Problems with bounded, bounds and boundary, and with closed sets.

One of the first concepts I discuss in my second-year analysis module is boundedness for subsets of \mathbb{R}^d.

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.

  1. A subset of the real line can be bounded below (or above) without being bounded.
  2. An unbounded region in \mathbb{R}^2 may be bounded by two curves, as in:Consider the two unbounded regions bounded by the two hyperbolae xy = 1 and xy=4.
  3. 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 \mathbb{R} 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 f on an interval J” has caused problems, with students thinking that such a function must take values in J.

Now, as if all that wasn’t bad enough, our second-year students are simultaneously taught about surfaces in {\mathbb{R}}^3 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.

Joel Feinstein

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7 responses to “Problems with bounded, bounds and boundary, and with closed sets.

  1. I agree that bounded is a problem. The problem I’ve seen most often in students is that they first hit the word bounded (above, below…) when I tell them about the Axiom of Completeness at the start of the real analysis course, and they get used to that. They then don’t seem to listen carefully enough the next time it comes up. Boundedness in Euclidean spaces of dimension more than 1 is widely misunderstood. I like the finite diameter idea proposed by Leader. However, budding mathematicians are just going to have to learn the language. You can’t stop the experts writing about bounded sets.
    The same applies to “closed” manifolds. Personally, I try to avoid this kind of language, and I write about compact intervals and compact manifolds. When teaching beginners, this option is not available until you have dealt with the basic characterizations of compactness.

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  2. I have a stupid trick for getting closed intervals to appear in WordPress documents. Let me attempt to prove it by giving you \{x:0\leq x\leq 1\}: I’ll see whether this has worked when I submit the comment but I hope it will be \null [0,1]. Assuming that worked, then you may be interested to know that what I typed was a dollar sign followed by “latex” followed by “\null” followed by [0,1] followed by a dollar sign. Somehow adding the \null seems to make things OK.

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  3. Great trick … but how on earth did you discover it?
    Joel

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  4. I just thought there might be a problem I didn’t understand with putting a dollar and an open-square-brackets together; a space didn’t cure it; and from my days of using plain TeX (I switched to LaTeX very recently) I knew that \null was a good way of doing something nonempty but invisible. (There I needed it sometimes because plain TeX ignores \vskip unless it’s between two things, so if I wanted a \vskip after some text and wanted it to fill up the rest of the page I had to follow it by \null and then \eject. I did this for things like writing letters and having the text slightly above the middle of the page, say — with two \vskips before it and three after perhaps. I have no idea how I’d manage that with LaTeX.)

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  5. Have just found this blog. Joel: excellent that you are putting down in words some of your thoughts about teaching (and just when I have to write my first teaching award essay!)

    I would echo Tony’s view about. The problem with mathematical terminology is that while it (a) is often very far from standard English usage, and (b) can annoyingly vary between disciplines in mathematics, terminology does tend to have a precise, well-used meaning, at least within disciplines.

    To give an example: suppose you stopped using “bounded” and opted for “finite diameter”. This, I think, takes care of (a) and (b) above (I cannot think of another meaning of finite diameter, but perhaps I’m not being creative enough). However, students who opened any of scores of standard Calculus or Analysis textbooks in the library would instantly be confused (I shall resist making some unfair comment about how likely this is!) You would have to warn all your colleagues about your non-standard usage of terminology, and they would have to modify their lecture courses, tutorial teaching etc.

    Or maybe you would, near the end of your course, let on that “bounded” actually means the same as “finite diameter”, and that actually this is what you will see in a textbook, and this is what Dr Bloggs will use next semester and so forth. But isn’t this likely to just cause more confusion? Perhaps not: maybe students, once they have correctly internalised a concept, will be happy to have it called different things. But I am skeptical: after all, many students won’t see any problem with the use of “bounded” (suitably defined)…

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  6. I agree that serious students of mathematics will have to learn to pay careful attention to definitions and work with them. This is a whole topic in itself!

    Students need to have lots of practice at working with easy definitions. I had hoped that boundedness would be an easy concept to get them started on, because it is very easy to check the properties of a few examples of bounded sets and unbounded sets rigorously before moving on to being able to “spot the answer”. Unfortunately, the problem with the langauge appeared to get in the way a bit.

    You could argue that the difficulty with the language will only help to emphasize to the students how important it is to treat precise definitions with respect! However, I think that I might prefer to get the students to get used to working carefully with “non-controversial” definitions first, rather than coping with two new tasks at the same time.

    Students who are already used to working with formal definitions and checking properties of examples should then be able to handle the additional task of overcoming confusion caused by the language. I am pleased to say that I have reasonable evidence that a large majority of my students know the difference between a closed set and a set which is not open!

    I will return to the issue of working with definitions in future posts. (It is one of the important issues that I have to address at the start of my second-year mathematical analysis module.) For the moment, you may be interested to look at my various presentations on “Why do we do proofs?” and “How do we do proofs?”, available online from my teaching presentations page.

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  7. I just got to the definition of boundedness again for the second years.
    Today I gave them a quote from Hamlet (Shakespeare) that might possibly help. Time will tell!

    … I could be bounded in a nutshell, and count myself a king of infinite space, were it not that I have bad dreams.

    Joel
    30/9/10

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