discrete mathematics for the mathematics journal see discrete mathematics journal graphs like this are among the objects studied by discrete mathematics for their interesting mathematical properties their usefulness as models of realworld problems and their importance in developing computer algorithms discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous in contrast to real numbers that have the property of varying smoothly the objects studied in discrete mathematics such as integers graphs and statements in logic indeed discrete mathematics is described less by what is included than by what is excluded continuously varying quantities and related notions the set of objects studied in discrete mathematics can be finite or infinite the term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets particularly those areas relevant to business research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science such as computer algorithms programming languages cryptography automated theorem proving and software development conversely computer implementations are significant in applying ideas from discrete mathematics to realworld problems such as in operations research although the main objects of study in discrete mathematics are discrete objects analytic methods from continuous mathematics are often employed as well in university curricula discrete mathematics appeared in the 1980s initially as a computer science support course its contents were somewhat haphazard at the time the curriculum has thereafter developed in conjunction with efforts by acm and maa into a course that is basically intended to develop mathematical maturity in freshmen therefore it is nowadays a prerequisite for mathematics majors in some universities as well the fulkerson prize is awarded for outstanding papers in discrete mathematics grand challenges past and present much research in graph theory was motivated by attempts to prove that all maps like this one can be colored using only four colors so that no areas of the same color share an edge kenneth appel and wolfgang haken proved this in 1976 the history of discrete mathematics has involved a number of challenging problems which have focused 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 until 1976 by kenneth appel and wolfgang haken using substantial computer assistance in logic the second problem on david hilberts list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent gdels second incompleteness theorem proved in 1931 showed that this was not possible at least not within arithmetic itself hilberts 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 englands bletchley park with the guidance of alan turing and his seminal work on computable numbers at the same time military requirements motivated advances in operations research the cold war meant that cryptography remained important with fundamental advances such as publickey 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 formal verification of statements in logic has been necessary for software development of safetycritical systems and advances in automated theorem proving have been driven by this need computational geometry has been an important part of the computer graphics incorporated into modern video games and computeraided design tools several fields of discrete mathematics particularly theoretical computer science graph theory and combinatorics are important in addressing the challenging bioinformatics problems associated with understanding the tree of life 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 usd prize for the first correct proof along with prizes for six other mathematical problems topics in discrete mathematics theoretical computer science theoretical computer science complexity studies the time taken by algorithms such as this sorting routine theoretical computer science includes areas of discrete mathematics relevant to computing it draws heavily on graph theory and mathematical 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 space and other resources taken by computations automata theory and formal language theory are closely related to computability petri nets and process algebras are used to model computer systems and methods from discrete mathematics are used in analyzing vlsi electronic circuits computational geometry applies algorithms to geometrical problems while computer image analysis applies them to representations of images theoretical computer science also includes the study of various continuous computational topics information theory information theory 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 informationprocessing algorithms 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 information theory also includes continuous topics such as analog signals analog coding analog encryption logic mathematical logic logic is the study of the principles of valid reasoning and inference as well as of consistency soundness and completeness for example in most systems of logic but not in intuitionistic logic peirces law pqpp is a theorem for classical logic it can be easily verified with a truth table the study of mathematical proof is particularly important in logic and has applications to automated theorem proving and formal verification of software logical formulas are discrete structures as are proofs which form finite trees eg infinitary logic set theory set theory set theory is the branch of mathematics that studies sets which are collections of objects such as blue white red or the infinite set of all prime numbers partially ordered sets and sets with other relations have applications in several areas in discrete mathematics countable sets including finite sets are the main focus the beginning of set theory as a branch of mathematics is usually marked by georg cantors work distinguishing between different kinds of infinite set motivated by the study of trigonometric series and further development of the theory of infinite sets is outside the scope of discrete mathematics indeed contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics combinatorics combinatorics combinatorics studies the way in which discrete structures can be combined or arranged enumerative combinatorics concentrates on counting the number of certain combinatorial objects eg the twelvefold way provides a unified framework for counting permutations combinations and partitions analytic combinatorics concerns the enumeration ie determining the number of combinatorial structures using tools from complex analysis and probability theory in contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results analytic combinatorics aims at obtaining asymptotic formulae design theory is a study of combinatorial designs which are collections of subsets with certain intersection properties partition theory studies various enumeration and asymptotic problems related to integer partitions and is closely related to qseries special functions and orthogonal polynomials originally a part of number theory and analysis partition theory is now considered a part of combinatorics or an independent field order theory is the study of partially ordered sets both finite and infinite graph theory graph theory graph theory has close links to group theory this truncated tetrahedron graph is related to the alternating group a4 graph theory the study of graphs and networks is often considered part of combinatorics but has grown large enough and distinct enough with its own kind of problems to be regarded as a subject in its own right graphs are one of the prime objects of study in discrete mathematics they are among the most ubiquitous models of both natural and humanmade structures they can model many types of relations and process dynamics in physical biological and social systems in computer science they can represent networks of communication data organization computational devices the flow of computation etc in mathematics they are useful in geometry and certain parts of topology eg knot theory algebraic graph theory has close links with group theory there are also continuous graphs however for the most part research in graph theory falls within the domain of discrete mathematics probability discrete probability theory discrete probability theory deals with events that occur in countable sample spaces for example count observations such as the numbers of birds in flocks comprise only natural number values 0 1 2 on the other hand continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal discrete probability distributions can be used to approximate continuous ones and vice versa for highly constrained situations such as throwing dice or experiments with decks of cards calculating the probability of events is basically enumerative combinatorics number theory the ulam spiral of numbers with black pixels showing prime numbers this diagram hints at patterns in the distribution of prime numbers number theory number theory is concerned with the properties of numbers in general particularly integers it has applications to cryptography and cryptanalysis particularly with regard to modular arithmetic diophantine equations linear and quadratic congruences prime numbers and primality testing other discrete aspects of number theory include geometry of numbers in analytic number theory techniques from continuous mathematics are also used topics that go beyond discrete objects include transcendental numbers diophantine approximation padic analysis and function fields algebra abstract algebra algebraic structures occur as both discrete examples and continuous examples discrete algebras include boolean algebra used in logic gates and programming relational algebra used in databases discrete and finite versions of groups rings and fields are important in algebraic coding theory discrete semigroups and monoids appear in the theory of formal languages calculus of finite differences discrete calculus or discrete analysis finite difference a function defined on an interval of the integers is usually called a sequence a sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system such a discrete function could be defined explicitly by a list if its domain is finite or by a formula for its general term or it could be given implicitly by a recurrence relation or difference equation difference equations are similar to a differential equations but replace differentiation by taking the difference between adjacent terms they can be used to approximate differential equations or more often studied in their own right many questions and methods concerning differential equations have counterparts for difference equations for instance where there are integral transforms in harmonic analysis for studying continuous functions or analogue signals there are discrete transforms for discrete functions or digital signals as well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces geometry computational geometry applies computer algorithms to representations of geometrical objects discrete geometry and computational geometry discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects a longstanding topic in discrete geometry is tiling of the plane computational geometry applies algorithms to geometrical problems topology although topology is the field of mathematics that formalizes and generalizes the intuitive notion of continuous deformation of objects it gives rise to many discrete topics this can be attributed in part to the focus on topological invariants which themselves usually take discrete values see combinatorial topology topological graph theory topological combinatorics computational topology discrete topological space finite topological space topology chemistry operations research operations research pert charts like this provide a project management technique based on graph theory operations research provides techniques for solving practical problems in engineering business and other fields problems such as allocating resources to maximize profit or scheduling project activities to minimize risk operations research techniques include linear programming and other areas of optimization queuing theory scheduling theory network theory operations research also includes continuous topics such as continuoustime markov process continuoustime martingales process optimization and continuous and hybrid control theory game theory decision theory utility theory social choice theory cooperate defect cooperate 1 1 10 0 defect 0 10 5 5 payoff matrix for the prisoners dilemma a common example in game theory one player chooses a row the other a column the resulting pair gives their payoffs decision theory is concerned with identifying the values uncertainties and other issues relevant in a given decision its rationality and the resulting optimal decision utility theory is about measures of the relative economic satisfaction from or desirability of consumption of various goods and services social choice theory is about voting a more puzzlebased approach to voting is ballot theory game theory deals with situations where success depends on the choices of others which makes choosing the best course of action more complex there are even continuous games see differential game topics include auction theory and fair division discretization 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 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 difference equations discrete dynamical systems and discrete vectormeasures 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 relation in algebraic geometry the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field and letting subvarieties or spectra of other rings provide the curves that lie in that space although the space in which the curves appear has a finite number of points the curves are not so much sets of points as analogues of curves in continuous settings for example every point of the form v x c spec k x a 1 displaystyle vxcsubset operatorname spec kmathbb a 1 for k displaystyle k a field can be studied either as spec k x x c spec k displaystyle operatorname spec kxccong operatorname spec k a point or as the spectrum spec k x x c displaystyle operatorname spec kxc of the local ring at xc a point together with a neighborhood around it algebraic varieties also have a welldefined notion of tangent space called the zariski tangent space making many features of calculus applicable even in finite settings hybrid discrete and continuous mathematics the time scale calculus is a unification of the theory of difference equations with that of differential equations which has applications to fields requiring simultaneous modelling of discrete and continuous data another way of modeling such a situation is the notion of hybrid dynamical system discrete mathematics portal outline of discrete mathematics cyberchase a show that teaches discrete mathematics to children norman l biggs 20021219 discrete mathematics oxford university press isbn9780198507178 john dwyer 2010 an introduction to discrete mathematics for business computing isbn9781907934001 susanna s epp 20100804 discrete mathematics with applications thomson brookscole isbn9780495391326 ronald graham donald e knuth oren patashnik concrete mathematics ralph p grimaldi 2004 discrete and combinatorial mathematics an applied introduction addison wesley isbn9780201726343 donald e knuth 20110303 the art of computer programming volumes 14a boxed set addisonwesley professional isbn9780321751041 ji matouek jaroslav neetil 1998 discrete mathematics oxford university press isbn9780198502081 obrenic bojana 20030129 practice problems in discrete mathematics prentice hall isbn9780130458032 kenneth h rosen john g michaels 2000 hand book of discrete and combinatorial mathematics crc pressi llc isbn9780849301490 kenneth h rosen 2007 discrete mathematics and its applications mcgrawhill college isbn9780072880083 andrew simpson 2002 discrete mathematics by example mcgrawhill incorporated isbn9780077098407 veerarajan t2007 discrete mathematics with graph theory and combinatorics tata mcgraw hill wikibooks has a book on the topic of discrete mathematics media related to discrete mathematics at wikimedia commons discrete mathematics at the utkedu mathematics archives providing links to syllabi tutorials programs etc iowa central electrical technologies program discrete mathematics for electrical engineering vteareas of mathematics outline topic lists branchesalgebra linear multilinear abstract elementary arithmetic number theory calculus analysis differential equations dynamical systems numerical analysis optimization functional analysis geometry discrete algebraic analytic differential finite topology trigonometry foundations philosophy of mathematics mathematical logic set theory category theory applied mathematical physics probability mathematical statistics statistics game theory information theory computer science computation control theory others history of mathematics recreational mathematics mathematics and art mathematics education order theory graph theory divisions pure applied discrete computational category portal commons wikiproject vteindustrial and applied mathematicscomputational computational logic algorithms design analysis information theory coding theory cryptography discrete computer algebra computational number theory combinatorics graph theory discrete geometry mathematicalphysicsalgebraic operator algebra representation theory renormalization group feynman integral mtheory analytic differential geometry fourier analysis harmonic analysis functional analysis operator theory analysis approximation theory numerical analysis differential equations dynamical systems control theory variational calculus probability theory distributionsrandom variables stochastic processes analysis path integral stochastic variational calculus decision sciences statistics operations research optimization game theory mathematical economics mathematical finance other applications biology chemistry psychology sociology the unreasonable effectiveness of mathematics in the natural sciences category mathematics portal outline topics list 