I am a teaching assistant for a first year linear algebra course for mathematics and physics students and I think that it is not so clear for my students what "well-defined" means. Therefore, I would like to dedicate a part of the next session to giving them an idea of what this notion is.
From my own experience as a first year student, I remember that somehow seeing examples of functions or operations that are not well-defined was much more helpful to understand what this concept really means.
So my question is: What are cool (not too complicated) example of definitions of mathematical objects that are not well-defined ? The examples can relate to both analysis or linear algebra but need to be quite simple. I would also be interested in not well-defined constructions that are not necessarily related to functions directly (along the lines of my last example).
I've come up with a few example but I was wondering if there were better ones :). Here are the example I have come up with.
- Under what condition is the "identity function" $\varphi : \mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m \mathbb{Z}, [x] \mapsto [x]$ well-defined?
- Is the function $f: \mathbb{Q} \rightarrow \mathbb{Z}, \frac{a}{b} \mapsto a + b$ well-defined ?
- Why do we care about associativity in groups/rings/fields ?