What is Discrete Mathematics?

discrete / dis'krët.

Adjective: Individually separate and distinct.

Synonyms: separate - detached - distinct - abstract.

Defining discrete mathematics is hard because defining mathematics is hard. What is mathematics? The study of numbers? In part yes, but you also study functions and lines and triangles and parallelepipeds and vectors and . Or perhaps you want to say that mathematics is a collection of tools that allow you to solve problems. What sort of problems? Well, those that involve numbers, functions, lines, triangles, . Whatever your conception of what mathematics is, try applying the concept of discrete to it, as defined above. Some math fundamentally deals with stuff that is individually separate and distinct.

In an algebra or calculus class, you might have found a particular set of numbers (perhaps they constitute the range of a function). You would represent this set as an interval: \([0,\infty)\) is the range of \(f(x) = x^2\) since the set of outputs of the function are all real numbers 0 and greater. This set of numbers is NOT discrete. The numbers in the set are not separated by much at all. In fact, take any two numbers in the set and there are infinitely many more between them that are also in the set.

Discrete math could still ask about the range of a function, but the set would not be an interval. Consider the function that gives the number of children of each person reading this. What is the range? I'm guessing it is something like \(\{0, 1, 2, 3, 4\}\). Maybe 5 or 6 is in there too.Even larger natural numbers for old ladies who live in shoes. But certainly nobody reading this has 1.32419 children. This output set is discrete because the elements are separate. The inputs to the function also form a discrete set because each input is an individual person.

There are many discrete mathematical objects besides sets of numbers; we will introduce some of these in . Studying these discrete structures is the main focus of discrete mathematics and this book. However, the reason we want to study these structures is because they provide a way to model real-world problems.Many of the problems discussed in this book are admittedly contrived and clearly fictional, but hopefully you will see how these toy problems can be generalized to actually represent problems that people would care about in reality.

To get a feel for the subject, let's consider the types of problems you solve in discrete math. Here are a few simple examples:

Condensed: the full section is in Levin, Discrete Mathematics: An Open Introduction.