One of the first concepts I discuss in my second-year analysis module is boundedness for subsets of .
Here, the concept is a simple one, but I have found that the word “bounded” appears to give trouble. (Have others come across this?)
It is, perhaps, quite hard for some of us to remember what it was like before we knew the mathematical meaning of the word “bounded”. The fact is that the English word does not really correspond to its mathematical meaning. (A familiar problem!)
From what I have seen of the struggle students have with this, it appears that one of the issues is that if a set “has bounds”, shouldn’t the bounds be in the set? Add to this some confusion with the word “boundary”, and you end up with some of my mathematical analysis FAQ’s “Isn’t every closed set bounded?”, “Isn’t every bounded set closed?”, perhaps via the chain of ideas “bounded”, “has bounds”, “has its bounds”, “has its boundary”, “contains its boundary”.
Now it is easy for us to comfort ourselves with the idea that students should just read and work with the definition given, and that any confusion is not our fault. However, standard terminology and expression does contribute to the problem. Here are some examples.
- A subset of the real line can be bounded below (or above) without being bounded.
- An unbounded region in may be bounded by two curves, as in:Consider the two unbounded regions bounded by the two hyperbolae and .
- The boundedness theorem for closed intervals is sometimes stated as follows:Every continuous, real-valued function on a closed interval [a,b] is bounded and attains its bounds.
[By the way, can someone tell me how to get square brackets into latex markup in wordpress?]
I find the second of these examples rather disturbing myself!
The third example has some sub-problems attached to it.
- Is it safer to say “closed and bounded interval”, in view of the fact that some unbounded intervals in are closed? Is it safe to follow the convention that “closed interval” means “closed and bounded interval”, when there are other related issues that can combine with this to cause confusion?
- Obviously a bounded function can not attain ALL of “its bounds”, but only (at most) two of them (the least upper bound and the greatest lower bound of the set of values taken by the function).
- Even the notion of “a bounded, real-valued function on an interval ” has caused problems, with students thinking that such a function must take values in .
Now, as if all that wasn’t bad enough, our second-year students are simultaneously taught about surfaces in in another module. There the term bounded means the same thing (thank goodness!), but “closed surface” and “boundary of the surface” do NOT agree with the usage in my mathematical analysis module. So both lecturers involved have to warn the students about the clash in terminology, and that the terms used should be understood in context.
What can we do to address this problem?
Obviously, I have added some comments and warnings to my lecture notes and to my Mathematical Analysis FAQ document
Is there an alternative to the term “bounded”? Imre Leader suggested that the term “finite diameter” might be useful. (Depending on your definition, you may have to watch out for the empty set!)
I think that students who work with enough examples of each type of set and solve enough problems should soon learn to tell the difference between closed sets and bounded sets. Nevertheless, I think that it is a pity that the English can get in the way of understanding the maths.