• Source: Outline of discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and 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 terms of art that may be encountered.

Logic – Study of correct reasoning
Modal logic – Type of formal logic
Set theory – Branch of mathematics that studies sets
Number theory – Mathematics of integer properties
Combinatorics – Branch of discrete mathematics
Finite mathematics – Syllabus in college and university mathematics
Graph theory – Area of discrete mathematics
Digital geometry – Deals with digitized models or images of objects of the 2D or 3D Euclidean space
Digital topology – Properties of 2D or 3D digital images that correspond to classic topological properties
Algorithmics – Sequence of operations for a taskPages displaying short descriptions of redirect targets
Information theory – Scientific study of digital information
Computability – Ability to solve a problem in an effective manner
Computational complexity theory – Inherent difficulty of computational problems
Probability theory – Branch of mathematics concerning probability
Probability – Branch of mathematics concerning chance and uncertainty
Markov chains – Random process independent of past history
Linear algebra – Branch of mathematics
Functions – Association of one output to each input
Partially ordered set – Mathematical set with an ordering
Proofs – Reasoning for mathematical statements
Relation – Relationship between two sets, defined by a set of ordered pairs




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.

Automata theory – Study of abstract machines and automata
Coding theory – Study of the properties of codes and their fitness
Combinatorics – Branch of discrete mathematics
Computational geometry – Branch of computer science
Digital geometry – Deals with digitized models or images of objects of the 2D or 3D Euclidean space
Discrete geometry – Branch of geometry that studies combinatorial properties and constructive methods
Graph theory – Area of discrete mathematics a study of graphs – Vertices connected in pairs by edges
Mathematical logic – Subfield of mathematics
Discrete optimization – Branch of mathematical optimization
Set theory – Branch of mathematics that studies sets
Number theory – Mathematics of integer properties
Information theory – Scientific study of digital information
Game theory – Mathematical models of strategic interactions




Concepts in discrete mathematics






= Sets

=
Set (mathematics) – Collection of mathematical objects
Element (mathematics) – Any one of the distinct objects that make up a set in set theory
Venn diagram – Diagram that shows all possible logical relations between a collection of sets
Empty set – Mathematical set containing no elements
Subset – Set whose elements all belong to another set
Union (set theory) – Set of elements in any of some sets
Disjoint union – In mathematics, operation on sets
Intersection (set theory) – Set of elements common to all of some sets
Disjoint sets – Sets with no element in common
Complement (set theory) – Set of the elements not in a given subset
Symmetric difference – Elements in exactly one of two sets
Ordered pair – Pair of mathematical objects
Cartesian product – Mathematical set formed from two given sets
Power set – Mathematical set of all subsets of a set
Simple theorems in the algebra of sets
Naive set theory – Informal set theories
Multiset – Mathematical set with repetitions allowed




= Functions

=
Function – Association of one output to each input
Domain of a function – Mathematical concept
Codomain – Target set of a mathematical function
Range of a function – Subset of a function's codomain
Image (mathematics) – Set of the values of a function
Injective function – Function that preserves distinctness
Surjection – Mathematical function such that every output has at least one inputPages displaying short descriptions of redirect targets
Bijection – One-to-one correspondence
Function composition – Operation on mathematical functions
Partial function – Function whose actual domain of definition may be smaller than its apparent domain
Multivalued function – Generalized mathematical function
Binary function – Function that takes two inputs
Floor function – Nearest integers from a numberPages displaying short descriptions of redirect targets
Sign function – Mathematical function returning -1, 0 or 1
Inclusion map – Set-theoretic function
Pigeonhole principle – If there are more items than boxes holding them, one box must contain at least two items
Relation composition – Mathematical operationPages displaying short descriptions of redirect targets
Permutations – Mathematical version of an order changePages displaying short descriptions of redirect targets
Symmetry – Mathematical invariance under transformations




= Arithmetic

=
Decimal – Number in base-10 numeral system
Binary numeral system – Number expressed in the base-2 numeral systemPages displaying short descriptions of redirect targets
Divisor – Integer that is a factor of another integer
Division by zero – Class of mathematical expression
Indeterminate form – Expression in mathematical analysis
Empty product – Result from multiplying no factors
Euclidean algorithm – Algorithm for computing greatest common divisors
Fundamental theorem of arithmetic – Integers have unique prime factorizations
Modular arithmetic – Computation modulo a fixed integer
Successor function – Elementary operation on a natural number




= Elementary algebra

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Elementary algebra – Basic concepts of algebra

Left-hand side and right-hand side of an equation – Mathematical nomenclaturePages displaying short descriptions of redirect targets
Linear equation – Equation that does not involve powers or products of variables
Quadratic equation – Polynomial equation of degree two
Solution point – Mathematical formula expressing equalityPages displaying short descriptions of redirect targets
Arithmetic progression – Sequence of equally spaced numbers
Recurrence relation – Pattern defining an infinite sequence of numbers
Finite difference – Discrete analog of a derivative
Difference operator – Pattern defining an infinite sequence of numbersPages displaying short descriptions of redirect targets
Groups – Set with associative invertible operation
Group isomorphism – Bijective group homomorphism
Subgroups – Subset of a group that forms a group itselfPages displaying short descriptions of redirect targets
Fermat's little theorem – A prime p divides a^p–a for any integer a
Cryptography – Practice and study of secure communication techniques
Faulhaber's formula – Expression for sums of powers




