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Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP.More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.
Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.
The Clay Mathematics Institute has offered a $1 million USD prize for the first correct proof, along with prizes for six other mathematical problems.
Theoretical computer science includes areas of discrete mathematics relevant to computing.
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 Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry.
Discrete objects can often be enumerated by integers.Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself.Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits.At the same time, military requirements motivated advances in operations research.The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades.Kenneth Appel and Wolfgang Haken proved this in 1976.The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.It draws heavily on graph theory and mathematical logic.Included within theoretical computer science is the study of algorithms for computing mathematical results.