What is Discrete Mathematics Anyway?

by Joseph G. Rosenstein

The participants in the programs with discrete mathematics that I direct — students and teachers at all grade levels — are often asked the question "What is discrete mathematics, anyway?" Unfortunately, there is no easy answer to this question. Nevertheless, I have to come up with an answer. And so I actually give three answers.

The first answer is that discrete mathematics includes a lot of interesting problems. Discrete mathematics deals with a variety of topics that arise in everyday experience and in important applications, and many of the problems in discrete mathematics can be understood by students at all grade levels. Some of the problems are so simply formulated that many people may not realize that there are actually mathematical, nor that the solutions to these problems emerge from mathematical theories. Here are a few examples of such problems:

how many different pizzas can you have if each pizza must have at most three of the eight available toppings?
what’s a good strategy for dividing up a pie among three people so that each is satisfied with the portion he or she receives?
what’s the quickest way of alphabetizing a list of 1000 names — on index cards, or in a database?
which way of connecting a number of sites into a network involves the least cable?
what’s the best way for a robot to pick up items stored in an automated warehouse, or for a courier to collect deposits at all ATM machines?

My second answer is designed for the person who really needs a one-paragraph description of discrete mathematics. Following is the brief description that I developed for New Jersey’s Core Curriculum Content Standards.

Discrete mathematics is the branch of mathematics that deals with arrangements of distinct objects. It includes a wide variety of topics and techniques that arise in everyday life, such as how to find the best route from one city to another, where the objects are cities arranged on a map. It also includes how to count the number of different combinations of toppings for pizzas, how best to schedule a list of tasks to be done, and how computers store and retrieve arrangements of information on a screen. Discrete mathematics is the mathematics used by decision-makers in our society, from workers in government to those in health care, transportation and telecommunications. Its various applications help students see the relevance of mathematics in the real world.

My third answer is designed for the person who really wants more information, a more comprehensive idea of the various topics encompassed in the term "discrete mathematics". In our work on New Jersey’s Core Curriculum Content Standards, we organized the various topics within discrete mathematics into five major themes. These five themes are enumerated below. (It should be noted that not everyone would agree that all of these topics belong in discrete mathematics, and that some people would feel that other topics, like matrices, also belong in discrete mathematics.) The five themes are:

Systematic listing and counting of the number of ways there are to complete a particular task.
Using discrete mathematical models such as graphs and trees to represent and solve a variety of problems based on real-world situations.
Applying iterative patterns and processes in a variety of settings in nature and art, as well as in mathematics.
Arranging, organizing, analyzing, transforming, and communicating information, and understanding how these methods are used in a variety of settings.
Exploring different methods of solving real-world problems, and determining what is the best solution using a variety of algorithms — where best may be defined, for example as most cost-effective or most equitable.

For further information about these themes, please check out the chapter on discrete mathematics in the New Jersey Mathematics Curriculum Framework (starting with the K-12 Overview) at http://dimacs.rutgers.edu/nj_math_coalition/framework/ch14/ch14_toc.html

I hope that these three answers to the question "What is discrete mathematics?" give you enough of an idea about the area so that you will want to explore it further.

Joseph G. Rosenstein is Associate Director for Education of the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) and Director of the Leadership Program in Discrete Mathematics.