Discrete Mathematics

MA3354

DISCRETE MATHEMATICS

COURSE OBJECTIVES:

• To extend student’s logical and mathematical maturity and ability to deal with abstraction.

• To introduce most of the basic terminologies used in computer science courses and  application of ideas to solve practical problems.

• To understand the basic concepts of combinatorics and graph theory.

• To familiarize the applications of algebraic structures.

• To understand the concepts and significance of lattices and boolean algebra which are  widely used in computer science and engineering.

UNIT I LOGIC AND PROOFS

Propositional logic – Propositional equivalences - Predicates and quantifiers – Nested quantifiers – Rules of inference - Introduction to proofs – Proof methods and strategy. 

UNIT II COMBINATORICS

Mathematical induction – Strong induction and well ordering – The basics of counting – The  pigeonhole principle – Permutations and combinations – Recurrence relations – Solving linear  recurrence relations – Generating functions – Inclusion and exclusion principle and its applications.

UNIT III GRAPHS

Graphs and graph models – Graph terminology and special types of graphs – Matrix representation  of graphs and graph isomorphism – Connectivity – Euler and Hamilton paths.

UNIT IV ALGEBRAIC STRUCTURES

Algebraic systems – Semi groups and monoids - Groups – Subgroups – Homomorphism’s – Normal  subgroup and cosets – Lagrange’s theorem – Definitions and examples of Rings and Fields.

UNIT V LATTICES AND BOOLEAN ALGEBRA

Partial ordering – Posets – Lattices as posets – Properties of lattices - Lattices as algebraic systems  – Sub lattices – Direct product and homomorphism – Some special lattices – Boolean algebra – Sub  Boolean Algebra – Boolean Homomorphism.

COURSE OUTCOMES:

At the end of the course, students would :

CO1:Have knowledge of the concepts needed to test the logic of a program. CO2:Have an understanding in identifying structures on many levels.

CO3:Be aware of a class of functions which transform a finite set into another finite set which   relates to input and output functions in computer science. 

CO4:Be aware of the counting principles.

CO5:Be exposed to concepts and properties of algebraic structures such as groups, rings and   fields.

TEXT BOOKS:

1. Rosen. K.H., "Discrete Mathematics and its Applications", 7th Edition, Tata McGraw  Hill Pub. Co. Ltd., New Delhi, Special Indian Edition, 2017. 

2. Tremblay. J.P. and Manohar. R, "Discrete Mathematical Structures with Applications to  Computer Science", Tata McGraw Hill Pub. Co. Ltd, New Delhi, 30th Reprint, 2011.

REFERENCES:

1. Grimaldi. R.P. "Discrete and Combinatorial Mathematics: An Applied Introduction",  5thEdition, Pearson Education Asia, Delhi, 2013.

2. Koshy. T. "Discrete Mathematics with Applications", Elsevier Publications, 2006.

3. Lipschutz. S. and Mark Lipson., "Discrete Mathematics", Schaum’s Outlines, Tata McGraw  Hill Pub. Co. Ltd., New Delhi, 3rd Edition, 2010.