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Discrete mathematics

For the mathematics journal, see Discrete Mathematics (journal).
"Finite math" redirects here. For the syllabus, see Finite mathematics.
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Graphs such as these 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.

Discrete 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). Objects studied in discrete mathematics include integers, graphs, and statements in logic.[1][2][3] By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[4] (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics".[5]

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 digital computers 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 computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems.

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 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.[6][7] Some high-school-level discrete mathematics textbooks have appeared as well.[8] At this level, discrete mathematics is sometimes seen as a preparatory course, like precalculus in this respect.[9]

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

  1. ^ Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008.
  2. ^ Franklin, James (2017). "Discrete and continuous: a fundamental dichotomy in mathematics". Journal of Humanistic Mathematics. 7 (2): 355–378. doi:10.5642/jhummath.201702.18. S2CID 6945363. Retrieved 30 June 2021.
  3. ^ "Discrete Structures: What is Discrete Math?". cse.buffalo.edu. Retrieved 16 November 2018.
  4. ^ Biggs, Norman L. (2002), Discrete mathematics, Oxford Science Publications (2nd ed.), The Clarendon Press Oxford University Press, p. 89, ISBN 9780198507178, MR 1078626, Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets.
  5. ^ Hopkins, Brian, ed. (2009). Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles. Mathematical Association of America. ISBN 978-0-88385-184-5.
  6. ^ Levasseur, Ken; Doerr, Al. Applied Discrete Structures. p. 8.
  7. ^ Geoffrey Howson, Albert, ed. (1988). Mathematics as a Service Subject. Cambridge University Press. pp. 77–78. ISBN 978-0-521-35395-3.
  8. ^ Rosenstein, Joseph G. Discrete Mathematics in the Schools. American Mathematical Society. p. 323. ISBN 978-0-8218-8578-9.
  9. ^ "UCSMP". uchicago.edu.

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