Note for Discrete Mathematics - DMS By Dharmendra Kumar

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S Pati DR A K Lal AF T Lecture Notes on Discrete Mathematics November 20, 2016

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2 General Instructions Throughout this book, an item with a label 2.1.3 means the 3-rd item of the 1-st section of Chapter 2. While defining a new term or a new notation we shall use bold face letters. The symbol := is used at places we are defining a term using equality symbol. A symbol !! at the end of a statement reminds the reader to verify the statement by writing a proof, if necessary. We assume that the reader is familiar with the very basic of counting. A reader who is not, may avoid the counting items in the initial parts till we start to discuss counting. We also assume that the reader is familiar with some very basic definitions involving sets. This book is written with the primary purpose of making the reader understand the discussion. We do not intend to write elaborate proofs for the reader to read, as there is no end to elaboration. We request the reader to take each statement in the book with the best possible natural meaning. Here are a few collected quotes, mainly intended to inspire the authors. Albert Einstein • The value of a college education is not the learning of many facts but the training of the mind to think. Imagination is more important than knowledge. For knowledge is limited, whereas AF T • imagination embraces the entire world, stimulating progress, giving birth to evolution. DR It is, strictly speaking, a real factor in scientific research. • Everything should be made as simple as possible, but no simpler. • Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

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Contents 1 Basic Set Theory 7 1.1 Common Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 More on the Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . 14 1.4 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Advanced Topics in Set Theory 31 Families of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 More on Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 More on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Supplying Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 DR AF T 2.1 3 Countability, cardinal numbers* and partial order 45 3.1 Countable-Uncountable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Introduction to Logic 59 4.1 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Lattices and Boolean Algebra 79 5.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6 Counting 6.1 91 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.1.1 Multinomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 Circular Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Solutions in Non-negative Integers . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Set Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.5 Lattice Paths and Catalan Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.6 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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4 CONTENTS 7 Advanced Counting Principles 117 7.1 Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2 Principle of Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.4 Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.5 Generating Function from Recurrence Relation . . . . . . . . . . . . . . . . . . . 138 8 Graphs 147 8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3 Isomorphism in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5 Connectivity 8.6 Eulerian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.7 Hamiltonian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.8 Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.9 Matching in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 AF T 8.10 Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.11 Degree Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 DR 8.12 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.13 Vertex Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.14 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.15 More Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Index 188