List of basic discrete mathematics topics

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally , in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as integers, finite graphs, and formal languages. Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, while discrete mathematics courses emphasize concepts for computer science majors. For contrast, see continuum, topology, and mathematical 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.

Essence of discrete mathematics

The discrete mathematics subjects

* Logic - a study of reasoning
* Set theory - a study of collections of elements
* Number theory
* Combinatorics - a study of counting
* Graph theory
* Digital geometry and digital topology
* Algorithmics - a study of methods of calculation
* Information theory
* Computability and complexity theories - dealing with theoretical and practical limitations of algorithms
* Elementary probability theory and Markov chains
* Linear algebra - a study of related linear equations
* Functions
* Partially Ordered Sets
* Probability
* Proofs
* Counting and Relations
* Collections

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 - Combinatorics - Combinatorial geometry - Computational geometry - Digital geometry - Discrete geometry - Graph theory - Mathematical logic - Combinatorial optimization - Set theory - Combinatorial topology - Number theory - Information theory - Game theory

Concepts in discrete mathematics

ets

*Set (mathematics)
**Element (mathematics)
**Venn diagram
**Empty set
**Subset
**Union (set theory)
***Disjoint union
**Intersection (set theory)
***Disjoint sets
**Complement (set theory)
**Symmetric difference
*Ordered pair
*Cartesian product
*Power set
*Simple theorems in the algebra of sets
*Naive set theory
*Multiset

Functions

:Function - How to specify a function - Domain of a function - Codomain - Range of a function - Image (mathematics) - Injective function - Surjection - Bijection - Function composition - Partial function - Multivalued function - Binary function - Floor function - Sign function - Inclusion map - Pigeonhole principle - Relation composition - Permutations - Symmetry

Operations

:Binary operator - Associativity - Commutativity - Distributivity

Arithmetic

:Decimal - Binary numeral system - Divisor - Division by zero - Indeterminate form - Empty product - Euclidean algorithm - Fundamental theorem of arithmetic - Modular arithmetic - Successor function

Elementary algebra

:Left-hand side and right-hand side of an equation - Linear equation - Quadratic equation - Solution point - Arithmetic progression - Recurrence relation - Finite difference - Difference operator - Groups - Group isomorphism - Subgroups - Fermat's little theorem - Cryptography - Faulhaber's formula

Mathematical relations

*Binary relation
*Mathematical relation
*Reflexive relation
*Reflexive property of equality
*Symmetric relation
*Symmetric property of equality
*Antisymmetric relation
*Transitivity (mathematics)
**Transitive closure
**Transitive property of equality
*"Equivalence and identity"
**Equivalence relation
**Equivalence class
**Equality (mathematics) (Inequation, Inequality)
**Similarity (mathematics)
**Congruence (geometry)
**Equation
**Identity (mathematics) (Identity element, Identity function)
**Substitution property of equality
**Graphing equivalence
**Extensionality
**Uniqueness quantification

Mathematical phraseology

:If and only if - Necessary and sufficient (Sufficient condition) - Distinct - Difference - Absolute value - Up to - Modulo - Characterization (mathematics) - Normal form - Canonical form - Without loss of generality - Vacuous truth - Contradiction, Reductio ad absurdum - Counterexample - Sufficiently large - Pons asinorum - Table of mathematical symbols - Contrapositive - Mathematical induction

Combinatorics

*Permutations and combinations
*Permutation
*Combination
*Factorial
**Empty product
*Pascal's triangle
*Combinatorial proof
**Bijective proof
**Double counting (proof technique)

Probability

:Average - Expected value - Discrete random variable - Sample space - Events - Conditional Probability - Independence - Random variables

Propositional logic

:Logical operator - Truth table - De Morgan's laws - Open sentence - List of topics in logic

Discrete mathematicians

:Paul Erdős - Ronald Graham

External links

* [http://archives.math.utk.edu/topics/discreteMath.html Archives]