Are 'converse domain' and 'co-domain' the same? If not, why not?

Question

Consider the function $f(x) = 1/x$, where I am focusing on the real numbers (as opposed to, say, the complex numbers).

Most texts would say that, given that I focus on all real numbers, the 'co-domain' is $\mathbb{R}$ and that its 'range' is $\mathbb{R} \setminus \{ 0 \}$, and that these two are not the same.

But Russell and Whitehead's define the'converse domain' as the domain of the converse, which in this case is $\mathbb{R} \setminus \{ 0 \}$.

So: the 'converse domain' of $f$ is $\mathbb{R} \setminus \{ 0 \}$, but its 'co-domain' (which I assume is just short-hand for 'converse domain') is $\mathbb{R}$?!? What gives?

If we have changed the use or meaning of 'converse domain' when we started using 'co-domain' ... why? ... Is there any good reason for this? Why not say that for this particular function, the 'domain of discourse', as well as its 'co-domain of discourse' is $\mathbb{R}$ (and so we can characterize this function as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ without having to worry about what specific values can go into the input or come out as its output) and that its 'domain of definition', as well as its 'co-domain of definition' is $\mathbb{R} \setminus \{ 0 \}$? Wouldn't that be much more in the spirit of Russell and Whitehead? Indeed, why are we confusing all our high school students by insisting that the 'domain' is the function's 'domain of definition', while its 'co-domain' is its 'co-domain of discourse'? Why the asymmetry? Is it because we assume all functions have to be total (so that the 'domain of discourse' and 'domain of definition' are one and the same?) But again, why do this? If we say that functions can be surjective/onto or not, why not also say that functions can be total or partial? And yes, I realize that some treatments of functions do this, but most treatments of functions do not. What happened here and why?

Answer 1

The "terminological confusion" has a quite long history.

For today's definitions, you can see e.g. :

• David Lawrence Johnson, Elements of Logic via Numbers and Sets (1998), page 90 :

Given set $A$ and $B$, a map $\theta$ from $A$ to $B$ is a rule that assigns to each element $a \in A$ an element of $B$, called the image of $a$ under $\theta$ and written $\theta(a)$. It is customary to express this by writing

$$\theta : A \to B,$$

and to visualize $\theta$ as an agent that acts on each element $a \in A$ to produce an element $\theta(a) \in B$.

The set $A$ is called the domain of $\theta$, and $B$ the codomain. [...] The set of values of $\theta$ on all $a \in A$ is a subset of $B$ called the image of $\theta$,

$$\text {Im} \theta = \{b \in B \mid \exists a \in A : \theta(a) = b \}.$$

An "early occurrence" is into :

• Garrett Birkhoff & Saunders Mac Lane, A Survey of Modern Algebra (4th ed - 1977), page 33 :

A function $\phi : A \to B$ is also called a mapping, a transformation, or a correspondence from $A$ to $B$. The set $A$ is called the domain of the function $\phi$, and $В$ its codomain.

The image (or "range") of a function $\phi : A \to B$ is the set of all the "values" of the function; that is, all $а \phi$ for $a$ in $A$. The image is a subset of the codomain $B$, but need not be all of $B$.

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You can see the post : who popularized the modern definition of domain and codomain of function for the W&R original definition of converse domain and subsequent evolution.

Their convese domain is the equivalent of modern image (or range).