ambiguous between authors.

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OK, here's the deal.

I would say the fourth bullet point is exactly as I would define range.

Now, the second bullet point is a fair definition: the domain is usually assumed to be the set on which the function is defined. BUT in common practice, when writing a function in the form $f:?\to Y$, you almost always put the domain in for the question mark, even if the domain is a proper subset of some other set $X$.

So the upshot is that when writing functions in the arrow format, it is (almost?) universally assumed that the set in your third bullet is what appears to the left of the arrow in the arrow notation.

People do study functions defined on subset of $X$, and the definitions you gave for domain of a function on a subset of a set to another set are valid. The usage for the arrow notation and terms domain and range are still applied this way:

  • They normally say "Suppose $f$ is a function $U\to V$ where $U\subseteq M$." and the fact that $U$ is the domain is implicit.
  • But they don't normally say "Suppose $f$ is a function $M\to V$ where $U$ is the domain of $f$."

If you are studying relations rather than functions, you won't find the arrow notation used as much. It would still be reasonable to call out the sets you've defined as domain and range above (aka "the left coordinates of things in the relation" and "the right coordinates of things in the relation") and I can't immediately recall any other names I've heard for these sets.

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I'm not clear on what should be a comment vs. what should be an answer, but I'll add to what's already been said the following.

From a computer science perspective, which is probably influenced by category theory, I think your $Y$ (bullet 2) is commonly called the codomain. Also, at least in the context of functional languages, the $\rightarrow$ in $f : X \rightarrow Y$ indicates "function from". In text (if not in code) the notation mentioned by @dtldarek (which I don't know how to encode in latex) is often used to indicate a partial function and functions are usually assumed to be total (modulo termination) unless the context is made clear.

You can read ":" to mean "has type". So you can read $f : X \rightarrow Y$ as $f$ has the type of (total, depending on context) function from domain $X$ to codomain $Y$. In this sense, you might see (general) relations as $R : X \times Y$ or $R \subseteq X \times Y$ -- $R$ has the type (or is a subset of) $X \times Y$. I have been corrected on the ":" notation for relations in mathematical contexts, so it is probably peculiar to computer science.

EDIT: As pointed out in the comments, I am no longer so sure about the $\colon$ notation for relations. At any rate, I agree that $\subseteq$ is probably clearer. The point I was inelagantly trying to make was that I do not think the $\rightarrow$ notation is usually used for (general) relations.

Finally, I believe even the ordered triple notation $(f,X,Y)$ for a function $f$ usually assumes a total function (i.e. $X$ is the domain as in bullet 3) into codomain $Y$, unless it is clear from the context.

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  • $\begingroup$ Just a quick comment: I don't think $R : X \times Y$ is good notation for '$R$ is a relation with source $X$ and target $Y$,' because it could be confused for the statement that $R$ is an ordered pair $(x,y)$ with $x : X$ and $y : Y.$ So better notation would be $R : \mathcal{P}(X \times Y).$ But, I think the arrow notation $R : X \rightarrow Y$ is even better. You just have to be clear that $R$ is a relation. $\endgroup$ – goblin Jul 18 '13 at 15:20
  • $\begingroup$ Also, you're right that $Y$ is usually called a codomain. Thus, $X$ should be called a domain. This is yet further evidence that my nomenclature is bad and needs help!! $\endgroup$ – goblin Jul 18 '13 at 15:22
  • $\begingroup$ You may be right about $R : X \times Y$ as it was just from memory (i.e. I didn't look up any sources before typing that). However, I don't think I've ever seen the arrow notation for (general) relations. Nor do I think I've seen "source" and "target" used in the context of (general) relations. $\endgroup$ – joeA Jul 18 '13 at 15:27
  • $\begingroup$ It seems like $R\subseteq X\times Y$ and $R\in \mathcal{P}(X\times Y)$ are even clearer than using the colon. $\endgroup$ – rschwieb Jul 18 '13 at 16:13

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