Discrete Mathematics: An Introduction

The world of mathematics can be divided roughly into two realms: the continuous and the discrete. Continuous mathematics studies concepts that are infinite in scope, where one object can blend smoothly into the next. The real-number system lies at the core of continuous mathematics and - just like the watch - between any two real numbers, there is an infinity of real numbers. Continuous mathematics provides excellent models and tools for analyzing real-world phenomena that change smoothly over time, including the motion of planets around the sun or the flow of blood through the body. Discrete mathematics provides excellent models and tools for analyzing real-world phenomena that change abruptly and that lie clearly in one state or another. Discrete mathematics is the tool of choice in a host of applications, from computers to telephone call routing and from personnel assignments to genetics. Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term “discrete mathematics” is therefore used in contrast with “continuous mathematics,” which is the branch of mathematics dealing with objects that can vary smoothly. Whereas discrete objects can often be characterized by integers, continuous objects require real numbers. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities. The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics. The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.