Discrete mathematics

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Graphs like this are among the objects studied by discrete mathematics, both for their interesting properties and because of their manipulation by computer algorithms.

For the mathematics journal, see Discrete Mathematics (journal).

Discrete mathematics is the study of mathematical structures that are fundamentally discrete in the sense that their objects can assume only distinct, separate values, rather than values on a continuum.^[1] This excludes topics in "continuous mathematics" such as calculus and analysis. Objects studied in discrete mathematics are largely elements of countable sets such as integers, graphs, statements in logic, and formal languages. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics, particularly areas relevant to business.

Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in computer algorithms and programming languages, and have applications in cryptography, automated theorem proving, and software development.

Computer implementation is an important aspect of discrete mathematics. Here a computer is being used to verify statements in logic, which is helpful in developing software for safety-critical systems.

The distinction between discrete mathematics and other mathematics is somewhat artificial as analytic methods are often used to study discrete problems and vice versa. Number theory in particular sits on the boundary between discrete and continuous mathematics.

[edit] Grand Challenges, Past and Present

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colours. Kenneth Appel and Wolfgang Haken finally proved this in 1976.^[2]

The history of discrete mathematics has involved a number of challenging problems which have focussed attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved till 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).^[2]

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Kurt Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.

The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory.

Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million US prize for the first correct proof, along with prizes for six other mathematical problems.^[3]

[edit] Topics in Discrete Mathematics

Discrete mathematics includes several different topics, listed below

The ASCII codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.

[edit] Logic

Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. The study of mathematical proofs is particularly important, and has applications to automated theorem proving and software development.

[edit] Set Theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Partially ordered sets and sets with other relations have applications in several areas.

[edit] Information Theory

Computational geometry applies computer algorithms to representations of geometrical objects.

Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods.

[edit] Number Theory

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to prime numbers and primality testing. In analytic number theory, techniques from continuous mathematics are also used.

[edit] Combinatorics

Combinatorics covers topics such as design theory, enumerative combinatorics, counting, combinatorial geometry, combinatorial topology, and graph theory. In analytic combinatorics and algebraic graph theory, techniques from continuous mathematics are also used.

[edit] Theoretical Computer Science

Complexity studies the time taken by computations, such as this sorting routine.

Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images.

[edit] Operations Research

Operations research provides techniques for solving practical problems in fields such as business. It includes linear programming, queuing theory, and a continuously growing list of other techniques.

[edit] Discretization

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.

[edit] Discrete analogues of continuous mathematics

There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, and discrete dynamical systems.

In applied mathematics, discrete modelling is the discrete analogue of continuous modelling . In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations.

[edit] See also

Discrete mathematics portal

• Topic outline of discrete mathematics
• Important publications in discrete mathematics

[edit] Footnotes

[edit] Further reading

Wikibooks has more on the topic of Discrete mathematics

• 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 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, 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). 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
• Jiří Matoušek & Jaroslav Nešetřil, Introduction aux mathematiques discretes
• C.L. Liu, Elements of Discrete Math