Discrete mathematics for computer science - IEEE Xplore

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Discrete mathematics computer science Murali ft Varanasi Oscar /V. Garcia

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he i m p o r t a n c e of mathematics to the study of both electrical engineering and com­ puter science and engi­ neering is well under­ stood by educators and students alike. The topics learned in mathe­ matics courses are often reviewed in engineering courses that partly focus on relevant mathematical concepts. It is almost impossible to imagine any engineering topic that can be studied without the aid of mathema­ tics; today it is common practice to include at least half a year of mathe­ matics in the engineering curriculum. Typical mathematical foundations developed in such a curriculum in­ clude courses in differential and in­ tegral calculus, probability and sta­ tistics, and differential equations. Additional topics such as linear alge­ bra and matrices, numerical analysis, and discrete mathematics are often included as électives. The branch of mathematics re­ ferred to as discrete mathematics, or discrete structures, is critical to the study of computer science and en­ gineering, as well as to the study of computer-related topics in electrical engineering. Discrete mathematics provides a foundation for the design and implementation of hardware and software in digital computers. The need for such study is recognized by professional organizations such as the IEEE Computer Society and the Association for Computing Machin­ ery (ACM). These organizations pe­ riodically make curricular recom­ mendations through their respective educational activities boards, as well as through joint committees. They also work in collaboration with the IEEE and the Accreditation Board for Engineering and Technology to define criteria for accreditation of computer-related programs. All these

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for Studying this branch of mathematics can provide you with a solid background for computer theory and applications

eminent groups have endorsed the need for a course in discrete mathe­ matics in addition to endorsing courses that cover the mathematics of the continuum, such as calculus and differential equations. The emphasis of a course in dis­ crete mathematics should be directed toward developing theoretical tools for describing algorithmic applica­ tions and processes. The material can be broadly divided into two tracks, one based on the concepts of modern algebra as applied to sequential ma­ chines and computer system design, and the other based on graphs and trees as applied to data structures and algorithms. In addition, combinatoric mathematics is often included, covering techniques for enumerating the number of elements having a cer­ tain property. The key challenge, then, is to weave a unifying thread through sometimes dissimilar topics and organize a cohesive body of knowledge that will provide a firm foundation for theory and applica­ tions in the computer area. The best teaching approach is to introduce each concept and follow it with inter­ esting applications to reinforce its im­ portance. A student can then appreci­ ate the purpose of each theoretical tool and gain some understanding of its use. With this approach, the stu­ dent avoids having to learn complex mathematical tools in the middle of studying other advanced topics. A number of important topics should be included in a discrete mathematics course. Because of the variety of topics, some can only be treated in an introductory fashion. A more advanced treatment requires advanced undergraduate or graduate courses. The pertinent topics can be classi­ fied into five major themes:

• mathematical reasoning • set theory • algebraic structures • graphs and trees • combinatorics At times the topics of generating functions and recurrence relations are also included under the heading of discrete mathematics. Mathematical reasoning includes such concepts as logical connectives, well-formed formulas, rules of infer­ ence, induction, methods of proof, predicates and quantifiers, and ap­ plications to computer logic and proofs of program correctness. These concepts breed familiarity with the terminology and methodology of for­ mal logic. The study of set theory includes sets and set algebra, relations of sets, recursive definition of sets, binary re­ lations, ordering, composition of re­ lations, equivalence relations and partitions, and applications. Set theory is basic in learning how to deal with objects and their relations from an abstract point of view. Concepts of algebraic structures provide the background essential to the study of finite state machines, switching theory, logic design, and general concepts in modern algebra. These algebraic structures include sets with defined operations; func­ tions; composition; inverses; associa­ tive, commutative, and distributive operations; congruence relations; applications such as modular arith­ metic, hashing functions, and ma­ chine equivalence and simulation; semigroups, monoids; groups; morphisms; lattices; Boolean algebra; and applications in switching theory and logic design. Another topic in discrete mathe­ matics, graphs and trees, includes the study of directed and undirected graphs, paths, circuits, trees, reach­ ability and connectedness, matrix

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IEEE POTENTIALS · FEBRUARY 1985

Topics in d i s c r e t e m a t h e m a t i c s

representation, applications to deci­ sion trees, balanced trees, Polish notation and trees, graph scheduling programs, flows in networks, data structure representation, theorem proving, and state graphs. These con­ cepts form the basis for the study of data structures and algorithms in a systematic manner. Combinatorics includes basic counting techniques, permutations and combinations, enumeration by recursion, the principle of inclusion and exclusion, and elements of Polya's theory, which provide a foun­ dation in counting techniques that can be applied to algorithm analysis. Useful textbooks

