Discrete mathematics, also called finite mathematics or decision mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Objects studied in finite mathematics are largely countable sets such as integers, finite graphs, and formal languages. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics and Computer science, a graph is the basic object of study in Graph theory. A formal language is a set of words, ie finite strings of letters, or symbols.
Discrete mathematics has become popular in recent decades because of its applications to computer science. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. 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. In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Analysis has its beginnings in the rigorous formulation of Calculus.
Discrete mathematics includes the following topics:
- Logic - a study of reasoning
- Set theory - a study of collections of elements
- Number theory
- Combinatorics, including
- Algorithmics - a study of methods of calculation
- Information theory
- Digital geometry
- Computability and complexity theories - dealing with theoretical and practical limitations of algorithms
- Partially ordered sets
- Proofs
- Counting and relations
See also
Applications
References and further reading
- Donald E. Knuth, The Art of Computer Programming
- Kenneth H. Logic is the study of the principles of valid demonstration and Inference. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Combinatorial design theory is the part of combinatorial Mathematics that deals with the existence and construction of systems of finite sets whose Combinatorial enumeration is a subfield of Enumeration that deals with the counting of objects whose symmetries do not exist or if they exist are combinatorial in In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. Digital geometry deals with discrete sets (usually discrete point sets considered to be digitized models or Images of objects of the In general usage complexity often tends to be used to characterize something with many parts in intricate arrangement In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting Algebra Theory of equations Hisab Image analysis is the extraction of meaningful information from Images mainly from Digital images by means of Digital image processing techniques Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" A formal language is a set of words, ie finite strings of letters, or symbols. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or Combinatorial, either In Mathematics, combinatorial topology was an older name for Algebraic topology, dating from the time when topological invariants of spaces (for example In Mathematics, linear programming (LP is a technique for optimization of a Linear Objective function, subject to Linear equality Operations Research (OR in North America South Africa and Australia and Operational Research in Europe is an interdisciplinary branch of applied Mathematics and Queueing theory is the mathematical study of waiting lines (or queues ' The theory of computation is the branch of Computer science that deals with whether and how efficiently problems can be solved on a Model of computation, using an Donald Ervin Knuth (kəˈnuːθ (born 10 January 1938) is a renowned computer scientist and Professor Emeritus of the Art of Computer The Art of Computer Programming is a comprehensive Monograph written by Donald Knuth that covers many kinds of Programming Algorithms 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/
- Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction 5th ed. Addison Wesley. ISBN 0-20-172634-3
- 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.
- Klette, R. , and A. Rosenfeld (2004). Professor Dr Azriel Rosenfeld ( February 19, 1931 - February 22, 2004) was an American Research Professor a Distinguished University Professor Digital Geometry. Morgan Kaufmann. ISBN 1-55860-861-3. 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
- Discrete Mathematics AJ Sadler
Dictionary
discrete mathematics
-noun
- (mathematics) A blanket term which includes most discrete or computer science related branches of mathematics, such as graph theory and combinatorics.
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