Discrete Mathematics About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities. This tutorial explains the fundamental concepts of Sets, Relations and Functions, Mathematical Logic, Group theory, Counting Theory, Probability, Mathematical Induction and Recurrence Relations, Graph Theory, Trees and Boolean Algebra. Audience This tutorial has been prepared for students pursuing a degree in any field of computer science and mathematics. It endeavors to help students grasp the essential concepts of discrete mathematics. Prerequisites This tutorial has an ample amount of both theory and mathematics. The readers are expected to have a reasonably good understanding of elementary algebra and arithmetic. 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If you discover any errors on our website or in this tutorial, please notify us at [email protected] i Discrete Mathematics Table of Contents About the Tutorial ............................................................................................................................................ i Audience ........................................................................................................................................................... i Prerequisites ..................................................................................................................................................... i Copyright & Disclaimer ..................................................................................................................................... i Table of Contents ............................................................................................................................................ ii 1. Discrete Mathematics – Introduction ........................................................................................................ 1 PART 1: SETS, RELATIONS, AND FUNCTIONS ............................................................................... 2 2. Sets ........................................................................................................................................................... 3 Set – Definition ................................................................................................................................................ 3 Representation of a Set ................................................................................................................................... 3 Cardinality of a Set .......................................................................................................................................... 4 Types of Sets .................................................................................................................................................... 5 Venn Diagrams ................................................................................................................................................ 7 Set Operations ................................................................................................................................................. 7 Power Set ........................................................................................................................................................ 9 Partitioning of a Set ......................................................................................................................................... 9 3. Relations ................................................................................................................................................. 11 Definition and Properties .............................................................................................................................. 11 Domain and Range ........................................................................................................................................ 11 Representation of Relations using Graph ...................................................................................................... 11 Types of Relations ......................................................................................................................................... 12 4. Functions ................................................................................................................................................ 14 Function – Definition ..................................................................................................................................... 14 Injective / One-to-one function ..................................................................................................................... 14 Surjective / Onto function ............................................................................................................................. 14 Bijective / One-to-one Correspondent .......................................................................................................... 14 Inverse of a Function ..................................................................................................................................... 15 Composition of Functions .............................................................................................................................. 15 PART 2: MATHEMATICAL LOGIC ................................................................................................ 17 5. Propositional Logic .................................................................................................................................. 18 Prepositional Logic – Definition ..................................................................................................................... 18 Connectives ................................................................................................................................................... 18 Tautologies .................................................................................................................................................... 20 Contradictions ............................................................................................................................................... 20 Contingency ................................................................................................................................................... 20 Propositional Equivalences ............................................................................................................................ 21 Inverse, Converse, and Contra-positive ......................................................................................................... 22 Duality Principle ............................................................................................................................................. 22 Normal Forms ................................................................................................................................................ 22 ii Discrete Mathematics 6. Predicate Logic ........................................................................................................................................ 24 Predicate Logic – Definition ........................................................................................................................... 24 Well Formed Formula .................................................................................................................................... 24 Quantifiers ..................................................................................................................................................... 24 Nested Quantifiers ........................................................................................................................................ 25 7. Rules of Inference ................................................................................................................................... 26 What are Rules of Inference for? .................................................................................................................. 26 Table of Rules of Inference ............................................................................................................................ 26 Addition ......................................................................................................................................................... 27 Conjunction ................................................................................................................................................... 27 Simplification ................................................................................................................................................. 27 Modus Ponens ............................................................................................................................................... 27 Modus Tollens ............................................................................................................................................... 28 Disjunctive Syllogism ..................................................................................................................................... 28 Hypothetical Syllogism .................................................................................................................................. 28 Constructive Dilemma ................................................................................................................................... 29 Destructive Dilemma ..................................................................................................................................... 29 PART 3: GROUP THEORY ........................................................................................................... 30 8. Operators and Postulates ....................................................................................................................... 31 Closure ........................................................................................................................................................... 31 Associative Laws ............................................................................................................................................ 31 Commutative Laws ........................................................................................................................................ 32 Distributive Laws ........................................................................................................................................... 32 Identity Element ............................................................................................................................................ 32 Inverse ........................................................................................................................................................... 33 De Morgan’s Law ........................................................................................................................................... 33 9. Group Theory .......................................................................................................................................... 34 Semigroup ..................................................................................................................................................... 34 Monoid .......................................................................................................................................................... 34 Group ............................................................................................................................................................. 34 Abelian Group ................................................................................................................................................ 35 Cyclic Group and Subgroup ........................................................................................................................... 35 Partially Ordered Set (POSET) ........................................................................................................................ 36 Hasse Diagram ............................................................................................................................................... 37 Linearly Ordered Set ...................................................................................................................................... 