Editing Discrete mathematics (section)

{{Short description|Study of discrete mathematical structures}}
{{For|the mathematics journal|Discrete Mathematics (journal){{!}}''Discrete Mathematics'' (journal)}}
{{Redirect|Finite math|the syllabus|Finite mathematics}}
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[[File:6n-graf.svg|thumb|250px|[[Graph (discrete mathematics)|Graphs]] such as these are among the objects studied by discrete mathematics, for their interesting [[graph property|mathematical properties]], their usefulness as models of real-world problems, and their importance in developing computer [[algorithm]]s.]]

'''Discrete mathematics''' is the study of [[mathematical structures]] that can be considered "discrete" (in a way analogous to [[discrete variable]]s, having a [[bijection]] with the set of [[natural numbers]]) rather than "continuous" (analogously to [[continuous function]]s). Objects studied in discrete mathematics include [[integer]]s, [[Graph (discrete mathematics)|graphs]], and [[Statement (logic)|statements]] in [[Mathematical logic|logic]].<ref>[[Richard Johnsonbaugh]], ''Discrete Mathematics'', Prentice Hall, 2008.</ref><ref>{{cite journal |last1=Franklin |first1=James |authorlink=James Franklin (philosopher) |date=2017 |title=Discrete and continuous: a fundamental dichotomy in mathematics |url=http://philsci-archive.pitt.edu/16561/1/Discrete%20and%20Continuous.pdf |journal=Journal of Humanistic Mathematics |volume=7 |issue=2 |pages=355–378 |doi=10.5642/jhummath.201702.18 |s2cid=6945363 |access-date=30 June 2021|doi-access=free }}</ref><ref>{{cite web |url=https://cse.buffalo.edu/~rapaport/191/S09/whatisdiscmath.html |title=Discrete Structures: What is Discrete Math? |website=cse.buffalo.edu |access-date=16 November 2018}}</ref> By contrast, discrete mathematics excludes topics in "continuous mathematics" such as [[real number]]s, [[calculus]] or [[Euclidean geometry]]. Discrete objects can often be [[enumeration|enumerated]] by [[integers]]; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with [[countable set]]s<ref>{{citation
 | last = Biggs | first = Norman L. | author-link = Norman L. Biggs
 | edition = 2nd
 | isbn = 9780198507178
 | mr = 1078626
 | page = 89
 | publisher = The Clarendon Press Oxford University Press
 | series = Oxford Science Publications
 | title = Discrete mathematics
 | url = https://books.google.com/books?id=Mj9gzZMrXDIC&pg=PA89
 | year = 2002
 | quote = Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets.}}</ref> (finite sets or sets with the same [[cardinality]] as the natural numbers). However, there is no exact definition of the term "discrete mathematics".<ref>{{cite book |editor-first=Brian |editor-last=Hopkins |title=Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles |publisher=Mathematical Association of America |location= |date=2009 |isbn=978-0-88385-184-5 |pages= |url={{GBurl|05DEJ8Kh67AC|pg=PR11}}}}</ref>

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 algorithm]]s, [[programming language]]s, [[cryptography]], [[automated theorem proving]], and [[software development]]. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems.

Although the main objects of study in discrete mathematics are discrete objects, [[Analysis (mathematics)|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 [[Association for Computing Machinery|ACM]] and [[Mathematical Association of America|MAA]] into a course that is basically intended to develop [[mathematical maturity]] in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well.<ref name="LevasseurDoerr">{{cite book |first1=Ken |last1=Levasseur |first2=Al |last2=Doerr|title=Applied Discrete Structures|url=https://discretemath.org/ads/index-ads.html|page=8}}</ref><ref name="Howson1988">{{cite book|editor-first=Albert |editor-last=Geoffrey Howson|title=Mathematics as a Service Subject|year=1988|publisher=Cambridge University Press|isbn=978-0-521-35395-3|pages=77–78}}</ref> Some high-school-level discrete mathematics textbooks have appeared as well.<ref name="Rosenstein">{{cite book|first=Joseph G. |last=Rosenstein|title=Discrete Mathematics in the Schools|publisher=American Mathematical Society|isbn=978-0-8218-8578-9|page=323}}</ref> At this level, discrete mathematics is sometimes seen as a preparatory course, like [[precalculus]] in this respect.<ref>{{cite web|url=http://ucsmp.uchicago.edu/secondary/curriculum/precalculus-discrete/|title=UCSMP|work=uchicago.edu}}</ref>

The [[Fulkerson Prize]] is awarded for outstanding papers in discrete mathematics.