What is a good prototypical example of a construction that is not well-defined?

Question

In the question Why do students have problems with showing that something is well-defined? How can this be improved?, it was suggested that perhaps students have never seen something that is not well-defined.

I agree with this and would like to hear what you think is a good starting example for a not-well-defined object. I think a good starting example should require as little background knowledge as possible while avoiding the pitfalls of the following examples:

• Too fixable: You could claim that "the antiderivative of a function" is not well-defined, since there are lots of them. But students will easily protest that the antiderivative can just be something with a +C on it, which seems to clear up the issue. This doesn't help anyone who doesn't already understand the idea.

• Too obvious: You could say that "the denominator of a rational function" is not well-defined (or similar things in the link above), but the students see it immediately. This gives the wrong impression; in fact checking well-definedness carefully is critical and you can almost never see it immediately.

Answer 1

The naive "addition" on rational numbers, denote it $\oplus$, where you simply add numerator with numerator and denominator with denominator.

Example: $\tfrac 1 3 \oplus \tfrac 3 4 = \tfrac 4 7$.

I think this is less obviously non-well-defined than the other examples involving fractions.

Answer 2

Supposedly defining a ring homomorphism $f:\mathbb Z/m \to \mathbb Z/n$ by $f(k)=k$ is a popular, traditional attempt to define a map ... initially $\mathbb Z\to \mathbb Z/n$, that may be imagined to "factor through" the quotient $\mathbb Z/m$, but which does not without assumptions on $m,n$. Yet, in my experience, students look at the "formula" for it, and cannot see the difficulty, since the formula has such a straightforward appearance. :)

Answer 3

The linked question has a decent example, from mweiss:

"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.

Answer 4

Here is a troubling one that requires very little background:

Given $x \in \mathbf{R}$ and $q \in \mathbf{Q}$, define $x^q$ by:

• Write q as a fraction $\frac{a}{b}$.
• Evaluate $\sqrt[b]{x^a}$.

Then, for example, we compute something unpleasant like

$i = \sqrt{-1} = (-1)^{1/2} = (-1)^{2/4} = \sqrt[4]{(-1)^2} = 1$

or even

$-1 = \sqrt[3]{-1} = (-1)^{1/3} = (-1)^{2/6} = \sqrt[6]{(-1)^2} = 1$

Answer 5

You can't get much more prototypical than Euclid. Book I, proposition 1 assumes that the two circles intersect, but there is no postulate that guarantees that they intersect. From a modern point of view, the hidden assumption is that you're working in a space like $\mathbb{R}^2$, with completeness, rather than, say, $\mathbb{Q}^2$.

Answer 6

The simple example I use is an operation on rational numbers, call it $\star$ where

$$ \frac{a}{b}\star\frac{c}{d} := \frac{\max(a,c)}{\max(b,d)} $$

This then motivates the need usual proof that one needs to check that $+$ and $\cdot$ are still well defined modulo an ideal.