Editing Discrete mathematics (section)

==Topics==
{{See also|Outline of discrete mathematics}}

===Theoretical computer science===
{{Main|Theoretical computer science}}
[[File:Sorting quicksort anim.gif|right|thumb|210px|[[Computational complexity theory|Complexity]] studies the time taken by [[algorithm]]s, such as this [[Quicksort|sorting routine]].]]

[[File:SimplexRangeSearching.svg|left|thumb|150px|[[Computational geometry]] applies computer [[algorithm]]s to representations of [[geometry|geometrical]] objects.]]

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 and data structures. [[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 net]]s and [[process algebra]]s 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 and representations of [[geometry|geometrical]] objects, while [[computer image analysis]] applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.

===Information theory===
{{Main|Information theory}}
[[File:WikipediaBinary.svg|thumb|150px|The [[ASCII]] codes for the word "Wikipedia", given here in [[Binary numeral system|binary]], provide a way of representing the word in [[information theory]], as well as for information-processing [[algorithm]]s.]]
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 signal]]s, [[analog coding]], [[analog encryption]].

===Logic===
{{Main|Mathematical logic}}
Logic is the study of the principles of valid reasoning and [[inference]], as well as of [[consistency]], [[soundness]], and [[Completeness (logic)|completeness]]. For example, in most systems of logic (but not in [[intuitionistic logic]]) [[Peirce's law]] (((''P''→''Q'')→''P'')→''P'') 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 accumulated to [[automated theorem proving]] and [[formal verification]] of software.

[[Well-formed formula|Logical formulas]] are discrete structures, as are [[Proof theory|proofs]], which form finite [[tree structure|trees]]<ref>{{cite book| first1 = A.S. |last1=Troelstra|first2=H. |last2=Schwichtenberg| title = Basic Proof Theory| url = https://books.google.com/books?id=x9x6F_4mUPgC&pg=PA186| date = 2000-07-27| publisher = Cambridge University Press| isbn = 978-0-521-77911-1| page = 186 }}</ref> or, more generally, [[directed acyclic graph]] structures<ref>{{cite book| first = Samuel R. |last=Buss| title = Handbook of Proof Theory| url = https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA13| year = 1998| publisher = Elsevier| isbn = 978-0-444-89840-1| page = 13 }}</ref><ref>{{cite book| first1 = Franz |last1=Baader|first2=Gerhard |last2=Brewka|first3=Thomas |last3=Eiter| title = KI 2001: Advances in Artificial Intelligence: Joint German/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001. Proceedings| url = https://books.google.com/books?id=27A2XJPYwIkC&pg=PA325| date = 2001-10-16| publisher = Springer| isbn = 978-3-540-42612-7| page = 325 }}</ref> (with each [[Rule of inference|inference step]] combining one or more [[premise]] branches to give a single conclusion). The [[truth value]]s of logical formulas usually form a finite set, generally restricted to two values: ''true'' and ''false'', but logic can also be continuous-valued, e.g., [[fuzzy logic]]. Concepts such as infinite proof trees or infinite derivation trees have also been studied,<ref>{{cite journal | title = Cyclic proofs of program termination in separation logic | first1 = J. | last1 = Brotherston | first2 = R. | last2 = Bornat | first3 = C. | last3 = Calcagno | journal = ACM SIGPLAN Notices | volume = 43 | issue = 1 |date=January 2008 | pages = 101–112 | doi = 10.1145/1328897.1328453 }}</ref> e.g. [[infinitary logic]].

===Set theory===
{{Main|Set theory}}
Set theory is the branch of mathematics that studies [[set (mathematics)|sets]], which are collections of objects, such as {blue, white, red} or the (infinite) set of all [[prime number]]s. [[Partially ordered set]]s and sets with other [[Relation (mathematics)|relations]] have applications in several areas.

