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Discrete mathematics is concerned with the study of finite or countably infinite objects. It is a m´elange of topics from logic, set theory, algebra, combinatorics, number theory, and other areas of mathematics rather than a mathematical discipline itself. Computer scientists pragmatically characterize it as the sub- ject that provides the mathematical foundation for computing. In particular, it supplies the tools to analyze algorithms and data structures. Since discrete mathematics consists of many different subjects, it uses a great variety of methods to solve problems. The aim of this book is to bring some coherency into these seemingly incongruent subjects. We begin by laying a solid foundation for the study of discrete structures by discussing logic, sets, and mathematical proofs. We systematically develop methods to find the closed form for finite sums. We deduce combinatorial methods from the foundations that we have laid. The methods of discrete mathematics are often seemingly simple. For ex- ample, anyone can immediately grasp the pigeonhole principle, but may be dazzled by the sophisticated applications of the principle. Generally, it re- quires a lot of practice until one is able to master the methods. We have included 690 exercises of various levels of difficulty in this book. We encour- age the reader to study the examples in the text and solve many of these exercises.