Definition of the domain of a partial function

Question

I have seen various places define the domain of a partial function $f$ on $S$ to be the set $S'\subseteq S$ of elements that $f$ is defined on. So then what do you call $S$ in terms of $f$? You can't call it the domain of $f$ because that word is taken already.

Answer 1

You call it... $S$. There's no name for it in terms of $f$, because there's always a still larger set $T\supsetneq S$, and then what would that be called in terms of $f$? Typically, $S$ is fixed in context -- it's, for example, the reals $\Bbb R$, or the positive reals $\Bbb R^+$, and so on. $f$ simply doesn't carry information about possible supersets of its domain; the context does.

Answer 2

Maybe it's wise first to establish a definition of a partial function that we can all accept, and then address your question from there. This definition is motivated by a wikipedia article about "Partial Functions":

A partial function $f$ on $S$ is a function whose domain is contained in $S$.

From this, and, relative to your question, the only way to relate $f$ back to $S$ is simply by saying $\text{dom}(f) \subseteq S$.

That is, a function $f$ is a partial function on a set $S$ iff $\text{dom}(f) \subseteq S$.

Answer 3

In "Mathematical Analysis I" by V.A. Zorich (section 1.3.1) the author uses the term domain of departure of $f$ for your set $S$ (he also uses the term "domain of arrival of $f$" as a synonym for the codomain of $f$):

We see that it is sometimes necessary to consider a function $\varphi: A \to Y$ defined on a subset $A$ of some set $X$ while the range $\varphi(A)$ of $\varphi$ may also turn out be a subset of $Y$ that is different from $Y$. In this connection, we sometimes use the term domain of departure of the function to denote any set $X$ containing the domain of a function, and domain of arrival to denote any subset of $Y$ containing its range.