Completely elementary motivation of the definition of the concept of "codomain"?

Question

Among the most prominent sins against Reason that are widespread in the mathematical community (but not the only one) is presenting definitions without motivations. (What if presenting theorems without proofs were so widespread?) Thus, for example, undergraduates are told that matrices are multiplied by taking dot-products of rows of one matrix and columns of another, without saying that that is just what is needed to make matrix multiplication correspond to composition of linear transformations. And many other such examples exist. (E.g. how many who read this posting can explain why the definition of "characteristic zero"? Would it be rash to think that maybe at least fifty-percent cannot?).

If $f:X\to Y$ and $V\subseteq Y,$ than many mathematicians in many contexts use the notation $f^{-1}(V)$ to mean $\{x\in X:f(x)\in V\},$ but some, especially set-theorists, use the notation $f^{-1}[V]$ with square brackets for that concept and reserve the notation with round brackets to mean the unique member of the domain mapped to the specified member of the codomain, when such a unique element exists. That can matter when the domain is something like the von Neumann ordinals, or any of many in set theory in which a set can be both a member and a subset of the domain.)

So a mathematician teaches a course for undergraduate math majors that treats elementary logic and set theory and leads into the topic of how to write proofs, including such things as the fact that if you want to prove $A\subseteq B,$ you don't start by saying $\text{“Let $x\in B$.”},$ etc. And this instructor is fastidious about the distinction between round and square brackets described above, leading me to suspect he is interested in foundational issues. And he expects students to learn the distinction between the image and the codomain of a function. If $f:\mathbb R\to\mathbb R$ and $\forall x\in\mathbb R\,\,f(x)\in[0,1],$ then students who think $[0,1]$ is the codomain are considered to have failed to learn that point.

This afternoon I told a student in that course that one reason for this distinction is that in some cases it is trivial to see that every image of a point under a function is in $\mathbb R,$ but identifying the image of the function may be a challenging research problem. If the student had wanted more depth than that, I might not have had an answer that I could have presented in a manner worthy of framing it and hanging it from the wall of a museum. Or to use what may be a better metaphor in this context, imagine a companion volume to Proofs from the Book called Motivations from the Book.

So my question is: What motivation for that concept fits into that companion volume and can readily be presented to that kind of student?

Answer 1

I'd say the concept of codomains is sort of type-theoretic, motivated by a desire to decouple a function's "type" (its domain and codomain) from its "value" (the actual assignment of output values to input values). As you suggested, we often want to work abstractly with arbitrary functions without assuming anything about their evaluation rules; the notation $f:\Bbb R\to\Bbb R$ enables this.

The tempting alternative might be to avoid attaching a particular codomain to $f$ and instead consider any superset of $f$'s range to be "a codomain" of it. Nothing breaks if we do this, and the notation $f:\Bbb R\to\Bbb R$ can still mean something, but then "$f:\Bbb R\to[0,1]$" is a statement whose truth depends on $f$'s values, and two functions given with different codomains can be equal (again depending on their values). Consequences like this seem unclarifying and not particularly useful. Organizing functions into types based on domain and codomain provides a clean mental picture—essentially the picture of functions as morphisms in the category of sets—that facilitates more abstract, "value-independent" reasoning.

The discussion on this related question also looks relevant.