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Note for Discrete Mathematics - DMS by Masti adda With vikas

  • Discrete Mathematics - DMS
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  • Future/AKTU - F.I.E.T
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Masti Adda With Vikas
Masti Adda With Vikas
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Contents 1 Basic Set Theory 1.1 1.2 1.3 5 Basic Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Union and Intersection of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Set Difference, Set Complement and the Power Set . . . . . . . . . . . . . . . . 8 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.2 Equivalence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Advanced topics in Set Theory and Relations∗ . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 Families of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.2 More on Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Peano Axioms and Countability 23 2.1.1 Addition, Multiplication and its properties . . . . . . . . . . . . . . . . . . . . 24 2.1.2 Well Ordering in N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Countable and Uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 Cantor’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Creating Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Schr¨ oder-Bernstein Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Integers and Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 2.5 AF T Peano Axioms and the set of Natural Numbers . . . . . . . . . . . . . . . . . . . . . . DR 2.1 23 Construction of Integers and Rationals∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.1 Construction of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.2 Construction of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Partial Orders, Lattices and Boolean Algebra 57 3.1 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Basic Counting 4.1 77 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.1 Multinomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Circular Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Solutions in Non-negative Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Set Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5 Lattice Paths and Catalan Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3

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4 CONTENTS 4.6 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Advanced Counting Principles 5.1 Pigeonhole Principle . . . . . . . . . . . . . . . 5.2 Principle of Inclusion and Exclusion . . . . . . 5.3 Generating Functions . . . . . . . . . . . . . . . 5.4 Recurrence Relation . . . . . . . . . . . . . . . 5.5 Generating Function from Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 101 101 104 107 116 119 6 Introduction to Logic 127 6.1 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 Predicate Logic∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T . . . . . . . . . . . . . . . AF . . . . . . . . . . . . . . . DR 7 Graphs 7.1 Basic Concepts . . . . . . . . . . . 7.2 Connectedness . . . . . . . . . . . 7.3 Isomorphism in Graphs . . . . . . 7.4 Trees . . . . . . . . . . . . . . . . . 7.5 Connectivity . . . . . . . . . . . . 7.6 Eulerian Graphs . . . . . . . . . . 7.7 Hamiltonian Graphs . . . . . . . . 7.8 Bipartite Graphs . . . . . . . . . . 7.9 Matching in Graphs . . . . . . . . 7.10 Ramsey Numbers . . . . . . . . . . 7.11 Degree Sequence . . . . . . . . . . 7.12 Planar Graphs . . . . . . . . . . . 7.13 Vertex Coloring . . . . . . . . . . . 7.14 Representing graphs with Matrices 7.14.1 More Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 145 151 154 156 161 163 166 169 170 173 174 175 178 179 180 184

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Chapter 1 Basic Set Theory We will use the following notation throughout the book. 1. The empty set, denoted ∅, is the set that has no element. 2. N := {1, 2, . . .}, the set of Natural numbers; 3. W := {0, 1, 2, . . .}, the set of whole numbers 4. Z := {. . . , −2, −1, 0, 1, 2, . . .}, the set of Integers; 5. Q := { pq : p, q ∈ Z, q 6= 0}, the set of Rational numbers; AF DR 7. C := the set of Complex numbers. T 6. R := the set of Real numbers; and For the sake of convenience, we have assumed that the integer 0, is also a natural number. This chapter will be devoted to understanding set theory, relations, functions and the principle of mathematical induction. We start with basic set theory. 1.1 Basic Set Theory Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment that the set received happened only in the 19th century due to the german mathematician Georg Cantor. He was the first person who was responsible in ensuring that the set had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. He developed the transfinite numbers of which the ordinals and cardinals are two types. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russell’s Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [8]. In this book, we will consider the intuitive or naive view point of sets. The notion of a set is taken as a primitive and so we will not try to define it explicitly. On the contrary, we will give it an informal description and then go on to establish the properties of a set. 5

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6 CHAPTER 1. BASIC SET THEORY A set can be described intuitively as a collection of distinct objects. The objects are called the elements or members of the set. Here, we will be able to say when an object/element belongs to a set or not. The objects can be just about anything from real physical things to abstract mathematical objects. The principal, distinguishable and an important feature of a set is that the objects are “distinct” or “uniquely identifiable.” Any object of the collection comprising a set is referred as an element of the set. So, if S is a set and x is an element of S, we denote it by x ∈ S. If x is not an element of S, we denote it by x 6∈ S. A set is typically denoted by curly braces, { }. Example 1.1.1. 1. X = {apple, tomato, orange}. Hence, orange ∈ X, but potato 6∈ X. 2. X = {a1 , a2 , . . . , a10 }. Then, a100 6∈ X. 3. Observe that the sets {1, 2, 3}, {3, 1, 2} and {digits in the number12321} are the same as the order in which the elements appear doesn’t matter. We now address the idea of distinctness of elements of a set, which comes with its own subtleties. Example 1.1.2. 1. Consider a collection of identical red balls in a basket. Is it a set? 2. Consider the list of digits 1, 2, 1, 4, 2. Is it a set? 3. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then X is the set of first 10 natural numbers. Or equivalently, X is the set of integers between 0 and 11. The set S that contains no element is called the empty set or the T Definition 1.1.3. [Empty Set] null set denoted by { } or ∅. DR AF An object x is an element or a member of a set S, written x ∈ S, if x satisfies the rule that defines the membership for S. With this notation, one has three main ways for specifying a set. They are: 1. Listing all its elements (list notation), e.g., X = {2, 4, 6, 8, 10}. Then X is the set of even integers between 0 and 12. 2. Stating a property with notation (predicate notation), e.g., (a) X = {x : x is a prime number}. This is read as “X is the set of all x such that x is a prime number”. Here x is a variable and stands for any object that meets the criteria after the colon. (b) The set X = {2, 4, 6, 8, 10} in the predicate notation can be written as i. X = {x : 0 < x ≤ 10, x is an even integer }, or ii. X = {x : 1 < x < 11, x is an even integer }, or iii. x = {x : 2 ≤ x ≤ 10, x is an even integer } etc. (c) X = {x : x is a student in IITK and x is older than 30}. Note that the above expressions are certain rules that help in defining the elements of the set X. In general, one writes X = {x : p(x)} or X = {x | p(x)} to denote the set of all elements x (variable) such that property p(x) holds. In the above note that “colon” is sometimes replaced by “—”. 3. Defining a set of rules which generate its members (recursive notation), e.g., let X = {x : x is an even integer greater than 3}. Then, X can also be written as (a) 4 ∈ X. (b) whenever x ∈ X then x + 2 ∈ X. (c) no other element different from those above belongs to X.

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