The discrete mathematics as a branch of mathematics concerned with mathematical operations on finite or at most countably infinite quantities, ie with discrete mathematical questions. In contrast to areas like analysis , which deals with continuous functions or curves on uncountable, infinite sets, continuity does not play a role in discrete mathematics.
The areas represented in discrete mathematics (such as number theory or graph theory ) are partly quite old, but discrete mathematics has long been overshadowed by "continuous" mathematics, which has been used in the natural sciences since the development of infinitesimal calculus (especially physics) has become the focus of interest. It was not until the 20th century that a multitude of new applications of discrete mathematics emerged due to the possibility of rapid digital data processing by computers (which naturally work with discrete states). At the same time, there was a rapid development of discrete mathematics, which was largely driven by issues related to computers ( algorithms , theoretical computer science , etc.).
An example of a field that lies at the intersection of analysis and discrete mathematics is numerical mathematics , which deals with the approximation of continuous by discrete quantities and with the estimation (and minimization) of the errors that occur.
The core areas of discrete mathematics include:
- Combinatorics ,
- Number theory ,
- Coding theory ,
- Graph theory ,
- Game theory ,
- Cryptography ,
- Information theory .
In addition, discrete mathematics has made additional contributions in the following areas:
- Further contributions of numerics to the improvement of discrete computing can be found in the areas of linear and discrete optimization (which goes beyond combinatorial tasks),
- discrete mathematics has many points of contact with algebra and mathematical logic ,
- in geometry there is the subfield of discrete geometry and
- In the computability theory , which is a branch of theoretical computer science , one needs finite automata , which are examined in discrete mathematics.
The Department of Discrete Mathematics German Mathematical Society awarded the two-year cycle to after the German mathematician Richard Rado named Richard Rado Prize for the best dissertation in discrete mathematics.
Studying discrete mathematics is possible at various universities by setting a corresponding focus within the mathematics course. Among others the Eidgenössische Technische Hochschule Zürich , the Technische Universität Berlin , the Technische Hochschule Mittelhessen , the Philipps-Universität Marburg , the Georg-August-Universität Göttingen , the Hochschule Bremerhaven , the RWTH Aachen , the Technische Universität München , the Fachhochschule Münster , the University of Applied Sciences Northwestern Switzerland , the Kempten University of Applied Sciences , the Hof University of Applied Sciences , the Friedrich Schiller University Jena and the Technical University of Cologne treat discrete mathematics as a compulsory module in the basic course in computer science . Among the universities of applied sciences, the Mittweida University of Applied Sciences offers this option as part of a specialized master's degree.
- Albrecht Beutelspacher , Marc-Alexander Zschiegner: Discrete Mathematics for Beginners . 4th edition. Vieweg Verlag, Wiesbaden 2011, ISBN 3-834-81248-X . 264 pp.
- Bernhard Ganter : Discrete Mathematics: Ordered Sets . Springer Spectrum, Berlin Heidelberg 2013, ISBN 978-3-642-37499-9 . 192 pp.
- Thomas Ihringer : Discrete Mathematics: An Introduction to Theory and Applications . 2nd Edition. Heldermann Verlag, Lemgo 2002, ISBN 3-88538-109-5 . 270 pp.
- Jiri Matoušek, Jaroslav Nešetřil; Hans Mielke (translator): Discrete mathematics: a journey of discovery . 2nd Edition. Springer textbook, Berlin 2007, ISBN 3-540-30150-X ; ISBN 978-3-540-30150-9 . 487 pp.
- Karl-Heinz Zimmermann : Discrete Mathematics . 1st edition. Books on Demand (BoD), Hamburg 2006, ISBN 3-8334-5529-2 . 412 pp.
- Angelika Steger : Discrete Structures 1: Combinatorics, Graph Theory, Algebra . 2nd Edition. Springer, Berlin 2007, ISBN 3-540-46660-6 . 270 pp.
- Angelika Steger, Thomas Schickinger: Discrete Structures 2: Probability Theory and Statistics . 1st edition. Springer, Berlin 2001, ISBN 3-540-67599-X . 249 pp.
- Official website of the Discrete Mathematics Section
- Video series "Discrete Optimizers" from DFG Science TV about mathematicians at the TU Berlin