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Outline of discrete mathematics
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<!--... Attention: THIS IS AN OUTLINE part of the set of 830+ outlines listed at [[Wikipedia:Contents/Outlines]]. Wikipedia outlines are a special type of list article. They make up one of Wikipedia's content navigation systems See [[Wikipedia: Outlines]] and [[Wikipedia:WikiProject Outlines]] for more details. --> {{Short description|1=Overview of and topical guide to discrete mathematics}} '''[[Discrete mathematics]]''' is the study of [[Mathematics|mathematical]] [[Mathematical structure|structures]] that are fundamentally [[discrete space|discrete]] rather than [[Continuous function|continuous]]. In contrast to [[real number]]s that have the property of varying "smoothly", the objects studied in discrete mathematics β such as [[integer]]s, [[Graph (discrete mathematics)|graphs]], and statements in [[Mathematical logic|logic]]<ref>[[Richard Johnsonbaugh]], ''Discrete Mathematics'', Prentice Hall, 2008; James Franklin, [http://scholarship.claremont.edu/jhm/vol7/iss2/18/ Discrete and continuous: a fundamental dichotomy in mathematics], ''Journal of Humanistic Mathematics'' 7 (2017), 355-378.</ref> β do not vary smoothly in this way, but have distinct, separated values.<ref>{{MathWorld |title=Discrete mathematics |urlname=DiscreteMathematics}}</ref> Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as [[calculus]] and [[Mathematical analysis|analysis]]. Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical ''[[term of art|terms of art]]'' that may be encountered. * {{annotated link|Logic}} * {{annotated link|Modal logic}} * {{annotated link|Set theory}} * {{annotated link|Number theory}} * {{annotated link|Combinatorics}} * {{annotated link|Finite mathematics}} * {{annotated link|Graph theory}} * {{annotated link|Digital geometry}} * {{annotated link|Digital topology}} * {{annotated link|Algorithmics}} * {{annotated link|Information theory}} * {{annotated link|Computability}} * {{annotated link|Computational complexity theory}} * {{annotated link|Probability theory}} * {{annotated link|Probability}} * {{annotated link|Markov chain|Markov chains}} * {{annotated link|Linear algebra}} * {{annotated link|Function (mathematics)|Functions}} * {{annotated link|Partially ordered set}} * {{annotated link|Mathematical proof|Proofs}} * {{annotated link|Relation (mathematics)|Relation}} ==Discrete mathematical disciplines== For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to [[computer science]]. * {{annotated link|Automata theory}} * {{annotated link|Coding theory}} * {{annotated link|Combinatorics}} * {{annotated link|Computational geometry}} * {{annotated link|Digital geometry}} * {{annotated link|Discrete geometry}} * {{annotated link|Graph theory}} a study of {{annotated link|graph (discrete mathematics)|graphs}} * {{annotated link|Mathematical logic}} * {{annotated link|Discrete optimization}} * {{annotated link|Set theory}} * {{annotated link|Number theory}} * {{annotated link|Information theory}} * {{annotated link|Game theory}} ==Concepts in discrete mathematics== ===Sets=== *{{annotated link|Set (mathematics)}} **{{annotated link|Element (mathematics)}} **{{annotated link|Venn diagram}} **{{annotated link|Empty set}} **{{annotated link|Subset}} **{{annotated link|Union (set theory)}} ***{{annotated link|Disjoint union}} **{{annotated link|Intersection (set theory)}} ***{{annotated link|Disjoint sets}} **{{annotated link|Complement (set theory)}} **{{annotated link|Symmetric difference}} *{{annotated link|Ordered pair}} *{{annotated link|Cartesian product}} *{{annotated link|Power set}} *{{annotated link|Simple theorems in the algebra of sets}} *{{annotated link|Naive set theory}} *{{annotated link|Multiset}} ===Functions=== * {{annotated link|Function (mathematics)|Function}} * {{annotated link|Domain of a function}} * {{annotated link|Codomain}} * {{annotated link|Range of a function}} * {{annotated link|Image (mathematics)}} * {{annotated link|Injective function}} * {{annotated link|Surjection}} * {{annotated link|Bijection}} * {{annotated link|Function composition}} * {{annotated link|Partial function}} * {{annotated link|Multivalued function}} * {{annotated link|Binary function}} * {{annotated link|Floor function}} * {{annotated link|Sign function}} * {{annotated link|Inclusion map}} * {{annotated link|Pigeonhole principle}} * {{annotated link|Relation composition}} * {{annotated link|Permutations}} * {{annotated link|Symmetry}} ===Arithmetic=== * {{annotated link|Decimal}} * {{annotated link|Binary numeral system}} * {{annotated link|Divisor}} * {{annotated link|Division by zero}} * {{annotated link|Indeterminate form}} * {{annotated link|Empty product}} * {{annotated link|Euclidean algorithm}} * {{annotated link|Fundamental