# Portal:Discrete mathematics

**Discrete mathematics** is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. Discrete objects can be enumerated by integers. Topics in discrete mathematics include number theory (which deals mainly with the properties of integers), combinatorics, logic, graphs, algorithms, and formal languages.

Discrete mathematics has become popular in recent decades because of its applications to computer science. Discrete mathematics is the mathematical language of computer science. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in all branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are tremendously significant in applying ideas from discrete mathematics to real-world applications, such as in operations research.

The set of objects studied in discrete mathematics can be finite or infinite. In real-world applications, the set of objects of interest are mainly finite, the study of which is often called **finite mathematics**. In some mathematics curricula, the term "finite mathematics" refers to courses that cover discrete mathematical concepts for business, while "discrete mathematics" courses emphasize discrete mathematical concepts for computer science majors.

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**Combinatorics** is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (*enumerative combinatorics*), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in *combinatorial designs and matroid theory*), finding "largest", "smallest", or "optimal" objects (*extremal combinatorics* and *combinatorial optimization*), and finding algebraic structures these objects may have (*algebraic combinatorics*).

Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century (see the page List of combinatorics topics for details of the more recent development of the subject). One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

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**Penrose tiling**, an example of a tiling that can completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic).

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## Did you know?

- ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
- ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with
*n*+1 factors?

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*Topics in Discrete mathematics*

Major areas | Combinatorics | Graph Theory | Game theory |
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## Related portals

Algebra |
Analysis |
Categorytheory |
Computerscience |
Cryptography |
Discretemathematics |
Geometry |

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Mathematics |
Numbertheory |
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Statistics |
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