Portal:Discrete mathematics

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Discrete mathematics is the study of mathematical structures that are fundamentally discrete, not involving 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 as a mathematical language for computer science, including computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Computer implementations thus serve as a bridge to apply ideas from discrete mathematics to real-world problems, as in operations research.

The set of objects studied in discrete mathematics can be finite or countably infinite. Real-world applications usually involve finite sets, the study of which is often called finite mathematics. In some mathematics curricula, courses on "finite mathematics" cover discrete mathematical topics for business students, while "discrete mathematics" courses emphasize topics useful for computer science majors.

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Four Colour Map Example.svg
Example of a four color map

The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point.

It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.

The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.

The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!"

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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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