For a one- or two-term introductory course in discrete mathematics. Chapters 2 (The Language of Mathematics) and 3 (Relations) in the sixth edition have been combined into Chapter 3 (Functions, Sequences, and Relations) in this revision. 13 Computational Geometry. 13.1 The Closest-Pair Problem. 13.2 An Algorithm to Compute the Convex.

Like this are among the objects studied by discrete mathematics, for their interesting, their usefulness as models of real-world problems, and their importance in developing computer. Discrete mathematics is the study of that are fundamentally rather than. In contrast to that have the property of varying 'smoothly', the objects studied in discrete mathematics – such as,, and in – do not vary smoothly in this way, but have distinct, separated values.

Discrete mathematics therefore excludes topics in 'continuous mathematics' such as. Discrete objects can often be by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with (finite sets or sets with the same as the natural numbers). However, there is no exact definition of the term 'discrete mathematics.' Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of, such as,,,,. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

In university curricula, 'Discrete Mathematics' appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by and into a course that is basically intended to develop in freshmen; therefore it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike in this respect. The is awarded for outstanding papers in discrete mathematics. Much research in was motivated by attempts to prove that all maps, like this one, can be using so that no areas of the same color share an edge. And proved this in 1976.

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance). In, the on 's list of open presented in 1900 was to prove that the of are., proved in 1931, showed that this was not possible – at least not within arithmetic itself.

Was to determine whether a given polynomial with integer coefficients has an integer solution. In 1970, proved that this. The need to German codes in led to advances in and, with the being developed at England's with the guidance of and his seminal work, On Computable Numbers. At the same time, military requirements motivated advances in.

The meant that cryptography remained important, with fundamental advances such as being developed in the following decades. Operations research remained important as a tool in business and project management, with the being developed in the 1950s. The industry has also motivated advances in discrete mathematics, particularly in graph theory. Of statements in logic has been necessary for of, and advances in have been driven by this need. Has been an important part of the incorporated into modern and tools.