= Mathematical relations

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Binary relation – Relationship between elements of two sets
Heterogeneous relation – Relationship between elements of two setsPages displaying short descriptions of redirect targets
Reflexive relation – Binary relation that relates every element to itself
Reflexive property of equality – Basic notion of sameness in mathematicsPages displaying short descriptions of redirect targets
Symmetric relation – Type of binary relation
Symmetric property of equality – Basic notion of sameness in mathematicsPages displaying short descriptions of redirect targets
Antisymmetric relation – Binary relation such that if A is related to B and is different from it then B is not related to A
Transitivity (mathematics) – Type of binary relation
Transitive closure – Smallest transitive relation containing a given binary relation
Transitive property of equality – Basic notion of sameness in mathematicsPages displaying short descriptions of redirect targets
Equivalence and identity
Equivalence relation – Mathematical concept for comparing objects
Equivalence class – Mathematical concept
Equality (mathematics) – Basic notion of sameness in mathematics
Inequation – Mathematical statement that two values are not equal
Inequality (mathematics) – Mathematical relation expressed with < or ≤
Similarity (geometry) – Property of objects which are scaled or mirrored versions of each other
Congruence (geometry) – Relationship between two figures of the same shape and size, or mirroring each other
Equation – Mathematical formula expressing equality
Identity (mathematics) – Equation that is satisfied for all values of the variables
Identity element – Specific element of an algebraic structure
Identity function – In mathematics, a function that always returns the same value that was used as its argument
Substitution property of equality – Basic notion of sameness in mathematicsPages displaying short descriptions of redirect targets
Graphing equivalence – Mathematical concept for comparing objectsPages displaying short descriptions of redirect targets
Extensionality – Logic principle
Uniqueness quantification – Logical property of being the one and only object satisfying a condition




= Mathematical phraseology

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If and only if – Logical connective
Necessary and sufficient – Terms to describe a conditional relationship between two statementsPages displaying short descriptions of redirect targets
Distinct – Basic notion of sameness in mathematicsPages displaying short descriptions of redirect targets
Difference – One of the four basic arithmetic operations
Absolute value – Distance from zero to a number
Up to – Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)
Modular arithmetic – Computation modulo a fixed integer
Characterization (mathematics) – Term in mathematics
Normal form – Standard representation of a mathematical objectPages displaying short descriptions of redirect targets
Canonical form – Standard representation of a mathematical object
Without loss of generality – Expression in mathematics
Vacuous truth – Conditional statement which is true because the antecedent cannot be satisfied
Contradiction – Logical incompatibility between two or more propositions, Reductio ad absurdum – Argument that leads to a logical absurdity
Counterexample – Exception to a proposed general rule
Sufficiently large – mathematical conceptPages displaying wikidata descriptions as a fallback
Pons asinorum – Statement that the angles opposite the equal sides of an isosceles triangle are themselves equal
Table of mathematical symbols – Meanings of symbols used in mathematicsPages displaying short descriptions of redirect targets
Contrapositive – Mathematical logic conceptPages displaying short descriptions of redirect targets
Mathematical induction – Form of mathematical proof




= Combinatorics

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Combinatorics – Branch of discrete mathematics

Permutations and combinations – Selection of items from a set
Permutation – Mathematical version of an order change
Combination – Selection of items from a set
Factorial – Product of numbers from 1 to n
Empty product – Result from multiplying no factors
Pascal's triangle – Triangular array of the binomial coefficients in mathematics
Combinatorial proof – proofs in enumerative combinatorics based on bijections or double countings of combinatorial objectsPages displaying wikidata descriptions as a fallback
Bijective proof – Technique for proving sets have equal size
Double counting (proof technique) – Type of proof technique




= Probability

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Probability – Branch of mathematics concerning chance and uncertainty

Average – Number taken as representative of a list of numbers
Expected value – Average value of a random variable
Discrete random variable – Variable representing a random phenomenonPages displaying short descriptions of redirect targets
Sample space – Set of all possible outcomes or results of a statistical trial or experiment
Event – In statistics and probability theory, set of outcomes to which a probability is assigned
Conditional Probability – Probability of an event occurring, given that another event has already occurredPages displaying short descriptions of redirect targets
Independence – When the occurrence of one event does not affect the likelihood of another
Random variables – Variable representing a random phenomenonPages displaying short descriptions of redirect targets




= Propositional logic

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Logical operator – Symbol connecting sentential formulas in logicPages displaying short descriptions of redirect targets
Truth table – Mathematical table used in logic
De Morgan's laws – Pair of logical equivalences
Open sentence – formula that contains at least one free variablePages displaying wikidata descriptions as a fallback
List of topics in logic – Overview of and topical guide to logicPages displaying short descriptions of redirect targets




Mathematicians associated with discrete mathematics



Paul Erdős – Hungarian mathematician (1913–1996)
Leonhard Euler - Swiss mathematician (1707-1783)
Claude Shannon - American mathematician (1916-2001)
Donald Knuth - American mathematician and computer scientist (b. 1938)
Aristotle – Ancient Greek philosopher and polymath (384–322 BC)




See also






References






External links


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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