In discrete mathematics, [[countable set]]s (including [[finite set]]s) 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===
{{Main|Combinatorics}}
Combinatorics studies the ways in which discrete structures can be combined or arranged.
[[Enumerative combinatorics]] concentrates on counting the number of certain combinatorial objects - e.g. the [[twelvefold way]] provides a unified framework for counting [[permutations]], [[combinations]] and [[Partition of a set|partitions]].
[[Analytic combinatorics]] concerns the enumeration (i.e., 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 analysis|asymptotic formulae]].
[[Topological combinatorics]] concerns the use of techniques from [[topology]] and [[algebraic topology]]/[[combinatorial topology]] in [[combinatorics]].
Design theory is a study of [[combinatorial design]]s, which are collections of subsets with certain [[Set intersection|intersection]] properties.
[[Partition theory]] studies various enumeration and asymptotic problems related to [[integer partition]]s, 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===
{{Main|Graph theory}}
[[File:TruncatedTetrahedron.gif|thumb|right|200px|[[Graph theory]] has close links to [[group theory]]. This [[truncated tetrahedron]] graph is related to the [[alternating group]] ''A''<sub>4</sub>.]]
Graph theory, the study of [[Graph (discrete mathematics)|graphs]] and [[network theory|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.<ref>{{cite book |author1-link=Bojan Mohar |author2-link=Carsten Thomassen (mathematician) |first1=Bojan |last1=Mohar |first2=Carsten |last2=Thomassen |title=Graphs on Surfaces |publisher=Johns Hopkins University Press |date=2001 |isbn=978-0-8018-6689-0 |oclc=45102952 |url=https://www.press.jhu.edu/books/title/1675/graphs-surfaces }}</ref> Graphs are one of the prime objects of study in discrete mathematics. 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.g. [[knot theory]]. [[Algebraic graph theory]] has close links with group theory and [[topological graph theory]] has close links to [[topology]]. There are also [[continuous graph]]s; however, for the most part, research in graph theory falls within the domain of discrete mathematics.

===Number theory===
[[File:Ulam 1.png|thumb|200px|right|The [[Ulam spiral]] of numbers, with black pixels showing [[prime number]]s. This diagram hints at patterns in the [[Prime number#Distribution|distribution]] of prime numbers.]]
{{Main|Number theory}}
Number theory is concerned with the properties of numbers in general, particularly [[integer]]s. It has applications to [[cryptography]] and [[cryptanalysis]], particularly with regard to [[modular arithmetic]], [[diophantine equations]], linear and quadratic congruences, prime numbers and [[primality test]]ing. 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 number]]s, [[diophantine approximation]], [[p-adic analysis]] and [[function field of an algebraic variety|function fields]].

=== Algebraic structures ===
{{Main|Abstract algebra}}

[[Algebraic structure]]s occur as both discrete examples and continuous examples. Discrete algebras include: [[Boolean algebra (logic)|Boolean algebra]] used in [[logic gate]]s and programming; [[relational algebra]] used in [[databases]]; discrete and finite versions of [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]] are important in [[algebraic coding theory]]; discrete [[semigroup]]s and [[monoid]]s appear in the theory of [[formal languages]].

===Discrete analogues of continuous mathematics===
There are many concepts and theories in continuous mathematics which have discrete versions, such as [[discrete calculus]], [[discrete Fourier transform]]s, [[discrete geometry]], [[discrete logarithm]]s, [[discrete differential geometry]], [[discrete exterior calculus]], [[discrete Morse theory]], [[discrete optimization]], [[discrete probability theory]], [[discrete probability distribution]], [[difference equation]]s, [[discrete dynamical system]]s, and [[Shapley–Folkman lemma#Probability and measure theory|discrete vector&nbsp;measures]].

==== Calculus of finite differences, discrete analysis, and discrete calculus ====
In [[discrete calculus]] and the [[calculus of finite differences]], a [[function (mathematics)|function]] defined on an interval of the [[integer]]s 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 [[differential equation]]s, but replace [[derivative|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 transform]]s for discrete functions or digital signals. As well as [[discrete metric space]]s, there are more general [[discrete topological space]]s, [[finite metric space]]s, [[finite topological space]]s.

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 system|hybrid dynamical system]]s.

==== Discrete geometry ====
[[Discrete geometry]] and combinatorial geometry are about combinatorial properties of ''discrete collections'' of geometrical objects. A long-standing topic in discrete geometry is [[tessellation|tiling of the plane]].

In [[algebraic geometry]], the concept of a curve can be extended to discrete geometries by taking the [[Spectrum of a ring|spectra]] of [[polynomial ring]]s over [[finite field]]s to be models of the [[affine space]]s over that field, and letting [[Algebraic variety|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 <math>V(x-c) \subset \operatorname{Spec} K[x] = \mathbb{A}^1</math> for <math>K</math> a field can be studied either as <math>\operatorname{Spec} K[x]/(x-c) \cong \operatorname{Spec} K</math>, a point, or as the spectrum <math>\operatorname{Spec} K[x]_{(x-c)}</math> of the [[Localization of a ring|local ring at (x-c)]], a point together with a neighborhood around it. 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.

==== Discrete modelling ====

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]]. [[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.