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{{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 – 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> – 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 Erdős}}
* '''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}}.

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