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Note for Discrete Mathematics - DMS By Ayush Agrawal

  • Discrete Mathematics - DMS
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Ayush Agrawal
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The objectives of Discrete Mathematical Structures are: • To introduce a number of Discrete Mathematical Structures (DMS) found to be serving as tools even today in the development of theoretical computer science. • Course focuses on of how Discrete Structures actually helped computer engineers to solve problems occurred in the development of programming languages. • Also, course highlights the importance of discrete structures towards simulation of a problem in computer science and engineering. • Introduction of a number of case studies involving problems of Computer Technology. Outcomes of this course are: • A complete knowledge on various discrete structures available in literature. • Realization of some satisfaction of having learnt that discrete structures are indeed useful in computer science and engineering and thereby concluding that no mistake has been done in studying this course. • Gaining of some confidence on how to deal with problems which may arrive in computer science and engineering in near future. • Above all, students who studied this course are found to be better equipped in a relative sense as far as preparation for entrance examinations involving placement opportunities. What is Discrete Mathematics then? • Mathematics is broadly divided into two parts; (i) the continuous mathematics and (ii) the discrete mathematics depending upon the presence or absence of the limiting processes. • In the case of continuum Mathematics, there do exists some relationship / linkage between various topics whereas Discrete Mathematics is concerned with study of distinct, or different, or un-related topics of mathematics curriculum; it embraces

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several topical areas of mathematics some of which go back to early stages of mathematical development while others are more recent additions to the discipline. The present course restricts only to introducing discrete structures which are being used as tools in theoretical computer science. • A course on Discrete mathematics includes a number of topics such as study of sets, functions and relations, matrix theory, algebra, Combinatorial principles and discrete probability, graph theory, finite differences and recurrence relations, formal logic and predicate calculus, proof techniques - mathematical induction, algorithmic thinking, Matrices, Primes, factorization, greatest common divisor, residues and application to cryptology, Boolean algebra; Permutations, combinations and partitions; Recurrence relations and generating functions; Introduction to error-correcting codes; Formal languages and grammars, finite state machines. linear programming etc. Also, few computer science subjects such as finite automata languages, data structures, logic design, algorithms and analysis were also viewed as a part of this course. • Because of the diversity of the topics, it is perhaps preferable to treat Discrete Mathematics, simply as Mathematics that is necessary for decision making in noncontinuous situations. For these reasons, we advise students of CSE / ISE / MCA, TE (Telecommunication Engineering) to study this course, as they needs to know the procedure of communicating with a computer may be either as a designer, programmer, or, at least a user. • Of course, in today’s situation, this is true for all, although we do not teach to students of other branches of engineering. In some autonomous engineering colleges, DMS is being offered s an elective. Considering these view points, you are informed to undertake a course on Discrete Mathematical Structures so that you will be able to function as informed citizens of an increasingly technological society. • Also, Discrete Mathematics affords students, a new opportunity to experience success and enjoyment in Mathematics classes. If you have encountered numerous difficulties with computation and the complexities of Mathematics in the past, then may I say that this course is soft and a study requires very few formal skills as prerequisites.

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• In case if you are discouraged by the routine aspects of learning Mathematics, Discrete Structures provides you a unique opportunity to learn Mathematics in a much different way than the one employed by your teachers previously. Above all, Discrete Mathematics is vital, exciting, and no doubt is useful otherwise you would not have been suggested to register for this course. • Further, Discrete Mathematics course serves as a gateway for a number of subjects in computer science and engineering. With these motivations, here, we initiate a detailed discussion on some of the topics: These include Basic set theory, Counting techniques, Formal Logic and Predicate calculus, Relations and functions in CSE, Order relations, Groups and Coding etc. • Before, continuing, let me mention the difference between Discrete Mathematics and other Mathematics; consider a bag of apples and a piece of wire. In the former, the apples sit apart discretely from each other while in the latter, the points on a wire spread themselves continuously from one end to the other. • Thus, the numbers 0, 1, 2, 3, . . . are sufficient to handle DMS, where as a real variable taking values continuously over a range of values is required to deal with continuum Mathematics. Hence, • Discrete Mathematics + Limiting Processes = Continuum Mathematics. • Prescribed text book: • Discrete and Combinatorial Mathematics by R. P. Grimaldi, PHI publications, 5th edition (2004). • Reference Books: • Discrete Mathematical Structures by Kenneth Rosen, Tata McGraw Hill Publications • Discrete Mathematical Structures by Kolman, Busby and Ross, PHI publications

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Basic set theory A set is a well defined collection of well defined distinct objects. A set is usually denoted by using upper case letters like A, B, G, T, X etc. and arbitrary elements of the set are denoted using lower case letters such as a, b, g, v etc. Universal set: The set of all objects under some investigation is called as universe or universal set, denoted by the symbol U. Consider a set A. Let x be an element of A. This we denote symbolically by xÎ A . On the other hand, if y is not an element of the set, the same is written as y Ï A . Thus, it is clear that with respect to a set A, and an element of the universal set U, there are only two types of relationship possible; (i) the element x under question is a member of the set A or the element x need not be a member of A. This situation may well be described by using binary numbers 0 and 1. indicate that the element in question is a member of the set A. We set x : 1 to The notation x : 0 means that the element under study is not a member of the set A. There are a number of ways of do this task. (i) Writing the elements of a set within the braces. For example, consider A = {dog, apple, dead body, 5, Dr. Abdul Kalam, rose}. Certainly, A qualifies as a set. (ii) A set may be explained by means of a statement where elements satisfying some conditions. Consider V = {x | x is a Engineering College Affiliated to VTU, Belgaum} (iii) Definition of a statement may be given by means of a statement like Z denotes the set of all integers. Thus, Z = {. . . -2, -1, 0, 1, 2, 3,. . }. A null set is a one not having any elements at all. It is denoted by the symbol { } or as . Give few examples of null set or empty set. Compliment of a set: Let A be a set. The compliment of A is defined as a set containing elements of the universe but not the elements of the set A. Thus, A = { x Î U | x Ï A}

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