**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^{[1]} – do not vary smoothly in this way, but have distinct, separated values.^{[2]} 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 – 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 – a study of graphs
- 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 –

For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to computer science.

- Set (mathematics) –
- Ordered pair –
- Cartesian product –
- Power set –
- Simple theorems in the algebra of sets –
- Naive set theory –
- Multiset –

- 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 –

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

- 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 –

- Binary relation –
- Heterogeneous relation –
- Reflexive relation –
- Reflexive property of equality –
- Symmetric relation –
- Symmetric property of equality –
- Antisymmetric relation –
- Transitivity (mathematics) –
*Equivalence and identity*

- If and only if –
- Necessary and sufficient (Sufficient condition) –
- Distinct –
- Difference –
- Absolute value –
- Up to –
- Modular arithmetic –
- 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 –

- Permutations and combinations –
- Permutation –
- Combination –
- Factorial –
- Pascal's triangle –
- Combinatorial proof –

- Average –
- Expected value –
- Discrete random variable –
- Sample space –
- Event –
- Conditional Probability –
- Independence –
- Random variables –

This section needs expansion. You can help by adding to it. (January 2016)