Preliminaries

This chapter is intended as a resource. You may be familiar with its contents, or there may be topics you never learned or that you need to review. You should not feel that you need to master Chapter 0 before beginning Chapter 1; just refer back to it as needed. (A possible exception is Section 0.7 on complex numbers.)

Section 0.2 analyzes the rather tricky business of negating mathematical statements. (To a mathematician, the statement “All eleven-legged alligators are orange with blue spots” is an obviously true statement, not an obviously meaningless one.) We first use this material in Section 1.5.

Set theory notation is discussed in Section 0.3. The “eight words” of set theory are used beginning in Section 1.1. The discussion of Russell’s paradox is not necessary; we include it because it is fun and not hard.

Section 0.4 defines the word “function” and discusses the relationship between a function being “onto” or “one to one” and the existence and uniqueness of solutions. This material is first needed in Section 1.3.

The discussion of countable and uncountable sets in Section 0.6 is fun and not hard. These notions are fundamental to Lebesgue integration.

In our experience, most students studying vector calculus for the first time are comfortable with complex numbers, but a sizable minority have either never heard of complex numbers or have forgotten everything they once knew. If you are among them, we suggest reading at least the first few pages of Section 0.7 and doing some of the exercises.

Reading mathematics is different from other reading. We think the following guidelines can make it easier. There are two parts to understanding a theorem: understanding the statement, and understanding the proof. The first is more important than the second.

What if you don’t understand the statement? If there’s a symbol in the formula you don’t understand, perhaps a $𝛿$, look to see whether the next line continues, “where $𝛿$ is such and such.” In other words, read the whole sentence before you decide you can’t understand it.

If you’re still having trouble, skip ahead to examples. This may contradict what you have been told – that mathematics is sequential, and that you must understand each sentence before going on to the next. In reality, although mathematical writing is necessarily sequential, mathematical understanding is not: you (and the experts) never understand perfectly up to some point and not at all beyond. The “beyond”, where understanding is only partial, is an essential part of the motivation and the conceptual background of the “here and now”. You may often find that when you return to something you left half-understood, it will have become clear in the light of the further things you have studied, even though the further things are themselves obscure.

Many students are uncomfortable in this state of partial understanding, like a beginning rock climber who wants to be in stable equilibrium at all times. To learn effectively one must be willing to leave the cocoon of equilibrium. If you don’t understand something perfectly, go on ahead and then circle back.

In particular, an example will often be easier to follow than a general statement; you can then go back and reconstitute the meaning of the statement in light of the example. Even if you still have trouble with the general statement, you will be ahead of the game if you understand the examples. We feel so strongly about this that we have sometimes flouted mathematical tradition and given examples before the proper definition.

Read with pencil and paper in hand, making up little examples for yourself as you go on.

Some of the difficulty in reading mathematics is notational. A pianist who has to stop and think whether a given note on the staff is A or F will not be able to sight-read a Bach prelude or Schubert sonata. The temptation, when faced with a long, involved equation, may be to give up. You need to take the time to identify the “notes”.

Learn the names of Greek letters – not just the obvious ones like alpha, beta, and pi, but the more obscure psi, xi, tau, omega. The authors know a mathematician who calls all Greek letters “xi” ($𝜉$), except for omega ($𝜔$), which he calls “w”. This leads to confusion. Learn not just to recognize these letters, but how to pronounce them. Even if you are not reading mathematics out loud, it is hard to think about formulas if $𝜉$, $𝜓$, $𝜏$, $𝜔$, $𝜙$ are all “squiggles” to you.

Sum notation can be confusing at first; we are accustomed to reading in one dimension, from left to right, but something like

requires what we might call two-dimensional (or even three-dimensional) thinking. It may help at first to translate a sum into a linear expression:

Two $\sum$ placed side by side do not denote the product of two sums; one sum is used to talk about one index, the other about another. The same thing could be written with one $\sum$, with information about both indices underneath. For example,

this double sum is illustrated in Figure 0.1.1.

Rules for product notation $\prod$ are analogous to those for sum notation: