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Discrete mathematics

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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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Pseudoforest.svg
A 1-forest (a maximal pseudoforest), formed by three 1-trees

In graph theory, a pseudoforest is an undirected graph[1] in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two closed paths of consecutive edges share any vertex with each other, nor can any two such closed paths be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.

The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan[2] attribute the naming of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.[3] Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – they have very few edges relative to their number of vertices – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests.

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Penrose tiling
A 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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  1. ^ The kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph.
  2. ^ Gabow & Tarjan (1988).
  3. ^ Dantzig (1963).