37 Lattice ............................................................................................................................................................ 38 Properties of Lattices ..................................................................................................................................... 39 Dual of a Lattice ............................................................................................................................................. 39 PART 4: COUNTING & PROBABILITY .......................................................................................... 40 10. Counting Theory ..................................................................................................................................... 41 The Rules of Sum and Product ...................................................................................................................... 41 Permutations ................................................................................................................................................. 41 Combinations ................................................................................................................................................ 43 Pascal's Identity ............................................................................................................................................. 44 Pigeonhole Principle ...................................................................................................................................... 44 The Inclusion-Exclusion principle .................................................................................................................. 45 iii Discrete Mathematics 11. Probability .............................................................................................................................................. 46 Basic Concepts ............................................................................................................................................... 46 Probability Axioms ......................................................................................................................................... 47 Properties of Probability................................................................................................................................ 48 Conditional Probability .................................................................................................................................. 48 Bayes' Theorem ............................................................................................................................................. 49 PART 5: MATHEMATICAL INDUCTION & RECURRENCE RELATIONS ........................................... 51 12. Mathematical Induction .......................................................................................................................... 52 Definition ....................................................................................................................................................... 52 How to Do It .................................................................................................................................................. 52 Strong Induction ............................................................................................................................................ 54 13. Recurrence Relation ................................................................................................................................ 55 Definition ....................................................................................................................................................... 55 Linear Recurrence Relations .......................................................................................................................... 55 Non-Homogeneous Recurrence Relation and Particular Solutions ............................................................... 57 Generating Functions .................................................................................................................................... 59 PART 6: DISCRETE STRUCTURES ................................................................................................ 60 14. Graph and Graph Models ........................................................................................................................ 61 What is a Graph? ........................................................................................................................................... 61 Types of Graphs ............................................................................................................................................. 62 Representation of Graphs ............................................................................................................................. 66 Planar vs. Non-planar Graph ......................................................................................................................... 68 Isomorphism .................................................................................................................................................. 68 Homomorphism............................................................................................................................................. 69 Euler Graphs .................................................................................................................................................. 69 Hamiltonian Graphs ....................................................................................................................................... 70 15. More on Graphs ...................................................................................................................................... 71 Graph Coloring .............................................................................................................................................. 71 Graph Traversal ............................................................................................................................................. 72 16. Introduction to Trees .............................................................................................................................. 77 Tree and its Properties .................................................................................................................................. 77 Centers and Bi-Centers of a Tree ................................................................................................................... 77 Labeled Trees ................................................................................................................................................ 80 Unlabeled trees ............................................................................................................................................. 80 Rooted Tree ................................................................................................................................................... 81 Binary Search Tree ......................................................................................................................................... 82 17. Spanning Trees........................................................................................................................................ 84 Minimum Spanning Tree ............................................................................................................................... 85 Kruskal's Algorithm ........................................................................................................................................ 85 Prim's Algorithm ............................................................................................................................................ 89 iv Discrete Mathematics PART 7: BOOLEAN ALGEBRA ..................................................................................................... 92 18. Boolean Expressions and Functions ........................................................................................................ 93 Boolean Functions ......................................................................................................................................... 93 Boolean Expressions ...................................................................................................................................... 93 Boolean Identities.......................................................................................................................................... 93 Canonical Forms ............................................................................................................................................ 94 Logic Gates .................................................................................................................................................... 96 19. Simplification of Boolean Functions ........................................................................................................ 99 Simplification Using Algebraic Functions ....................................................................................................... 99 Karnaugh Maps ............................................................................................................................................ 100 Simplification Using K- map ......................................................................................................................... 101 v 1. Discrete Mathematics – IntroDdisucrectet Mioathnem atics Mathematics can be broadly classified into two categories:  Continuous Mathematics ─ It is based upon continuous number line or the real numbers. It is characterized by the fact that between any two numbers, there are almost always an infinite set of numbers. For example, a function in continuous mathematics can be plotted in a smooth curve without breaks.  Discrete Mathematics ─ It involves distinct values; i.e. between any two points, there are a countable number of points. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, and can be presented as a complete list of those pairs. Topics in Discrete Mathematics Though there cannot be a definite number of branches of Discrete Mathematics, the following topics are almost always covered in any study regarding this matter:  Sets, Relations and Functions  Mathematical Logic  Group theory  Counting Theory  Probability  Mathematical Induction and Recurrence Relations  Graph Theory  Trees  Boolean Algebra We will discuss each of these concepts in the subsequent chapters of this tutorial. 6 Discrete Mathematics Part 1: Sets, Relations, and Functions 7 2. Sets Discrete Mathematics German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. In this chapter, we will cover the different aspects of Set Theory. Set – Definition A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set. Some Example of Sets  A set of all positive integers  A set of all the planets in the solar system  A set of all the states in India  A set of all the lowercase letters of the alphabet Representation of a Set Sets can be represented in two ways:  Roster or Tabular Form  Set Builder Notation Roster or Tabular Form The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas. Example 1: Set of vowels in English alphabet, A = {a,e,i,o,u} Example 2: Set of odd numbers less than 10, B = {1,3,5,7,9} 8 Discrete Mathematics Set Builder Notation The set is defined by specifying a property that elements of the set have in common. The set is described as A = { x : p(x)} Example 1: The set {a,e,i,o,u} is written as: A = { x : x is a vowel in English alphabet} Example 2: The set {1,3,5,7,9} is written as: B = { x : 1≤x<10 and (x%2) ≠ 0} If an element x is a member of any set S, it is denoted by x∈ S and if an element y is not a member of set S, it is denoted by y ∉ S. Example: If S = {1, 1.2,1.7,2}, 1∈ S but 1.5 ∉S Some Important Sets N: the set of all natural numbers = {1, 2, 3, 4, .....} Z: the set of all integers = {....., -3, -2, -1, 0, 1, 2, 3, .....} Z+: the set of all positive integers Q: the set of all rational numbers R: the set of all real numbers W: the set of all whole numbers Cardinality of a Set Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Example: |{1, 4, 3,5}| = 4, |{1, 2, 3,4,5,…}| = ∞ If there are two sets X and Y,  |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.  | X| ≤ | Y | denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y. 9