Editing Mathematics (section)

=== Discrete mathematics ===
{{Main|Discrete mathematics}}
[[File:Markovkate 01.svg|right|thumb|A diagram representing a two-state [[Markov chain]]. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.|class=skin-invert-image]]
Discrete mathematics, broadly speaking, is the study of individual, [[countable]] mathematical objects. An example is the set of all integers.<ref>{{cite journal |last=Franklin |first=James |author-link=James Franklin (philosopher) |date=July 2017 |title=Discrete and Continuous: A Fundamental Dichotomy in Mathematics |journal=Journal of Humanistic Mathematics |volume=7 |issue=2 |pages=355–378 |url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1334&context=jhm |doi=10.5642/jhummath.201702.18 |doi-access=free |issn=2159-8118 |lccn=2011202231 |oclc=700943261 |s2cid=6945363 |access-date=February 9, 2024 |archive-date=March 10, 2024 |archive-url=https://web.archive.org/web/20240310122911/https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1334&context=jhm |url-status=live }}</ref> Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.{{efn|However, some advanced methods of analysis are sometimes used; for example, methods of [[complex analysis]] applied to [[generating series]].}} [[Algorithm]]s{{emdash}}especially their [[implementation]] and [[computational complexity]]{{emdash}}play a major role in discrete mathematics.<ref>{{cite book |last=Maurer |first=Stephen B. |editor1-last=Rosenstein |editor1-first=Joseph G. |editor2-last=Franzblau |editor2-first=Deborah S. |editor3-last=Roberts |editor3-first=Fred S. |editor3-link=Fred S. Roberts |year=1997 |chapter=What is Discrete Mathematics? The Many Answers |pages=121–124 |title=Discrete Mathematics in the Schools |series=DIMACS: Series in Discrete Mathematics and Theoretical Computer Science |volume=36 |publisher=[[American Mathematical Society]] |doi=10.1090/dimacs/036/13 |isbn=0-8218-0448-0 |issn=1052-1798 |lccn=97023277 |oclc=37141146 |s2cid=67358543 |chapter-url={{GBurl|id=EvuQdO3h-DQC|p=121}} |access-date=February 9, 2024}}</ref>

The [[four color theorem]] and [[Kepler conjecture|optimal sphere packing]] were two major problems of discrete mathematics solved in the second half of the 20th century.<ref>{{cite book |last=Hales |first=Thomas C. |title=Turing's Legacy |author-link=Thomas Callister Hales |editor-last=Downey |editor-first=Rod |editor-link=Rod Downey |year=2014 |pages=260–261 |chapter=Turing's Legacy: Developments from Turing's Ideas in Logic |publisher=[[Cambridge University Press]] |series=Lecture Notes in Logic |volume=42 |doi=10.1017/CBO9781107338579.001 |isbn=978-1-107-04348-0 |lccn=2014000240 |oclc=867717052 |s2cid=19315498 |chapter-url={{GBurl|id=fYgaBQAAQBAJ|p=260}} |access-date=February 9, 2024}}</ref> The [[P versus NP problem]], which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of [[Computationally expensive|computationally difficult]] problems.<ref>{{cite conference |last=Sipser |first=Michael |author-link=Michael Sipser |date=July 1992 |title=The History and Status of the P versus NP Question |conference=STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing |pages=603–618 |doi=10.1145/129712.129771 |s2cid=11678884}}</ref>

Discrete mathematics includes:<ref name=MSC />
* [[Combinatorics]], the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or [[subset]]s of a given [[set (mathematics)|set]]; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of [[geometric shape]]s.
* [[Graph theory]] and [[hypergraph]]s
* [[Coding theory]], including [[error correcting code]]s and a part of [[cryptography]]
* [[Matroid]] theory
* [[Discrete geometry]]
* [[Discrete probability distribution]]s
* [[Game theory]] (although [[continuous game]]s are also studied, most common games, such as [[chess]] and [[poker]] are discrete)
* [[Discrete optimization]], including [[combinatorial optimization]], [[integer programming]], [[constraint programming]]