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I see a lot of students struggling when they have to show that something is well-defined. I have the feeling that this is often not understood.

Two examples:

  • When defining a sequence $x_n= g(x_{n-1})$ (for some function $g$) and asking to show that $\lbrace x_n\rbrace$ is well-defined (since there is a root or a fraction somewhere in the definition), students often show that the sequence converges to something (under the assumption that it is well-defined) and do not see that they have to ensure that there is something bad.
  • When you want to define a function, say $g:A\to B$ (for some sets $A$ and $B$) with $g(x):=\ldots$, they forget to show that $g(x)$ holds the properties to be an element of $B$.

I have the feeling that some are not even aware of what well-defined means.

Two questions:

  • Why do students have this problem?
  • How can one (or the one who introduces functions) do to gain a better understanding of the concept of well-defined functions/sequences/...?
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Gowers has written around this subject (at least) twice see dpmms.cam.ac.uk/~wtg10/welldefined.html and gowers.wordpress.com/2009/06/08/… –  quid Mar 24 at 17:45

6 Answers 6

up vote 17 down vote accepted

Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. If you have never been in trouble since everything you were shown was well-defined, then you don't even understand the problem! (Even harder: after proving that something is well-defined, the world looks right like it was before, since they don't see the gain. Thus, they might try to do something that has more use for them.) You might try to give them explicitly not well-defined functions. Say, the sum of a fraction $\frac{a}{b}$ is $a+b$. Then let them check whether different representations of a rational number give the same value. Let them find a representation equal to 1/1 which has the sum 100. You could then discuss what typically can go wrong (problems with different representations, formulae that aren't defined for all values, values that do not lie within the range, ...?).

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Alas, the sum of a fraction is well defined, but the sum of a rational number is not. –  Andrej Bauer Mar 25 at 8:42
    
Thank you Andrej, I have corrected this. Obviously, you were right. –  Anschewski Mar 26 at 14:08
    
I think this is exactly right. "Well-defined" only becomes a meaningful concept if you have experience with cases in which something is not well-defined. Here is a simple case in which something seems entirely reasonable: Let $m, n$ be two integers and $[m], [n]$ their equivalence classes mod $p$. Define $[m] ^{[n]} = [m^n]$. Seems reasonable, especially because of the way we define the other arithmetic operations mod $p$. But as soon as you try to calculate particular examples you realize the definition is broken; different representatives of $[n]$ yield different results. –  mweiss Apr 8 at 21:04

I've never tried this in a classroom, but I suspect a lot of the trouble with functions is that students haven't been taught the vocabulary to deal with things that are weaker than functions. For example, they are trying to define a function $f: A \to B$. What they have written down probably defines SOMETHING: Perhaps a subset $R$ of $A \times B'$ for some $B' \supset B$. What do they then need to check: That for each $a$ in $A$, there is precisely one $b \in B$ so that $(a,b) \in R$. But if they can't use set theoretic language well enough to figure out what they have constructed, it will be hard for them to see what more they need to do.

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I've noticed the phenomenon mentioned in the question is widespread even in an "intro to proofs" course where we explicitly had just spent time showing that a function is a special kind of relation (i.e. a special subset of $A\times B$) and that it has to have all the right properties. I think this runs deeper than just familiarity with set-theoretic language. –  brendansullivan07 Mar 24 at 15:35

On the question why?, I think the answer is clear enough: mathematicians often speak and write as if all of their formulas are well-defined, and students emulate them. Usually we write the additive rule of derivatives as: $$f'+g'=(f+g)'$$ when the rigorous version would be: $$f'\negthinspace\negthinspace\downarrow \thinspace \& \thinspace \ g'\negthinspace\negthinspace\downarrow \ \rightarrow \ \ f'+g' = (f+g)'.$$ Solomon Feferman has shown how to rigorize the introduction of partial functions in this way. If the notation for "is well-defined" looks unfamiliar, that may prove my point.

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To get text in italic you can just type *the text* to get the text. (Using MathJax instead has several downsides.) –  quid Mar 24 at 16:21
    
I disagree that what you claim happens with the example you're given. I've never seen it not written like this "If $f$ and $g$ are differentiable, then $\ldots$" and this is OK. –  Git Gud Mar 25 at 11:10

I think that you hit the nail on the head, when you said some are not even aware of what well-defined means. As Anschewski suggests the problem may be that students have not encountered enough non-well-defined operations to fully appreciate the problem.

This Spring I was teaching freshman algebra, and while explaining equivalence relations (prior to getting started with congruences) I tried the following example. I had explained that being a namesake gives an equivalence relation among human beings. But if you try and define a function called weight from the set of equivalence classes Humans/Namesakes, you run into the difficulty of namesakes often having different weights.

Unfortunately the student response was a bit mixed. Some of them commented in their teacher evaluation forms that the example with namesakes was great, but I failed to follow up on it and give something similar when explaining all the other concepts they encountered. Well, if I think of something, I will use it next time.

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In my opinion this problem arises due to the fact that students are never told that, when introducing a new symbol which isn’t a variable (here I’m not using variable in the formal system sense of the word) and which does not depend on previously defined symbols, then one must prove that there exists exactly one object with the given property.

As an example take the square root of a non-negative real number. Showing it is well defined requires one to first prove $\forall x\in \left[0, +\infty\right[ \; \exists !y\in \left[0, +\infty\right[ \;\left(y^2=x\right)$, otherwise why whould $\sqrt x$ have any meaning at all?

An additional problem happens with equivalence relations. In this context, let us consider $\equiv _p$ over the integers. Given $m,n\in \mathbb Z$, the function $+_{\equiv _p}$ maps $\left([m]_{\equiv p}, [n]_{\equiv p}\right)$ to $[m+n]_{\equiv p}$. Even after proving the analogous property to the example above, this has another problem, namely that the function isn't defined with respect to variables, but rather with respect to the representation of variables. Ideally one would write $f\colon A\to B, a\mapsto f(a)$, but in this context we’re actually writing $f\colon A\to B, g(a)\mapsto f(a)$. Why it can be done is a trivial matter, but it should be observed.

I have the feeling that some are not even aware of what well-defined means.

I agree with Jyrki on this, but I don't think it is just ‘some’. Knowing what well-defined means is being aware of all of the above. I graduated only recently and I’d bet all my money (that doesn't mean much, unfortunately) that less than 6% of my former fellow undergraduate students are aware of all the subtleties.

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Part of the problem is that we often talk and write as if something is well-defined and then check well-definedness afterward. For example, we might write $f(x)=\dots$ and afterward check that "$\dots$" really does define a function. Unfortunately, in the (usually very brief) time between writing $f(x)$ and checking that it's well-defined, we have probably confused some students. We've used a notation $f(x)$ that (usually) presupposes that $f$ is a function, without yet having verified the presupposition. Strictly speaking, we should perhaps have used a notation for relations (see David Speyer's answer), like $R(x,\dots)$ or $(x,\dots)\in R$, then verified uniqueness, and only afterward used function notation. But that's not what we usually do.

For people who already understand the issue, it's no problem that we use function notation before checking its legitimacy, but for students who are just learning (or trying to learn) these things, it can cause confusion.

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