Several good textbooks on discrete mathematics cover the major themes identified here. Some of them are di­ rected at beginning undergraduates, and others are aimed at advanced un­ dergraduates. The books listed are suitable as texts for an undergraduate course in discrete mathematics. This list is not exhaustive; in addition to the books included here, a number of more advanced textbooks provide indepth treatment of one or more topics in discrete mathematics. Rath­ er, the list is a good starting point for study of this area: • Bavel, Z., Math Companion for Computer Science, Reston, 1982, presents a concise treatment of the subject in an accessible and useful manner. • Birkhoff, F., and Bartee, T. C , Modern Applied Algebra, McGrawHill, 1970, provides excellent cover­ age of applications for set theory, algebraic structures, and graphs and trees. • Bobrow, L. S., and Arbib, Μ. Α., Discrete Mathematics, Saunders, 1974, provides an advanced under­ graduate or graduate treatment of algebraic structures. • Gersting, J. L., Mathematical Structures for Computer Science, / Freeman, 1983, offers good ^ coverage of algebraic aspects of discrete mathematics. • Gill, Α., Modern Algebra and Ap­ plications to Computer Science, McGraw-Hill, 1975, covers algebraic structures and their application to computers. • Johnsonbaugh, R., Discrete Math­ ematics, Macmillan, 1984, provides an overview of a large number of topics in a concise and pedagogically sound manner. • Kolman, B., and Busby, R. C. Dis­

crete Mathematical Structures for Computer Science, Prentice-Hall, 1984, presents an introductory treat­ ment that covers topics carefully se­ lected to motivate students. • Korfhage, R. R., Discrete Compu­ tational Structures, Academic Press, 1974, supplies wide-ranging coverage of the topic, with emphasis on graph theory, including exercises in aspects of programming. • Levy, L. S., Discrete Structures of Computer Science, Wiley, 1980, pro­ vides a brief and concise treatment of the subject, with applications. • Liu, C. L., Elements of Discrete Mathematics, McGraw-Hill, 1977, covers set theory, graphs and trees, algebraic structures, and combina­ torics, including recurrence relations, in an introductory level text. • Manna, Z., Mathematical Theory of Computation, McGraw-Hill, 1974, uses mathematical reasoning as a basis for a formal treatment of computation. • Mott, J. L., Kandel, Α., and Baker, T. P . , Discrete Mathematics for Computer Scientists, Reston, 1983, is an undergraduate text that treats combinatorics and graph theoretical concepts with a mathematical flavor.

VARANASI A N D GARCIA—DISCRETE MATHEMATICS FOR COMPUTER SCIENCE

• Prather, R. E., Discrete Mathema­ tical Structures for Computer Sci­ ence, Houghton-Mifflin, 1976, inte­ grates algebraic concepts with an informal introduction to algorithms in a fairly comprehensive fashion. • Preparata, F. P., and Yeh, R. T., Introduction to Discrete Structures, Addison-Wesley, 1973, presents a complete coverage of the subject, in­ cluding extensive bibliography, his­ torical notes, and applications. • Stanat, D. E., and McAllister, D. F., Discrete Mathematics in Com­ puter Science, Prentice-Hall, 1977, provides mathematically sound cov­ erage of most subject areas of discrete mathematics. • Stone, H. S., Discrete Mathemati­ cal Structures, Science Research As­ sociates, 1973, is an undergraduate text that surveys algebraic structures and their applications to computers. • Tremblay, J. P., and Mahohar, R. P., Discrete Mathematical Struc­ tures with Applications to Computer Science, McGraw-Hill, 1975, offers a detailed treatment of mathematical reasoning, set theory, algebraic struc­ tures, and graphs and trees in a math­ ematically precise manner. These texts, and other noteworthy explorations of discrete mathematics, can assist in the proper blending of key concepts with interesting applica­ tions, thereby enabling students to appreciate the importance of these mathematical tools for analysis and design of computer hardware and software. About the authors

Murali R. Varanasi is on the facul­ ty of the department of computer sci­ ence and engineering at the Universi­ ty of South Florida, Tampa. Dr. Varanasi has been active in the Edu­ cational Activities Board of the IEEE Computer Society over the last sev­ eral years. Oscar N. Garcia is professor and chairman of the department of com­ puter science and engineering at the University of South Florida, Tampa. Dr. Garcia is a fellow of the IEEE and was president of the IEEE Computer Society from 1982 through 1983. Drs. Varanasi and Garcia based this article on the recommendations of the Committee on Computer Sci­ ences in Electrical Engineering of the National Academy of Engineering, the Curriculum Committee on Com­ puter Science of the ACM, and the Educational Activities Board of the IEEE Computer Society. •

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