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Discrete mathematics
 
 
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Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense that its objects can assume only distinct, separate values, rather than a values on a continuum
Continuum (mathematics)

In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum....
. Objects studied in discrete mathematics are largely countable sets such as integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s, finite graphs
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
, and formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
s.

Discrete mathematics has become popular in recent decades because of its applications to computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
.






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Encyclopedia


Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense that its objects can assume only distinct, separate values, rather than a values on a continuum
Continuum (mathematics)

In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum....
. Objects studied in discrete mathematics are largely countable sets such as integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s, finite graphs
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
, and formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
s.

Discrete mathematics has become popular in recent decades because of its applications to computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in computer algorithm
Algorithm

In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing....
s and programming language
Programming language

A programming language is a machine-readable artificial language designed to express computations that can be performed by a machine, particularly a computer....
s.

In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, whereas discrete mathematics courses emphasize concepts for computer science majors, and combinatorics
Combinatorics

Combinatorics is a branch of pure mathematics concerning the study of Countable set objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics....
 and other specialized courses emphasize the mathematical theory.

Discrete mathematics is contrasted with "continuous mathematics", in which the mathematical continuum
Continuum (mathematics)

In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum....
 and more sophisticated topologies
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 play a role. Mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
 is a major branch of continuous mathematics.

Discrete mathematics includes the following topics:

See also

conjunction and disjunction

Applications

  • Computer image analysis
  • Cryptanalysis
    Cryptanalysis

    Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so....
  • Cryptography
    Cryptography

    Cryptography is the practice and study of hiding information. In modern times cryptography is considered a branch of both mathematics and computer science and is affiliated closely with information theory, computer security and engineering....
  • Cryptology
  • Finite automata
  • Formal language
    Formal language

    A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
  • Graph theory
    Graph theory

    In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
  • Combinatorial geometry
  • Combinatorial topology
    Combinatorial topology

    In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when Topological invariant#T of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes....
  • Linear programming
    Linear programming

    In mathematics, linear programming is a technique for optimization of a linear objective function, subject to linear equality and linear inequality Constraint ....
  • Operations research
    Operations research

    Operations Research in the USA, South Africa and Australia, and Operational Research in Europe and Canada, is an interdisciplinary branch of applied mathematics and formal science that uses methods such as mathematical modeling, statistics, and algorithms to arrive at optimal or near optimal solutions to complex problems....
  • Queuing theory
  • Theory of computation
    Theory of computation

    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 algorithm....


Footnotes


Further reading

  • Donald E. Knuth
    Donald Knuth

    Donald Ervin Knuth is a renowned computer science and Emeritus of the Art of Computer Programming at Stanford University.Author of the seminal multi-volume work The Art of Computer Programming , Knuth has been called the "father" of the run-time analysis, contributing to the development of, and systematizing formal mathematical techn...
    , The Art of Computer Programming
    The Art of Computer Programming

    The Art of Computer Programming is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis....
  • Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics CRC Press. ISBN 0-8493-0149-1.
  • Kenneth H. Rosen, Discrete Mathematics and Its Applications 6th ed. McGraw Hill. ISBN 0-07-288008-2. Companion Web site: http://highered.mcgraw-hill.com/sites/0072880082/information_center_view0/
  • 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
    Ralph Grimaldi

    Ralph Peter Grimaldi is an United States mathematician specializing in discrete mathematics who is a full professor at Rose-Hulman Institute of Technology....
    , 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.
  • 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
    Ronald Graham

    Ronald Lewis Graham is a mathematician credited by the American Mathematical Society with being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years"....
    , Donald E. Knuth
    Donald Knuth

    Donald Ervin Knuth is a renowned computer science and Emeritus of the Art of Computer Programming at Stanford University.Author of the seminal multi-volume work The Art of Computer Programming , Knuth has been called the "father" of the run-time analysis, contributing to the development of, and systematizing formal mathematical techn...
    , Oren Patashnik
    Oren Patashnik

    File:Patashnik.jpegOren Patashnik is a computer scientist. He is notable for co-creating BibTeX, and co-writing Concrete Mathematics. He is a researcher at the Center for Communications Research, La Jolla....
    , Concrete Mathematics
    Concrete Mathematics

    Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, is a perennial textbook in university computer science departments....
  • Jirí Matoušek
    Jirí Matoušek (mathematician)

    Jir? Matou?ek is a Czech mathematician working in computational geometry. He is a professor at Charles University in Prague and is the author of several textbooks and research monographs....
     & Jaroslav Nešetril
    Jaroslav Nešetril

    Jaroslav Ne?etril is a Czech Republic mathematician, Charles University in Prague . His research areas include combinatorics , graph theory , algebra , posets , computer science ....
    , Introduction aux mathematiques discretes
  • C.L. Liu, Elements of Discrete Math