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'''Discrete mathematics''' is the study of [[mathematical structures]] that are fundamentally [[Discrete space|discrete]] rather than continuous.

== Description ==

The objects studied in discrete mathematics –- such as [[Integer|integers]], [[Graph (mathematics)|graphs]], and statements in logic -– have distinct, separated values. 

Discrete mathematics therefore excludes topics in "continuous mathematics" such as [[calculus]] and [[analysis]], where objects may vary "smoothly".

Discrete objects can often be enumerated by [[integers]].

== Definitions ==

More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers).

However, there is no exact definition of the term "discrete mathematics."

Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

== Finite and infinite ==

The set of objects studied in discrete mathematics can be finite or infinite.

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.

== Computers ==

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.

Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of [[computer science]], including:

* Computer [[Algorithm|algorithms]]
* [[Programming language|Programming languages]]
* [[Cryptography]]
* [[Automated theorem proving]]
* [[Software development]]

== Real-world problems ==

Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in [[operations research]].

== Continuous mathematics ==

Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

== Education ==

In the university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time.

The curriculum has thereafter developed in conjunction to efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in freshmen; as such it is nowadays a prerequisite for mathematics majors in some universities as well.

Some high-school-level discrete mathematics textbooks have appeared as well.

At this level, discrete mathematics it is sometimes seen a preparatory course, not unlike precalculus in this respect.

== See also ==

* [[Automated theorem proving]]
* [[Algorithm]]
* [[Combinatorics]]
* [[Computer programming]]
* [[Computer science]]
* [[Cryptography]]
* [[Discrete geometry]]
* [[Graph (mathematics)]]
* [[Mathematics]]
* [[Operations research]]
* [[Programming language]]
* [[Integer]]
* [[Software development]]
* [[Time-scale calculus]]

== External links == 

* [https://en.wikipedia.org/wiki/Discrete_mathematics Discrete mathematics] @ Wikipedia

[[Category:Mathematics]]