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Discrete mathematics is the theory or study of mathematical structures which are discrete rather than continuous. In short to real numbers that have the nature of varying "smoothly", the elements or objects studied in discrete mathematics – such as graphs, integers and statements in logic do not vary smoothly in same way, but they have distinct and separated values. Thus discrete mathematics excludes topics in "continuous mathematics" for example calculus and analysis. Discrete objects may often be enumerated by integers. In short, discrete mathematics has been characterized as the branch of math which is dealing with countable sets. Thus, there is no exact, universally agreed, definition of the "discrete mathematics. Indeed, discrete mathematics is defined less by what is included than by what is excluded: continuously varying quantities or related notions.

The set of objects learned in discrete mathematics may be finite or infinite. The term finite mathematics is applied to parts of the area of discrete mathematics that deals with finite sets, particularly areas relevant to business.

The Research of discrete mathematics widely used in the latter half of the twentieth century partly for the reason of the development of digital computers which is operated in discrete steps and data is stored in discrete bits. Notations and concepts from discrete mathematics are useful in describing and studying objects and problems in branches of computer science, i.e. programming languages, computer algorithms, automated theorem proving, cryptography, and software development. Conversely, the computer implementations are significant in applying ideas from discrete mathematics to real-world problems and solutions, i.e. in operations research.

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

Topics in Discrete Mathematics

1. Theoretical Computer Science: Theoretical computer science involves fields or areas of discrete mathematics relevant to computing and draws heavily on logic and graph theory. Included within the theoretical computer science is the learning or study of algorithms for computing mathematical results. Computability concerns what can be computed in principle and has close ties to logic, while complexity concerns the time taken by computations. Formal language theory and Automata theory are closely related to computability.

2. Information Theory: Information theory includes the quantification of information. It is related is coding theory which is used to construct or design efficient and reliable data transmission and methods of storage. Information theory involves continuous topics for example analog signals, analog encryption, analog coding.

3. Logic: It is the study of the principles of valid reasoning and inference, as well as of soundness, consistency and completeness.

4. Set Theory: Set theory is the area of mathematics that is concerned with sets, which are collections of objects, for example {blue, white, red} or the (infinite) set of all partially ordered sets, prime numbers. and sets with other relations have applications in several areas.

5. Combinatorics: Combinatorics concerns the way in which discrete structures may be combined and arranged. Enumerative combinatorics concerns with counting the number of certain combinatorial objects – for example the twelvefold way provides a unified framework for counting combinations, permutations and partitions. Analytic combinatorics is concerned the enumeration of combinatorial structures using tools from probability theory and complex analysis.

6. Graph Theory: Graph theory is the study of graphs and networks, is often covered in part of combinatorics, but it has grown large and distinct enough, with its own kind of problems, this is regarded as a subject in its own right. Graphs are one of the main objects of studying in Discrete Mathematics. In Mathematics, they are very useful in study of geometry and certain parts or area of topology, for example Knot Theory. Algebraic graph theory has very close links with group theory. There are also mentioned some continuous graphs, thus for the most part research in graph theory comes up within the domain of discrete mathematics.

7. Probability: Discrete probability theory includes with events that come in countable sample spaces. Discrete probability distributions may be used to approximate continuous ones and vice versa.

8. Number Theory: Number theory includes with the properties of numbers in general, particularly integers. It has applications to cryptanalysis, cryptography and cryptology, particularly with regard to diophantine, modular arithmetic, equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include In analytic number theory, geometry of numbers., techniques from continuous mathematics are also used. Topics that go beyond discrete objects involve diophantine approximation, transcendental numbers, p-adic analysis and function fields.

9. Algebra: Algebraic structures come up as both discrete examples or continuous examples. Discrete algebras involves: programming; relational algebra used in databases and boolean algebra used in logic gates; discrete and finite versions of rings , groups and fields are important in algebraic coding theory; discrete monoids and semi groups appear in the theory of formal languages.

Some other areas where discrete mathematics is concerned are

10. Calculus of finite differences, discrete calculus, discrete analysis

11. Geometry

12. Topology

13. Operation Research

14. Game theory, utility theory, decision theory, social choice theory

15. Discretization

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