I am wondering if anyone knows any more on the history of the term 'co-domain' as it relates to functions.
Two sources I found:
Russell and Whitehead, Principia Mathematica, 1915, page 34 :
the class of all terms to which something or other has the relation $R$ is called the converse domain of $R$; it is the same as the domain of the converse of $R$.
Cassius Keyser, Mathematical Philosophy, 1922, page 168:
A relation $R$ has what is called a domain, - the class of all the terms such that each of them has the relation to something or other, - and also a codomain - the class of all the terms such that, given any one of them, something has the relation to it.
It seems to me that when Keyser talks about a 'codomain', he is talking about the same thing as Russell and Whitehead's 'converse domain'. So, it looks like we went from 'converse domain' to 'codomain' .... to 'co-domain'? That would seem to make sense.
Also, both texts talk about relations, not functions. But, a function is of course a special kind of relation. So ... it still makes sense.
However! (and this is really why I am asking this question): the way these two texts talk about the 'converse domain' and 'codomain' is (when applied to functions) what we nowadays call the 'range' or 'image' of the function, and not what we nowadays call its 'co-domain'.
Concrete example:
Take a function $f$ whose domain is defined as $\mathbb{R} - \{ 0 \}$, whose co-domain is defined as $\mathbb{R}$, and whose mapping is defined as $f(x) =1/x$.
For this function, the range or image is $\mathbb{R} - \{ 0 \}$, and that is what (again, if we see this function as a relation) Russell & Whitehead would consider its 'converse domain' what Keyser would call its 'codomain'.
But the 'co-domain' of this function was defined as $\mathbb{R} - \{ 0 \}$
So I think there has been a shift in the use of the term ... That is, it seems like we got:
'converse domain' -> 'codomain' -> 'range'
... while 'co-domain' is something different!
This is weird! What happened? Does anyone have some insight into any of this?