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.
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.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
In university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time.Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations.Automata theory and formal language theory are closely related to computability.It draws heavily on graph theory and mathematical logic.Included within theoretical computer science is the study of algorithms for computing mathematical results.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.The set of objects studied in discrete mathematics can be finite or infinite.Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent.