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 Cantor 's 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 studies the way in which discrete structures can be combined or arranged.

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Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e. Analytic combinatorics concerns the enumeration i. 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 q-series , 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, 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. They are among the most ubiquitous models of both natural and human-made 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 , e. 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. Discrete probability theory deals with events that occur in countable sample spaces.

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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 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 , p-adic analysis and function fields.

### Discrete Structures, Logic, and Computability (Jones & Bartlett Computer Science)

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. 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.

Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems. 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 provides techniques for solving practical problems in engineering, business, and other fields — problems such as allocating resources to maximize profit, and scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization , queuing theory , scheduling theory , and network theory.

Operations research also includes continuous topics such as continuous-time Markov process , continuous-time martingales , process optimization , and continuous and hybrid control theory. 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 puzzle-based 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.

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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. In applied mathematics , discrete modelling is the discrete analogue of continuous modelling.

## Discrete structures, logic, and computability - CERN Document Server

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.

Algebraic varieties also have a well-defined notion of tangent space called the Zariski tangent space , making many features of calculus applicable even in finite settings.