A Summer 2015 Mathematics A To Z: unbounded
Something is unbounded if it is not bounded. To summon a joke from my college newspaper days, all things considered, this wasn’t too tough a case for Inspector Bazalo.
Admittedly that doesn’t tell us much until we know what “bounded” means. But that means nearly what you might expect from common everyday English. A set of numbers is bounded if you can identify a value that the set never gets larger than, or smaller than. Specifically it’s bounded above if there’s some number that nothing in the set is bigger than. It’s bounded below if there’s some number that nothing in the set is smaller than. If someone just says bounded, they might mean that the set is bounded above and below simultaneously. Or she might mean there’s just an upper or a lower bound. The context should make it clear. If she says something is unbounded, she means that it’s not bounded below, or it’s not bounded above, or it’s not bounded on both sides.
We speak of a function being unbounded if its smallest possible range is unbounded. For example, think of a function with domain of all the real numbers. Give it the rule “match every number in the domain with its square”. In high school algebra you’d write this “f(x) = x2”. Then the range has to be the real numbers from 0 up to … well, just keep going up. It’s unbounded above, although it is bounded below. 0 or any negative number is a valid lower bound.
That’s a fairly obvious example, though. Functions can be more intricate and still be unbounded. For example, consider a function whose domain is all the counting numbers — 1, 2, 3, and so on. (This domain is an unbounded set.) Let the rule be that you match every number in the domain with one divided by its sine. That is, “f(x) = 1 / sin(x)”. There’s no highest, or lowest, number in this set. Pick any possible bound and you can find at least one x for which f(x) is bigger, or smaller.
Regions of space can be bounded or unbounded, too. A region of space is what it sounds like, some blotch on the map. The blotch doesn’t have to be contiguous. If it’s possible to draw a circle that the whole region fits within, then the region is bounded. If it’s impossible to do this, then the region is unbounded. I write blotches on maps and circles as if I’m necessarily talking about two-dimensional spaces. That’s a good way to get a feeling for bounded and unbounded regions. It appeals to our sense of drawing stuff out on paper and of looking at maps. But there’s no reason it has to be two-dimensional. The same ideas apply for one-dimensional spaces and three-dimensional ones. They also apply for higher dimensions. Just change “circles” to “spheres” or “hyperspheres” and the idea carries over.
You might remember the talk about measure, and how it gives an idea of how big a set is. And in that case you might expect an unbounded region has to have an infinitely large measure. After all, imagine a rectangle that’s one unit wide, starts at the left side of your paper, and goes off forever to the right. That’s obviously got infinitely large area. But it’s not so. You can have regions that are unbounded, but have finite — even zero — measure.
It’s often possible to swap a bounded set (function, region) for an unbounded one, or vice-versa. For example, if your set was the range of “1 / sin(x)”, you might match that up with “sin(x)”, its reciprocal. That’s obviously bounded. It’s less obvious how you might make a bounded set out of the range of “x2”. One way would be to match it with the function whose rule is “1 / (x2 + 1)”, which is bounded, above and below. As with duals, this is a way we can turn one problem into another, that we might be able to solve more easily.