Discrete Mathematics

Unit I:Set Theory and Logic                                                                                              09 Hours

Discrete Mathematics, Significance of Discrete Mathematics in Computer Engineering, Sets–Naïve Set Theory (Cantorian Set Theory), Axiomatic Set Theory, Need for Sets, Representation of Sets, Set Operations, cardinality of set, principle of inclusion and exclusion, Types of Sets –Countable and Uncountable Sets, Finite and Infinite Sets, Countably Infinite and Uncountably Infinite Sets. Introduction to bounded and unbounded sets and multiset. Countability of Rational Numbers Using Cantor Diagonalization Argument, power set. Propositional Logic- logic, Propositional Equivalences, Application of Propositional Logic-Translating English Sentences, Proof by Mathematical Induction and Strong Mathematical Induction.


Unit II:Relations and Functions                                                                                         09 Hours

Relations and Their Properties, n-ary Relations and Their Applications, Representing Relations ,Closures of Relations, Equivalence Relations, Partial Orderings, partitions, Hasse Diagram,Lattices, Chains and Anti-Chains, Transitive Closure and Warshall‘s Algorithm, n-Ary Relationsand their Applications. Functions- Surjective, Injective and Bijective functions, Inverse Functions and Compositions of Functions, The Pigeonhole Principle.


Unit III:Counting                                                                                                               09 Hours

The Basics of Counting, rule of Sum and Product, Permutations and Combinations, Binomial Coefficients and Identities, Generalized Permutations and Combinations, Algorithms for generating Permutations and Combinations.


Unit IV:Graph Theory                                                                                                        09 Hours

Graphs and Graph Models, Graph Terminology and Special Types of Graphs, Representing Graphs and Graph Isomorphism, Connectivity, Euler and Hamilton Paths, Single source shortest path- Dijkstra's Algorithm, Planar Graphs, Graph Colouring. Case Study- Web Graph, Google map.


Unit V:Trees                                                                                                                     09 Hours

Introduction, properties of trees, Binary search tree, decision tree, prefix codes and Huffman coding, cut sets, Spanning Trees and Minimum Spanning Tree, Kruskal‘s and Prim‘s algorithms, The Max flow- Min Cut Theorem (Transport network). Case Study- Game Tree, Mini-Max Tree.


Unit VI:Algebraic Structures and Coding Theory                                                               09 Hours

The structure of algebra, Algebraic Systems, Semi Groups, Monoids, Groups, Homomorphism andNormal Subgroups, and congruence relations, Rings, Integral Domains and Fields, coding theory, Polynomial Rings and polynomial Codes, Case Study- Brief introduction to Galois Theory –Field Theory and Group Theory.

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