theorem of arithmetic}} * {{annotated link|Modular arithmetic}} * {{annotated link|Successor function}} ===Elementary algebra=== {{annotated link|Elementary algebra}} * {{annotated link|Left-hand side and right-hand side of an equation}} * {{annotated link|Linear equation}} * {{annotated link|Quadratic equation}} * {{annotated link|Solution point}} * {{annotated link|Arithmetic progression}} * {{annotated link|Recurrence relation}} * {{annotated link|Finite difference}} * {{annotated link|Difference operator}} * {{annotated link|Group (mathematics)|Groups}} * {{annotated link|Group isomorphism}} * {{annotated link|Subgroups}} * {{annotated link|Fermat's little theorem}} * {{annotated link|Cryptography}} * {{annotated link|Faulhaber's formula}} ===Mathematical relations=== *{{annotated link|Binary relation}} *{{annotated link|Heterogeneous relation}} *{{annotated link|Reflexive relation}} *{{annotated link|Reflexive property of equality}} *{{annotated link|Symmetric relation}} *{{annotated link|Symmetric property of equality}} *{{annotated link|Antisymmetric relation}} *{{annotated link|Transitive relation|Transitivity (mathematics)}} **{{annotated link|Transitive closure}} **{{annotated link|Transitive property of equality}} *''Equivalence and identity'' **{{annotated link|Equivalence relation}} **{{annotated link|Equivalence class}} **{{annotated link|Equality (mathematics)}} *** {{annotated link|Inequation}} *** {{annotated link|Inequality (mathematics)}} **{{annotated link|Similarity (geometry)}} **{{annotated link|Congruence (geometry)}} **{{annotated link|Equation}} **{{annotated link|Identity (mathematics)}} *** {{annotated link|Identity element}} *** {{annotated link|Identity function}} **{{annotated link|Substitution property of equality}} **{{annotated link|Graphing equivalence}} **{{annotated link|Extensionality}} **{{annotated link|Uniqueness quantification}} ===Mathematical phraseology=== * {{annotated link|If and only if}} * {{annotated link|Necessary and sufficient}} * {{annotated link|Distinct (mathematics)|Distinct}} * {{annotated link|Subtraction|Difference}} * {{annotated link|Absolute value}} * {{annotated link|Up to}} * {{annotated link|Modular arithmetic}} * {{annotated link|Characterization (mathematics)}} * {{annotated link|Normal form (mathematics)|Normal form}} * {{annotated link|Canonical form}} * {{annotated link|Without loss of generality}} * {{annotated link|Vacuous truth}} * {{annotated link|Contradiction}}, {{annotated link|Reductio ad absurdum}} * {{annotated link|Counterexample}} * {{annotated link|Sufficiently large}} * {{annotated link|Pons asinorum}} * {{annotated link|Table of mathematical symbols}} * {{annotated link|Contrapositive}} * {{annotated link|Mathematical induction}} ===Combinatorics=== {{annotated link|Combinatorics}} *{{annotated link|Combination|Permutations and combinations}} *{{annotated link|Permutation}} *{{annotated link|Combination}} *{{annotated link|Factorial}} **{{annotated link|Empty product}} *{{annotated link|Pascal's triangle}} *{{annotated link|Combinatorial proof}} **{{annotated link|Bijective proof}} **{{annotated link|Double counting (proof technique)}} ===Probability=== {{annotated link|Probability}} * {{annotated link|Average}} * {{annotated link|Expected value}} * {{annotated link|Discrete random variable}} * {{annotated link|Sample space}} * {{annotated link|Event (probability theory)|Event}} * {{annotated link|Conditional Probability}} * {{annotated link|Independence (probability theory)|Independence}} * {{annotated link|Random variables}} ===Propositional logic=== * {{annotated link|Logical operator}} * {{annotated link|Truth table}} * {{annotated link|De Morgan's laws}} * {{annotated link|Open sentence}} * {{annotated link|List of topics in logic}} ==Mathematicians associated with discrete mathematics== {{no references|section|date=August 2024}} * {{annotated link|Paul ErdΕs}} * '''Leonhard Euler - Swiss mathematician (1707-1783)''' * '''Claude Shannon - American mathematician (1916-2001)''' * '''Donald Knuth - American mathematician and computer scientist (b. 1938)''' * {{annotated link|Aristotle}} ==See also== {{Portal|Mathematics}} {{clear}} ==References== {{Reflist}} ==External links== *[https://web.archive.org/web/20180714042048/http://archives.math.utk.edu/topics/discreteMath.html Archives] * Jonathan Arbib & John Dwyer, ''Discrete Mathematics for Cryptography'', 1st Edition {{ISBN|978-1-907934-01-8}}. * John Dwyer & Suzy Jagger, ''Discrete Mathematics for Business & Computing'', 1st Edition 2010 {{ISBN|978-1-907934-00-1}}. {{Outline footer}} [[Category:Outlines of mathematics and logic|Discrete mathematics]] [[Category:Outlines|Discrete mathematics]] [[Category:Discrete mathematics|*]] [[Category:Mathematics-related lists|Discrete mathematics]]
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