Note for Discrete Structures - DS by vtu rangers

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DISCRETE MATHEMATICAL STRUCTURES DISCRETE MATHEMATICAL STRUCTURES [As per Choice Based Credit System (CBCS) scheme] SEMESTER - III 15CS 35 Subject Code 15CS35 IA Marks 20 Number of Lecture 04 Exam Marks 80 Hours/Week Total Number of 50 Exam Hours 03 Lecture Hours Course objectives: . This course will enable students to· Prepare for a background in abstraction, notation, and critical thinking for the mathematics most directly related to computer science. · Understand and apply logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, · Understand and apply mathematical induction, combinatorics, discrete probability, recursion, sequence and recurrence, elementary number theory · Understand and apply graph theory and mathematical proof techniques. Module -1 Teaching RBT Hours Levels Set Theory: Sets and Subsets, Set Operations and the Laws of Set Theory, Counting and Venn Diagrams, A First Word on Probability, Countable and Uncountable Sets. 10 Hours L2, L3 Fundamentals of Logic: Basic Connectives and Truth Tables, Logic Equivalence –The Laws of Logic, Logical Implication – Rules of Inference. Module -2 Fundamentals of Logic contd.: The Use of Quantifiers, Quantifiers, Definitions and the Proofs of Theorems, Properties of the Integers: 10 Hours L3, L4 Mathematical Induction, The Well Ordering Principle – Mathematical Induction, Recursive Definitions Module – 3 Relations and Functions: Cartesian Products and Relations, Functions – Plain and One-to-One, Onto Functions – Stirling L3,L4, 10 Hours Numbers of the Second Kind, Special Functions, The Pigeon-hole L5 Principle, Function Composition and Inverse Functions. Module-4 Relations contd.: Properties of Relations, Computer Recognition – L3,L4, Zero-One Matrices and Directed Graphs, Partial Orders – Hasse 10 Hours L5 Diagrams, Equivalence Relations and Partitions. Module-5 Groups: Definitions, properties, Homomrphisms, Isomorphisms, Cyclic Groups, Cosets, and Lagrange’s Theorem. Coding Theory and Rings: Elements of CodingTheory, The Hamming Metric, The Parity Check, and Generator Matrices. Group Codes: Decoding with Coset Leaders, Hamming Matrices. Rings and Modular Arithmetic: The Ring Structure – Definition and Examples, Ring Properties and Substructures, The Integer modulo n. DEPT. OF CSE, SJBIT 10 Hours L3,L4, L5 Page 1

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DISCRETE MATHEMATICAL STRUCTURES 15CS 35 Course outcomes: After studying this course, students will be able to: 1. Verify the correctness of an argument using propositional and predicate logic and truth tables. 2. Demonstrate the ability to solve problems using counting techniques and combinatorics in the context of discrete probability. 3. Solve problems involving recurrence relations and generating functions. 4. Perform operations on discrete structures such as sets, functions, relations, and sequences. 5. Construct proofs using direct proof, proof by contraposition, proof by contradiction, proof by cases, and mathematical induction. Graduate Attributes (as per NBA) 1. Engineering Knowledge 2. Problem Analysis 3. Conduct Investigations of Complex Problems 4. Design/Development of Solutions Question paper pattern: The question paper will have ten questions. There will be 2 questions from each module. Each question will have questions covering all the topics under a module. The students will have to answer 5 full questions, selecting one full question from each module. Text Books: 1.Ralph P. Grimaldi: Discrete and Combinatorial Mathematics, , 5th Edition, Pearson Education. 2004. (Chapter 3.1, 3.2, 3.3, 3.4, Appendix 3, Chapter 2, Chapter 4.1, 4.2, Chapter 5.1 to 5.6, Chapter 7.1 to 7.4, Chapter 16.1, 16.2, 16.3, 16.5 to 16.9, and Chapter 14.1, 14.2, 14.3). Reference Books: 1. Kenneth H. Rosen: Discrete Mathematics and its Applications, 6th Edition, McGraw Hill, 2007. 2. JayantGanguly: A TreatCSE on Discrete Mathematical Structures, Sanguine-Pearson, 2010. 3. D.S. Malik and M.K. Sen: Discrete Mathematical Structures: Theory and Applications, Thomson, 2004. 4. Thomas Koshy: Discrete Mathematics with Applications, Elsevier, 2005, Reprint 2008. DEPT. OF CSE, SJBIT Page 2

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DISCRETE MATHEMATICAL STRUCTURES INDEX SHEET Module s Contents 15CS 35 Page No. Set Theory: Sets and Subsets, Set Operations and the Laws of Set Theory, Counting and Venn Diagrams, A First Word on Probability, Module -1 Countable and Uncountable Sets. Fundamentals of Logic: Basic Connectives and Truth Tables, Logic 04-26 Equivalence –The Laws of Logic, Logical Implication – Rules of Inference. Fundamentals of Logic contd.: The Use of Quantifiers, Quantifiers, Module -2 Definitions and the Proofs of Theorems, Properties of the Integers: Mathematical Induction, The Well Ordering Principle – Mathematical 27-40 Induction, Recursive Definitions Relations and Functions: Cartesian Products and Relations, Functions Module -3 – Plain and One-to-One, Onto Functions – Stirling Numbers of the Second Kind, Special Functions, The Pigeon-hole Principle, Function 41-68 Composition and Inverse Functions. Relations contd.: Properties of Relations, Computer Recognition – Module -4 Zero-One Matrices and Directed Graphs, Partial Orders – Hasse 69-76 Diagrams, Equivalence Relations and Partitions. Groups: Definitions, properties, Homomrphisms, Isomorphisms, Cyclic Groups, Cosets, and Lagrange’s Theorem. Coding Theory and Rings: Elements of CodingTheory, The Hamming Metric, The Parity Module -5 Check, and Generator Matrices. Group Codes: Decoding with Coset 77-114 Leaders, Hamming Matrices. Rings and Modular Arithmetic: The Ring Structure – Definition and Examples, Ring Properties and Substructures, The Integer modulo n DEPT. OF CSE, SJBIT Page 3

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DISCRETE MATHEMATICAL STRUCTURES Module 1: Set Theory: 15CS 35  Sets and Subsets,  Set Operations and the Laws of Set Theory,  Counting and Venn Diagrams,  A First Word on Probability,  Countable and  Uncountable Sets Fundamentals of Logic:  Basic Connectives and Truth Tables,  Logic Equivalence –The Laws of Logic,  Logical Implication – Rules of Inference. DEPT. OF CSE, SJBIT Page 4