# DISCRETE MATHEMATICAL STRUCTURES (18CS36)

## STRUCTURES

#### Course Learning Objectives: This course (18CS36) will enable students to:

Provide theoretical foundations of computer science to perceive other courses in the programme.
• Illustrate applications of discrete structures: logic, relations, functions, set theory and counting.
Describe different mathematical proof techniques,
Illustrate the use of graph theory in computer science

### Module 1

Fundamentals of Logic: Basic Connectives and Truth Tables, Logic Equivalence – The Laws of Logic, Logical Implication – Rules of Inference. Fundamentals of Logic contd.: The Use of Quantifiers, Quantifiers, Definitions and the Proofs of Theorems.
Text book 1: Chapter2 ### Module 2

Properties of the Integers: The Well Ordering Principle – Mathematical Induction,
Fundamental Principles of Counting: The Rules of Sum and Product, Permutations, Combinations – The Binomial Theorem, Combinations with Repetition.
Text book 1: Chapter4 – 4.1, Chapter1 ### Module 3

Relations and Functions: Cartesian Products and Relations, Functions – Plain and One-to-One, Onto Functions. The Pigeon-hole Principle, Function Composition and Inverse Functions.
Relations: Properties of Relations, Computer Recognition – Zero-One Matrices and Directed Graphs, Partial Orders – Hasse Diagrams, Equivalence Relations and Partitions.

Text book 1: Chapter5, Chapter7 – 7.1 to 7.4 ### Module 4

The Principle of Inclusion and Exclusion: The Principle of Inclusion and Exclusion, Generalizations of the Principle, Derangements – Nothing is in its Right Place, Rook Polynomials.
Recurrence Relations: First Order Linear Recurrence Relation, The Second Order Linear Homogeneous Recurrence Relation with Constant Coefficients.
Text book 1: Chapter8 – 8.1 to 8.4, Chapter10 – 10.1, 10.2 ### Module 5

Introduction to Graph Theory: Definitions and Examples, Sub graphs, Complements, and Graph Isomorphism,
Trees: Definitions, Properties, and Examples, Routed Trees, Trees and Sorting, Weighted Trees and Prefix Codes
Text book 1: Chapter11 – 11.1 to 11.2 Chapter12 – 12.1 to 12.4 #### Course Outcomes: The student will be able to :

Use propositional and predicate logic in knowledge representation and truth verification.
Demonstrate the application of discrete structures in different fields of computer science.
Solve problems using recurrence relations and generating functions.
Application of different mathematical proofs techniques in proving theorems in the courses.
• Compare graphs, trees and their applications.

#### Question Paper Pattern:

The question paper will have ten questions.
Each full Question consisting of 20 marks
There will be 2 full questions (with a maximum of four sub questions) from each module.
Each full question will have sub questions covering all the topics under a module.
The students will have to answer 5 full questions, selecting one full question from each module.

#### Question Bank: #### Textbooks:

1. Ralph P. Grimaldi: Discrete and Combinatorial Mathematics, 5th Edition, Pearson Education. 2004. ### Reference Books:

1. Basavaraj S Anami and Venakanna S Madalli: Discrete Mathematics – A Concept based approach, Universities Press, 2016
2. Kenneth H. Rosen: Discrete Mathematics and its Applications, 6th Edition, McGraw Hill, 2007. 3. Jayant Ganguly: A Treatise on Discrete Mathematical Structures, Sanguine-Pearson, 2010.
4. D.S. Malik and M.K. Sen: Discrete Mathematical Structures: Theory and Applications, Thomson, 2004.
5. Thomas Koshy: Discrete Mathematics with Applications, Elsevier, 2005, Reprint 2008.