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

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 $\mathbf{R}$ and that its 'range' is $\mathbf{R} / \{ 0 \}$, 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 $\mathbf{R} / \{ 0 \}$.

So: the 'converse domain' of $f$ is $\mathbf{R} / \{ 0 \}$, but its 'co-domain' (which I assume is just short-hand for 'converse domain') is $\mathbf{R}$?!? 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 $\mathbf{R}$ (and so we can characterize this function as a function $f : \mathbf{R} \rightarrow \mathbf{R}$ 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 $\mathbf{R} / \{ 0 \}$? 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?

• As I always heard the story, "codomain" is not short for "converse domain"; rather, it comes from applying the prefix "co-" (meaning "dual") from category theory to the word "domain". – Eric Wofsey Oct 17 '16 at 2:13
• Also, related: math.stackexchange.com/questions/396459/…. As indicated by some of the answers there, the notion of codomain is extremely natural as soon as you start to talk about objects with more structure than just sets (e.g., group homomorphisms between groups, or smooth maps between smooth manifolds). By the same token, in many such contexts it is not natural to consider partial functions. – Eric Wofsey Oct 17 '16 at 2:23
• Do you have a source for this? Because at math.stackexchange.com/questions/719227/… they seemed to think that 'co-domain' is short for 'converse domain'. – Bram28 Oct 17 '16 at 2:44
• None of the quotes provided there actually use the term "codomain", so I'm not at all convinced by that claim. I don't have any source for the actual history, but this is certainly the widely understood etymology of "codomain" among people who do category theory, whether or not it is historically accurate. – Eric Wofsey Oct 17 '16 at 3:13
• @EricWofsey Here is a quote from "Mathematical Philosophy, a study of fate and freedom" by Cassius Keyser: "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". So here is one of the first uses of 'codomain', and Keyser uses 'codomain' in the exact same way Russell and Whitehead use their term 'converse domain'. – Bram28 Oct 18 '16 at 16:43

## 1 Answer

The "terminological confusion" has a quite long history.

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

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 :

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 $B$.

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).

• OK, so W&R's 'converse domain' is not the same as today's 'co-domain', right? – Bram28 Oct 17 '16 at 11:49
• @Bram28 - exactly; it is the same of todays' image (or range). – Mauro ALLEGRANZA Oct 17 '16 at 11:50
• OK, thanks! But why I understand that in the notation $f:A \rightarrow B$ we mean A to be the doamin, and B the codomain, I am still confused why mathematicians opted to do so. I mean, when I look at that notation, to me it seems to be a nice way to indicate the potential candidates of this mapping; it would be kind of like letting students know "OK, when you want to graph this function, these are the axes you want to use". – Bram28 Oct 17 '16 at 11:58
• So we could indicate that the function is meant to be from natural numbers to natural numbers, or from real numbers to real numbers. So with a function like $f(x) = 1/x$, I am inclined to say: this is meant to be a function from real numbers to real numbers. And then, when I do some analysis, I'll figure out for what values the function is actually determined, and what values the function actually takes on. So I see a definite distinction between 'discourse' and 'definition'. But in the way we write this notation, and how we distinguish 'domain' from 'codomain', we mix up the two. Why? – Bram28 Oct 17 '16 at 11:59
• FYI: Here is a quote from "Mathematical Philosophy, a study of fate and freedom" by Cassius Keyser: "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". So here is one of the first uses of 'codomain', and Keyser uses 'codomain' in the exact same way Russell and Whitehead use their term 'converse domain'. So at some point the meaning changed ... – Bram28 Oct 18 '16 at 16:49