My mission in this book is to encourage programmers to think mathematically as they develop programs.

This idea is nothing new to programmers in science and engineering fields, because much of their work is inherently based on numerical mathematics and the mathematics of real numbers. However, there is more to mathematics than numbers.

Some of the mathematics that is most relevant to programming is known as “discrete mathematics”. This is the mathematics of discrete elements, such as symbols, character strings, truth values, and “objects” (to use a programming term) that are collections of properties. Discrete mathematics is concerned with such elements; collections of them, such as sets and sequences; and connections among elements, in structures such as mappings and relations. In many ways discrete mathematics is more relevant to programming than numerical mathematics is: not just to particular kinds of programming, but to all programming.

Many experienced programmers approach the design of a program by describing its input, output, and internal data objects in the vocabulary of discrete mathematics: sets, sequences, mappings, relations, and so on. This is a useful habit for us, as programmers, to cultivate. It can help to clarify our thinking about design problems; in fact, solutions often become obvious. And we inherit a well-understood vocabulary for specifying and documenting our programs and for discussing them with other programmers.