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.
Subjects in discrete mathematics
- Logic - a study of reasoning
- Set theory - a study of collections of elements
- Number theory -
- Combinatorics - a study of Counting
- Finite mathematics - a course title
- 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 set -
- Probability -
- Proofs -
- Relation -
Discrete mathematical disciplines
- Automata theory -
- Coding theory -
- Combinatorics -
- Computational geometry -
- Digital geometry -
- Discrete geometry -
- Graph theory -
- Mathematical logic -
- Discrete_optimization -
- Set theory -
- Combinatorial topology -
- Number theory -
- Information theory -
- Game theory -
Concepts in discrete mathematics
Sets
- Set -
- *Element -
- *Venn diagram -
- *Empty set -
- *Subset -
- *Union -
- **Disjoint union -
- *Intersection -
- **Disjoint sets -
- *Complement -
- *Symmetric difference -
- Ordered pair -
- Cartesian product -
- Power set -
- Simple theorems in the algebra of sets -
- Naive set theory -
- Multiset -
Functions
- Function -
- Domain of a function -
- Codomain -
- Range of a function -
- Image -
- Injective function -
- Surjection -
- Bijection -
- Function composition -
- Partial function -
- Multivalued function -
- Binary function -
- Floor function -
- Sign function -
- Inclusion map -
- Pigeonhole principle -
- Relation composition -
- Permutations -
- Symmetry -
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 -
- Heterogeneous relation -
- Reflexive relation -
- Reflexive property of equality -
- Symmetric relation -
- Symmetric property of equality -
- Antisymmetric relation -
- Transitivity -
- *Transitive closure -
- *Transitive property of equality -
- Equivalence and identity
- *Equivalence relation -
- *Equivalence class -
- *Equality -
- ** Inequation -
- ** Inequality -
- *Similarity -
- *Congruence -
- *Equation -
- *Identity -
- ** Identity element -
- ** Identity function -
- *Substitution property of equality -
- *Graphing equivalence -
- *Extensionality -
- *Uniqueness quantification -
Mathematical phraseology
- If and only if -
- Necessary and sufficient -
- Distinct -
- Difference -
- Absolute value -
- Up to -
- Modular arithmetic -
- Characterization -
- 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 -
Probability
- Average -
- Expected value -
- Discrete random variable -
- Sample space -
- Event -
- Conditional Probability -
- Independence -
- Random variables -
Propositional logic
- Logical operator -
- Truth table -
- De Morgan's laws -
- Open sentence -
- List of topics in logic -
Mathematicians associated with discrete mathematics
- Paul Erdős
- Ronald Graham
- George Szekeres