Discrete Mathematics and Its Applications

Product Information

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Each problem sheet is marked out of 10 and the overall coursework mark will be calculated as the average of the five marked assignments.

Discrete Mathematics and Its Applications: | Guide books Discrete Mathematics and Its Applications: | Guide books

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. Discrete Math/Computer Science Pilot Open link in a new tab. (2022). Ohio Department of Education. Retrieved February 16, 2023. Return to footnote 1 reference in main content Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field.How to Become a Computer and Information Research Scientist Open link in a new tab. (2023). U.S. Bureau of Labor Statistics. Retrieved February 17, 2023. Return to footnote 7 reference in main content Understanding and using asymptotic notations in design and analysis of algorithms Transferable skills

Discrete Mathematics and Its Applications 2 CS147 Discrete Mathematics and Its Applications 2

An Introduction to Symbolic Logic Open link in a new tab. (2013). Mathematical Association of America. Retrieved February 17, 2023. Return to footnote 3 reference in main content As you begin your study of this field, many courses will focus on the fundamentals, such as applied discrete mathematics and discrete mathematics symbols, and then move on to more advanced topics as needed. Throughout your coursework, you will learn about real-world discrete mathematics examples and engage in practical work. 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. In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics. Logical formulas are discrete structures, as are proofs, which form finite trees [10] or, more generally, directed acyclic graph structures [11] [12] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied, [13] e.g. infinitary logic.

In addition to these foundational topics, an online discrete mathematics course often covers algorithms and data structures. Algorithms are step-by-step instructions that can be applied to solve problems. And data structures are ways of organizing data so that algorithms can be used effectively. Other computer science topics can include the design and analysis of algorithms, automata theory, and computability theory. Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms and data structures. 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. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems and representations of geometrical objects, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics. Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law ((( P→ Q)→ P)→ P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem proving and formal verification of software. Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption. Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane.