1.0: Introduction
Before we set out on our “cool, brisk walk," let’s get oriented. What is discrete mathematics, anyway? Why is it called that? What does it encompass? And what is it good for?
Let’s take the two words of the subject, in reverse order. First, math . When most people hear “math," they think “numbers." After all, isn’t math the study of quantity? And isn’t that the class where we first learned to count, add, and multiply?
Mathematics certainly has its root in the study of numbers — specifically, the “natural numbers" (the integers from 1 on up) that fascinated the ancient Greeks. Yet math is broader than this, almost to the point where numbers can be considered a special case of something deeper. In this book, when we talk about trees, sets, or formal logic, there might not be a number in sight.
Math is about abstract, conceptual objects that have properties, and the implications of those properties. An “object" can be any kind of “thought material" that we can define and reason about precisely. Much of math deals with questions like, “suppose we defined a certain kind of thing that had certain attributes. What would be the implications of this, if we reasoned it all the way out?" The “thing" may or may not be numerical, whatever it turns out to be. Like a number, however, it will be crisply defined, have certain known aspects to it, and be capable of combining with other things in some way.
Fundamental to math is that it deals with the abstract . Abstract, which is the opposite of concrete, essentially means something that can’t be perceived with the senses. A computer chip is concrete: you can touch it, you can see it. A number is not; nor is a function, a binary tree, or a logical implication. The only way to perceive these things is with the power of the mind. We will write expressions and draw pictures of many of our mathematical structures in order to help visualize them, and nearly everything we study will have practical applications whereby the abstractness gets grounded in concreteness for some useful purpose. But the underlying mathematical entity remains abstract and ethereal — only accessible to the mind’s eye. We may use a pencil to form the figure “5" on a piece of paper, but that is only a concrete manifestation of the underlying concept of “five-ness." Don’t mistake the picture or the symbol for the thing itself, which always transcends any mere physical representation.
The other word in the name of our subject is “discrete" (not to be confused with “discreet," which means something else entirely). The best way to appreciate what discrete means is to contrast it with its opposite, continuous. Consider the following list:
| Discrete | Continuous |
|---|---|
| whole numbers ( \(\mathbb{Z}\) ) | real numbers ( \(\mathbb{R}\) ) |
int
|
double
|
| digital | analog |
| quantum | continuum |
| counting | measuring |
| number theory | analysis |
| \(\Sigma\) | \(\int\) |
| – | \(\frac{d}{dx}\) |
What do the left-hand entries have in common? They describe things that are measured in crisp, distinct intervals, rather than varying smoothly over a range. Discrete things jump suddenly from position to position, with rigid precision. If you’re 5 feet tall, you might some day grow to 5.3 feet; but though there might be 5 people in your family, there will never be 5.3 (although there could be 6 someday).
The last couple of entries on this list are worth a brief comment. They are math symbols, some of which you may be familiar with. On the right side — in the continuous realm — are \(\int\) and \(\frac{d}{dx}\) , which you’ll remember if you’ve taken calculus. They stand for the two fundamental operations of integration and differentiation. Integration, which can be thought of as finding “the area under a curve," is basically a way of adding up a whole infinite bunch of numbers over some range. When you “integrate the function \(x^2\) from 3 to 5," you’re really adding up all the tiny, tiny little vertical slivers that comprise the area from \(x=3\) on the left to \(x=5\) on the right. Its corresponding entry in the left-column of the table is \(\Sigma\) , which is just a short-hand for “sum up a bunch of things." Integration and summation are equivalent operations, it’s just that when you integrate, you’re adding up all the (infinitely many) slivers across the real-line continuum. When you sum, you’re adding up a fixed sequence of entries, one at a time, like in a loop. \(\Sigma\) is just the discrete “version" of \(\int\) .
The same sort of relationship holds between ordinary subtraction (“–") and differentiation ( \(\frac{d}{dx}\) ). If you’ve plotted a bunch of discrete points on \(x\) - \(y\) axes, and you want to find the slope between two of them, you just subtract their \(y\) values and divide by the ( \(x\) ) distance between them. If you have a smooth continuous function, on the other hand, you use differentiation to find the slope at a point: this is essentially subtracting the tiny tiny difference between two supremely close points and then dividing by the distance between them. Thus subtraction is just the discrete “version" of \(\frac{d}{dx}\) .
Don’t worry, you don’t need to have fully understood any of the integration or differentiation stuff I just talked about, or even to have taken calculus yet. I’m just trying to give you some feel for what “discrete" means, and how the dichotomy between discrete and continuous really runs through all of math and computer science. In this book, we will mostly be focusing on discrete values and structures, which turn out to be of more use in computer science. That’s partially because as you probably know, computers themselves are discrete, and can only store and compute discrete values. There can be many of them — megabytes, gigabytes, terabytes — but each value stored is fundamentally comprised of bits, each of which has a value of either 0 or 1. This is unlike the human brain, by the way, whose neuronal synapses communicate based on the continuous quantities of chemicals present in their axons. So I guess “computer" and “brain" are another pair of entries we could add to our discrete vs. continuous list.
There’s another reason, though, why discrete math is of more use to computer scientists than continuous math is, beyond just the bits-and-bytes thing. Simply put, computers operate algorithmically. They carry out programs in step-by-step, iterative fashion. First do this, then do that, then move on to something else. This mechanical execution, like the ticking of a clock, permeates everything the computer can do, and everything we can tell it to do. At a given moment in time, the computer has completed step 7, but not step 8; it has accumulated 38 values, but not yet 39; its database has exactly 15 entries in it, no more and no less; it knows that after accepting this friend request, there will be exactly 553 people in your set of friends. The whole paradigm behind reasoning about computers and their programs is discrete, and that’s why we computer scientists find different problems worth thinking about than most of the world did a hundred years ago.
But it’s still math. It’s just discrete math. There’s a lot to come, so limber up and let me know when you’re ready to hit the road.