6

There are many questions about how best to teach functions, for example why don't we teach codomains in HS and should we teach them at all. On Math.SX there are questions about the "right" name for this concept, such as this one and that one.

My question is different. Assuming that one is going to introduce functions as a special type of relation between two sets $A$ and $B$, with $A$ the domain (there aren't other words for that, are there?), what should we call $B$, from a pedagogical point of view?

There is no right answer here, but I do feel like people have strong opinions about this issue that are rooted in good pedagogical practice, which is after all why we're lurking on this site, right? So perhaps this question can serve as a good place to gather the 'range' of good pedagogical reasons for choice of terminology in this case. Discussions about how this is resolved (or if it's unnecessary) in non-Anglophone contexts are also welcome.

For what it's worth, because of the ambiguity in the meaning of the term my main feeling is, as a professor of mine once said, to leave the range to the cattle.

Community
  • 1
kcrisman
  • 5,980
  • 20
  • 42

2 Answers2

6

I think "codomain" is great. If $f: A \to B$, then I have only ever heard "the range of $f$" defined as the subset of the codomain $\{b \in B : \exists a \in A \textrm{ with } f(a) = b \}$. It is helpful to have names for these things, and I think these are pretty much standard in higher level mathematics.

Steven Gubkin
  • 25,127
  • 4
  • 61
  • 110
  • on page 14 of Munkres Topology he defines "range" and "image set" to mean what I would rather term "codomain" and "range". It is unfortunate. – James S. Cook Jan 12 '16 at 21:37
  • Perhaps Munkres is just wrong? – Steven Gubkin Jan 12 '16 at 21:38
  • Yes. I would agree Munkres is wrong on this point. Also, some folks might write an arrow going from $A$ to $B$ with an $f$ over it to denote the idea that the domain of $f$ is some subset of $A$. We see this in diagrammatic calculations... – James S. Cook Jan 12 '16 at 21:39
  • I have never seen this convention for partial functions. Usually I have just seen this for regular functions. – Steven Gubkin Jan 12 '16 at 21:47
  • How can a definition be "wrong"? – Gerald Edgar Jan 13 '16 at 01:38
  • 3
    @GeraldEdgar Mathematically, there is "nothing wrong" with any definition. I could write an elementary number theory book with the definitions of "prime" and "composite" interchanged from their usual definitions, and everything in the book could be mathematically correct. I would still call the definition "wrong" since it conflicts with the normal use of the words. Using range for codomain is not as horrible as this, but I still think it is a mistake. – Steven Gubkin Jan 13 '16 at 02:38
  • 1
    Maybe Munkres used terminology that was common in 1974. – Gerald Edgar Jan 13 '16 at 14:17
  • 2
    @GeraldEdgar Sure. If you feel strongly about this, I could just say "I don't think that is in common usage" instead of "wrong". – Steven Gubkin Jan 13 '16 at 14:28
  • The notions of range and codomain are useful for distinguishing types of functions, e.g. if range = codomain, you have a surjective function, an important distinction in set theory. If range $\neq$ codomain, no inverse can exist on the codomain. – Dan Christensen Jan 15 '16 at 04:05
  • You can also more rigorously define a function $f$ saying that for every element $x$ of the domain $A$ there exists a unique image $f(x)$ in the codomain $B$. – Dan Christensen Jan 15 '16 at 04:26
3

In French, the most common terminology in my experience is "ensemble de départ" ("starting set" or maybe "departure set") for the domain and "ensemble d'arrivée" ("ending set" or maybe "arrival set") for the codomain. Sometimes, especially in high school where it is a common exercize to determine for which values of the variable a given expression makes sense, a function might only be defined on a subset of its "ensemble de départ" which is then called "ensemble de définition". This is a little confusing when in university we tend to mean that a function is defined on its whole "ensemble de départ" (especially if physicist use the other definition).

As far as I know, "domain" is almost only used for the "ensemble de définition" of an unbounded operator on a linear space.

Benoît Kloeckner
  • 9,089
  • 27
  • 57