Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. At the same time, military requirements motivated advances in operations research. The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. In 1970, Yuri Matiyasevich proved that this could not be done. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance). The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. Kenneth Appel and Wolfgang Haken proved this in 1976. Much research in graph theory was motivated by attempts to prove that all maps, like this one, can be colored using only four colors so that no areas of the same color share an edge. The Fulkerson Prize is awarded for outstanding papers in discrete mathematics. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect. Some high-school-level discrete mathematics textbooks have appeared as well.
The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students therefore, it is nowadays a prerequisite for mathematics majors in some universities 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. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.Īlthough the main objects of study in discrete mathematics are discrete objects, analytic methods from "continuous" mathematics are often employed as well.
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Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in "discrete" steps and store data in "discrete" bits. 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. The set of objects studied in discrete mathematics can be finite or infinite. However, there is no exact definition of the term "discrete mathematics". Discrete objects can often be enumerated by integers more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Objects studied in discrete mathematics include integers, graphs, and statements in logic.
Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.ĭiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).