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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.

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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.

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  • $\begingroup$ Well people talk about the codomain of a function. By an analogous argument to what you said, this is ill-formed, as the function $f: A\rightarrow B$ carries no information about $B$. $\endgroup$ – Joshua Meyers Nov 14 '15 at 6:14
  • $\begingroup$ It's not ill-formed, but you're right it's symmetric with what I said about "partial functions". What you observe about codomains is usually true. To address that problem, Bourbaki uses a different definition: in their volumes, a function is a triple $(G_f, A, B)$, where $G_f$ is the graph of the function [which we typically consider the function itself], $A = domain(G_f)$ and $B\supseteq image(G_f)$ is the codomain. This makes it possible to be more precise about some things, while making other things awkward that hadn't been. There are tradeoffs :) $\endgroup$ – BrianO Nov 14 '15 at 6:17
  • $\begingroup$ Right. In the case of functions, the condition $\textrm{im}f=B$ is important (surjectivity). I think the condition $S=S'$ is important for similar reasons so it would be nice to have a way to talk about it. $\endgroup$ – Joshua Meyers Nov 14 '15 at 6:21
  • $\begingroup$ Well, in context there always is: "$f$ is a total function [on $S$]". E.g. in computability theory, partial functions are in a sense the main objects of study, and in practice no confusion arises. $\endgroup$ – BrianO Nov 14 '15 at 6:23
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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 $\text{dom}(f) \subseteq S$.

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