One to one, domain, onto, codomain, and that sort of stuff
(Scroll to the bottom of this post for my math questions for you.)
There is an interesting discussion on the NYS Math Teachers listserve about “onto.”
It’s interesting to read. Some teachers are shakier on “onto” and some are quite confident. Some have set ways of teaching the concept to kids, and some are looking for ideas. I like, big time, teachers talking to teachers.
The discussion is animated by a brand new New York State Regents exam: Integrated Algebra II and Trigonometry. We’re not really sure what will be on the exam. And we’re nervous. Some of us about teaching things we haven’t taught before. Others about deciding which topics to teach (some, and here I criticize, want to teach exactly what is on the exam. More on that, a bit later). Some are even shaky on a topic or two.
The exam will cover far too much material from Algebra II, Trigonometry, Statistics, Probability, even some traditionally precalc topics (eg series and sequences). Teach with some depth? You’ll run out of time. Teach the full gamut? They will have a grand tour of many topics, without sufficient time devoted to many of them. And forget creating a coherent course.
You can see the only sample test we’ve seen here. You can see some discussion on the listserve, and some careful criticism at Kate(t). Not too easy, not too hard, but awful. Scattered to too many topics. Too many procedural questions. Too many push-calculator-keys but no-understanding-necessary questions. Too many unnecessarily tricky questions.
Me? My school? We’ll go a little too slow, much too deep. Some of the early skills will carry over, and we’ll make some of it up. It’s a possibility that is easier with our kids (very bright, but many lean towards the humanities) than at many other schools
Anyway, some observations and questions, based on the listserve discussion:
1. Do any of the higher math folks out there (John, Kibr, Susan, one of 360, who else? there’s lots of you!) have a strong feeling about “codomain?”
2. Puzzle (not for any of the above-mentioned people, either by name or not): Can you think of a function that is onto but not 1-1? Too easy? How about 1-1 but not onto?
3. Would anyone be annoyed if I mashed up the language, and called the “codomain” the “target set” and said that a function is “onto” a particular “target set” if it uses up all the values in that set? And used as my favorite example the step function (onto the integers, not onto the reals)
4. Why do we care whether or not a relation is a function? (I think I know, and I think I have just, as the books do always, asked the question backwards)
5. Kate(t) posted a pedagogical device for teaching these terms. Take a look. (I think it is ok for the first 3, but love the tears for “onto.”)
JD2718 
I’m teaching a freshman algebra/precalc/calculus course for the first time this semester. I did talk about 1-1, but I did not even mention onto. I also did not mention the idea of a codomain, because except when I first introduced functions I’ve always assumed the functions are real-valued functions, and thus the codomain is R but we’re more interested in the range. In this context I didn’t want to get too deep into the set-theoretic language or generalize too much in the limited time I have.
I suppose if I wanted to teach onto functions, I would approach it from a graphical standpoint. To test whether a function is one-to-one we have the horizontal line test, where we ensure that the graph crosses each horizontal line at most once. So to see if a function is onto we have a very natural parallel: check that the graph crosses each horizontal line at least once.
Hi Jonathan,
(1) What is a codomain? Covectors, cohomology, and co-NP are fine with me, but I’ve never heard of codomain before.
(2) I like the puzzle questions — students should know examples like those if you are teaching 1-1 and onto.
(3) Since I don’t like “codomain”, “target set” works for me. It is very clear.
(4) It depends on the context. I’m sure your students want their grades to be a function of their tests. Nobody wants to hear “Your grade for this test is the set containing A, B, C, and D.”
(5) I’ll take a look
they wanna hear “your grade is
*an element of* the set…”.
right?
My students find the concept of “codomain” to be a little tricky, because it has the word domain in it but it’s more related to the range. [I think it’s related linguistically to cohomology because it’s related to the domain of the inverse function.] With that in mind, I like “target set” quite a bit — but if the students need to know what a co-domain is for testing purposes, you’d want to occasionally remind them.
For #2, are you looking for a function [y=blahblahblah] or are you looking for a picture [or are you looking for a set of ordered pairs? Some of our books do, though I personally focus on equations and graphs.]
For #4, I can think of two times that I really think about relations. My own personal default interpretation is that I tend to view relations in contrast to operations: in both cases you take a couple of things in a set: if you’re comparing them it’s a relation [“less than” “divides” etc.], but if you’re putting them together to get a new thing it’s an operation [“subtraction” “division’]. When I’m thinking about relations in this context, the word “function” doesn’t really enter into my mind.
But the other (less common) time I think about relations is with something like the equation of a circle, like x^2+y^2=1. There is clearly a relation between the x and y, but it’s not a function because, as written, there are often more than one y that work for the same x. So if you want to write it as a function, to graph it on your calculator for example, you need to glue together two different functions y=sqrt(1-x^2) and y=-sqrt(1-x^2). I think that’s when you really need relations versus functions.
my first move in a discussion showing
a tendency to involve “codomain” is to
push for Target and Image.
f:A—-}B
has B as its target
and f(A) as its image.
thanks for asking. yrs in the faith. kibr.
what’s the codomain again?
category theory or something…
Yes, reverse the arrows and slap a “co” in front of everything and you’ll sound smart. Sometimes this gives pretty deep insight: homology studies spaces by mapping simple spaces into them, cohomology studies spaces by mapping them into simple spaces. In the domain/codomain case it just sounds pedantic IMO.
I also prefer target and image. But if the word codomain will be on the test, I guess it must be taught. It shouldn’t get in the way too much, people get used to plenty of other silly words.
And in the end, I chose “target set” and said it is usually one of our “big sets” (my generic name for reals, rationals, integers, whole numbers, natural numbers). And I added that the target set is also called the codomain.
The step function is onto the integers, but is not onto the reals.
I think they have it.
ps never say range if you can help it.
unless you mean, like, a stove or something.
too many people are sure they’re right
that can’t all be right. these situations
have to be routed around.
w’edia is good as usual.
I agree with Kibrolv above; I was raised to call a “range” what you’re calling a codomain, and the syllabus I currently teach requires me to use “range” to mean image. “Target” is a good word.
For a function f:A->B…
In the recent college math books I’ve seen, codomain is used to refer to (all of) B, while range refers to f(A), the image of A in B. In which case the range of f would be the same as the domain of the inverse function.
That is, range(f) = image(f) < codomain(f) (where "<" means "set containment").
I don't know how universal that language is. But I will say I first heard "range" used in this sense when I was in 8th or 9th grade (in a district still using new-math influenced curricula), and I first heard "codomain" when I was in graduate school. But my students (college juniors, math majors) don't bat an eye at the word "codomain".