Discrete mathematics

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as integers, finite graphs, and formal languages.

Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, while discrete mathematics courses emphasize concepts for computer science majors.

For contrast, see continuum, topology, and mathematical analysis.

Discrete mathematics includes the following topics:


 * Logic - a study of reasoning;
 * Set theory - a study of collections of elements;
 * Number theory;
 * Combinatorics - a study of counting;
 * Graph theory;
 * Digital geometry and digital topology;
 * Algorithmics - a study of methods of calculation;
 * Information theory;
 * Computability and complexity theories - dealing with theoretical and practical limitations of algorithms;
 * Elementary probability theory and Markov chains;
 * Linear algebra - a study of related linear equations.
 * Functions
 * Partially Ordered Sets
 * Probability
 * Proofs
 * Counting and Relations
 * Collections

References and further reading

 * Donald E. Knuth, The Art of Computer Programming
 * Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics CRC Press. ISBN 0-8493-0149-1.
 * Kenneth H. Rosen, Discrete Mathematics and Its Applications 5th ed. McGraw Hill. ISBN 0-07-293033-0.  Companion Web site: http://www.mhhe.com/math/advmath/rosen/
 * Richard Johnsonbaugh, Discrete Mathematics 6th ed. Macmillan. ISBN 0-13-045803-1.  Companion Web site: http://wps.prenhall.com/esm_johnsonbau_discrtmath_6/
 * Norman L. Biggs, Discrete Mathematics 2nd ed. Oxford University Press. ISBN 0-19-850717-8. Companion Web site: http://www.oup.co.uk/isbn/0-19-850717-8 includes questions together with solutions..
 * Neville Dean, Essence of Discrete Mathematics Prentice Hall. ISBN 0-13-345943-8. Not as in depth as above texts, but a gentle intro.
 * Also on (digital) topology, graph theory, combinatorics, axiomatic systems.
 * Mathematics Archives, Discrete Mathematics links to syllabi, tutorials, programs, etc. http://archives.math.utk.edu/topics/discreteMath.html
 * Ronald Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics

Applications

 * Atonal music
 * Computer image analysis
 * Cryptanalysis
 * Cryptography
 * Cryptology
 * Finite automata


 * Formal language
 * Game theory
 * Graph theory
 * Combinatorial geometry
 * Combinatorial topology


 * Linear programming
 * Musical analysis
 * Operations research
 * Queuing theory
 * Theory of computation

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