A Summer 2015 Mathematics A To Z: unbounded


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.

A Summer 2015 Mathematics A To Z: jump (discontinuity)


Jump discontinuity.

Analysis is one of the major subjects in mathematics. That’s the study of functions. These usually have numbers as the domain and the range. The domain and range might be the real numbers, or complex numbers, or they might be sets of real or complex numbers. But they’re all numbers. If you asked for an example of one of these functions you’d get something that looked more or less like a function out of high school.

Continuity is one of the things mathematicians look for in functions. To a mathematician continuity means almost what you’d imagine from the everyday definition of the term. You could draw a sketch of a continuous function without having to lift your pen off the paper. (Typically. If you want to, you can define functions that meet the proper mathematical definition of “continuous” but that you really can’t draw. Mathematicians use these functions to keep one another humble.)

Continuous functions tend to be nice ones to work with. Continuity usually makes it easier to prove a function has whatever other properties you’d like. Mathematicians will even talk about continuous functions as being nice and well-behaved and even normal, as though the functions being easier to work with bestowed on them some moral virtue. However, not every function is continuous. Properly speaking, most functions aren’t continuous. This is the same way that most numbers aren’t whole numbers.

There are different ways that a function can be discontinuous. One of the easiest to understand and to work with is called a jump discontinuity. If you draw a plot representing a function with a jump discontinuity, it looks rather like the plot of a nice, well-behaved, continuous function except that at the discontinuity it jumps. From one side of the discontinuity to the other the function suddenly hops upward, or drops downward.

If a function only has jump discontinuities we aren’t badly off. We can write a function with jump discontinuities as the sum of a continuous function and a function made up only of jumps. The continuous function will be easy to work with, since it’s continuous. The function made of jumps isn’t continuous, by definition, but it’s going to be “flat” — it’ll have the same value in-between any two jumps. That’s usually easy to work with, and while the details of these jump functions will be different they’ll all look about the same. They’ll have different heights and jump up or down at different points, but if you know how to understand a function that jumps from being equal to 0 to being equal to 1 when the input goes from just below to just above 2, then you know how to understand a function that jumps from being equal to 0 to being equal to 3 when the input goes from just below 2.5 to just above 2.5.

This won’t let us work with every function. Most functions are going to be discontinuous in ways that we can’t resolve with jump functions. But a lot of the functions we’re naturally interested in, because they model interesting problems, can be. And so we can divide tricky functions into sets of functions that are easier to deal with.

A Summer 2015 Mathematics A To Z: into


Into.

The definition of “into” will call back to my A to Z piece on “bijections”. It particularly call on what mathematicians mean by a function. When a mathematician talks about a functions she means a combination three components. The first is a set called the domain. The second is a set called the range. The last is a rule that matches up things in the domain to things in the range.

We said the function was “onto” if absolutely everything which was in the range got used. That is, if everything in the range has at least one thing in the domain that the rule matches to it. The function that has domain of -3 to 3, and range of -27 to 27, and the rule that matches a number x in the domain to the number x3 in the range is “onto”.

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A Summer 2015 Mathematics A To Z: characteristic


Characteristic function. (Not the probability one.)

Today’s entry in my mathematical A-To-Z challenge is easier than the bijection function was. This is the characteristic function. Its domain is any set, any collection of things you like. This can be real numbers, it can be regions of space, it can be houses in a neighborhood. Its range, however, is just the two numbers 0 and 1. Its rule — well, that’s the trick. It’s not right to say there’s “the” characteristic function. There are many characteristic functions. It’s just they all look alike. This is the way they look.

To define a characteristic function we need some subset of the domain. A subset is just a collection of things that are also all in another set. So we want a subset — let me give it the name D — of the domain. This subset D can have one or a couple of things in it; it could have everything in the domain that’s in it. The one rule is that D can’t have something in it which isn’t also in the domain. Otherwise, anything goes. (It’s even fine if D doesn’t have anything in it.)

Now, the rule for the characteristic function for D is that the function for any given item in the domain — use x as a name for that — is equal to 1 if x is in D, and is equal to 0 if x is not in D. The function is usually written as the Greek letter chi (\chi ), or the letter I, or the number 1 put in some kind of fancy heavy font, with the D as a subscript so we know which characteristic function it is.

For example. Suppose the domain is the counting numbers. Suppose the subset D is the prime numbers: 2, 3, 5, 7, 11, 13, and so on. Then the characteristic function looks like this:

For the number x \chi_D(x) is
1 0
2 1
3 1
4 0
5 1
6 0
7 1
8 0
9 0
10 0

… and so on.

Some might ask why create, much less care about, such a boring function? These are people who’ve never had to count how many rows on a large spreadsheet satisfied some complicated set of conditions. That trick where you create a column with a rule like ‘IF((C$2 > 80 AND C$2 ’01/01/2013’ AND C$4 < '05/01/2013'), 1, 0)', and then add up the column, to find out how many things had a value between 80 and 90 and a date between the start of January and the start of May, 2013? That's using a characteristic function to figure out how large a collection of things is.

Characteristic functions offer ways of breaking down a complicated set into smaller ones all of which share some property. This can be used just to work out how large are the collections of things that share different properties. It can also be a way to break a big problem into multiple smaller problems. We hope those smaller problems are simpler enough that we’re making overall less work for ourselves despite increasing the number of problems. And that’s a good trick, one mathematicians rely on a lot.

A Summer 2015 Mathematics A To Z: bijection


Bijection.

To explain this second term in my mathematical A to Z challenge I have to describe yet another term. That’s function. A non-mathematician’s idea a function is something like “a line with a bunch of x’s in it, and maybe also a cosine or something”. That’s fair enough, although it’s a bit like defining chemistry as “mixing together colored, bubbling liquids until something explodes”.

By a function a mathematician means a rule describing how to pair up things found in one set, called the domain, with the things found in another set, called the range. The domain and the range can be collections of anything. They can be counting numbers, real numbers, letters, shoes, even collections of numbers or sets of shoes. They can be the same kinds of thing. They can be different kinds of thing.

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