The Set Tour, Stage 1: Intervals
I keep writing about functions. I’m not exactly sure how. I keep meaning to get to other things but find interesting stuff to say about domains and ranges and the like. These domains and ranges have to be sets. There are some sets that come up all the time in domains and ranges. I thought I’d share some of the common ones. The first family of sets is known as “intervals”.
[ 0, 1 ]
This means all the real numbers from 0 to 1. Written with straight brackets like that means to include the matching point there — that is, 0 is in the domain, and so is 1. We don’t always want to include the ending points in a domain; if we want to omit them, we write parentheses instead. So (0, 1) would mean all the real numbers bigger than zero and smaller than one, but neither zero nor one. We can also include one but not both endpoints: [0, 1) is fine. It offends copy editors, by having its open bracket and its closed parenthesis be unmatched, but its meaning is clear enough. It’s the real numbers from zero to one, with zero allowed but one ruled out. We can include or omit either or both endpoints, and we have to keep that straight. But for most of our work it doesn’t matter what we choose, as long as we stay consistent. It changes proofs a bit, but in routine ways.
Zero to one is a popular interval. Negative 1 to 1 is another popular interval. They’re nice round numbers. And the intervals between -π and π, or between 0 and 2π, are also popular. Those match nicely with trigonometric functions such as sine and tangent. You can make an interval that runs from any number to any other number — [ a, b ] if you don’t want to pin down just what numbers you mean. But [ 0, 1 ] and [ -1, 1 ] are popular choices among mathematicians. If you can prove something interesting about a function with a domain that’s either of these intervals, you can then normally prove it’s true on whatever a function with a domain that’s whatever interval you want. (I can’t think of an exception offhand, but mathematics is vast and my statement sweeping. There may be trouble.)
Suppose we start out with a function named f that has its domain the interval [ -16, 48 ]. I’m not saying anything about its range or its rule because they don’t matter. You can make them anything you like, if you need them to be anything. (This is exactly the way we might, in high school algebra, talk about the number named `x’ without ever caring whether we find out what number that actually is.) But from this start we can talk about a related function named g, which has as its domain [ -1, 1 ]. A rule for g is ; that is, g(0) is whatever number you get from f(16), for example. So if we can prove something’s true about g on this standard domain, we can almost certainly show the same thing is true about f on the other domain. This is considered a kind of mapping, or as a composition of functions. It’s a composition because you can see it as taking your original number called x, seeing what one function does with it — in this case, multiplying it by 32 and adding 16 — and then seeing what a second function — the one named f — does with that.
It’s easy to go the other way around. If we know what g is on the domain [ -1, 1 ] then we can define a related function f on the domain [ -16, 48 ]. We can say . And again, if we can show something’s true on this standard domain, we can almost certainly show the same thing’s true on the other domain.
I gave examples, mappings, that are simple. They’re linear. They don’t have to be. We just have to match everything in one interval to everything in another. For example, we can match the domain (1, ∞) — all the numbers bigger than 1 — to the domain (0, 1). Let’s again call f the function with domain (1, ∞). Then we can say g is the function with domain (0, 1) and defined by the rule . That’s a nonlinear mapping.
Linear mappings are easier to deal with than nonlinear mappings. Usually, mathematically, if something is divided into “linear” and “nonlinear” the linear version is easier. Sometimes a nonlinear mapping is the best one to use to match a function on some convenient domain to a function on some other one. The hard part is often a matter of showing that something true for one function on a common domain like (0, 1) will also be true for the other domain, (a, b).
However, showing that the truth holds can often be done without knowing much about your specific function. You maybe need to know what kind of function it is, that it’s continuous or bounded or something like that. But the actual specific rule? Not so important. You can prove that the truth holds ahead of time. Or find an analysis textbook or paper or something where someone else has proven that. So while a domain might be any interval, often in practice you don’t need to work with more than a couple nice familiar ones.