Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "Discrete Mathematics" is therefore used in contrast with "Continuous Mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers.
The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruence’s and recurrence relations are also considered part of discrete mathematics.
The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.
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.
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, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
IMPORTANCE OF DISCRETE MATHEMATICS IN COMPUTER SCIENCEAchieving working knowledge of many principles of computer science requires mastery of certain relevant mathematical concepts and skills. For example, A grasp of Boolean algebra including DeMorgans Law is useful for understanding Boolean expressions and the basics of combinational circuits concepts surrounding the growth of functions and summations are useful for analysis of loop control structures exposure to solving recurrence relations is de rigeur for the analysis of recursive algorithms and an introduction to proof methods facilitates consideration of program correctness and thinking rigorously in general.
Students are introduced to proof techniques before they begin to consider the idea of proving programs correct. They learn about propositional logic and Boolean algebra before they study some very elementary circuits and learn decision control structures and Boolean variables. They are introduced to predicate logic near the time they are beginning programming and learning about variables. They learn about growth of functions big-O notation and summations before they analyze loops and nested loops and they have the tools to begin algorithm analysis from the time they first begin to learn about iterative constructs. In conjunction with an introduction to number theory they do laboratory and programming exercises involving an assortment of integer algorithms.
Students learn about recursive definitions recurrence relations, analyzing recursive algorithms and writing recursive algorithms and programs together in the same course. They study matrices and matrix manipulations in conjunction with the array data structure. They learn about permutations and combinations, relations, graphs, and trees at the same time that their programming knowledge and sophistication are improving and they can do increasingly interesting programming exercises involving these concepts.
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