MATHEMATICS: PROOFS, GEOMETRY AND DISCRETE MATHEMATICS

By I. NANID

CONTENTS CHAPTER 1: LOGIC PROPOSITIONS AND TRUTH VALUES LOGICAL CONNECTIVES AND TRUTH TABLES TAUTOLOGIES AND CONTRADICTIONS LOGICAL EQUIVALENCE AND LOGICAL IMPLICATION THE ALGEBRA OF PROPOSITIONS ARGUMENTS FORMAL PROOF OF THE VALIDITY OF ARGUMENTS PREDICATE LOGIC ARGUMENTS IN PREDICATE LOGIC CHAPTER 2: MATHEMATICAL PROOF THE NATURE OF PROOF AXIOMS AND AXIOM SYSTEMS METHODS OF PROOF MATHEMATICAL INDUCTION CHAPTER 3: SETS SETS AND MEMBERSHIP SUBSETS OPERATIONS ON SETS COUNTING TECHNIQUES THE ALGEBRA OF SETS FAMILIES OF SETS THE CARTESIAN PRODUCT TYPES AND TYPED SET THEORY CHAPTER 4: RELATIONS RELATIONS AND THEIR REPRESENTATIONS PROPERTIES OF RELATIONS INTERSECTIONS AND UNIONS OF RELATIONS EQUIVALENCE RELATIONS AND PARTITIONS ORDER RELATIONS HASSE DIAGRAMS APPLICATION: RELATIONAL DATABASES CHAPTER 5: FUNCTIONS DE?NITIONS AND EXAMPLES COMPOSITE FUNCTIONS INJECTIONS AND SURJECTIONS BIJECTIONS AND INVERSE FUNCTIONS MORE ON CARDINALITY DATABASES: FUNCTIONAL DEPENDENCE AND NORMAL FORMS CHAPTER 6: MATRIX ALGEBRA INTRODUCTION SOME SPECIAL MATRICES OPERATIONS ON MATRICES ELEMENTARY MATRICES THE INVERSE OF A MATRIX CHAPTER 7: SYSTEMS OF LINEAR EQUATIONS MATRIX INVERSE METHOD GAUSS–JORDAN ELIMINATION GAUSSIAN ELIMINATION CONTENTS CHAPTER 8: ALGEBRAIC STRUCTURES BINARY OPERATIONS AND THEIR PROPERTIES ALGEBRAIC STRUCTURES MORE ABOUT GROUPS SOME FAMILIES OF GROUPS SUBSTRUCTURES MORPHISMS GROUP CODES CHAPTER 9: INTRODUCTION TO NUMBER THEORY DIVISIBILITY PRIME NUMBERS LINEAR CONGRUENCES GROUPS IN MODULAR ARITHMETIC PUBLIC KEY CRYPTOGRAPHY CHAPTER 10: BOOLEAN ALGEBRA 101 INTRODUCTION 102 PROPERTIES OF BOOLEAN ALGEBRAS 103 BOOLEAN FUNCTIONS 104 SWITCHING CIRCUITS 105 LOGIC NETWORKS 106 MINIMIZATION OF BOOLEAN EXPRESSIONS CONTENTS CHAPTER 11: GRAPH THEORY 111 DE?NITIONS AND EXAMPLES 112 PATHS AND CYCLES 113 ISOMORPHISM OF GRAPHS 114 TREES 115 PLANAR GRAPHS 116 DIRECTED GRAPHS CHAPTER 12: APPLICATIONS OF GRAPH THEORY 121 INTRODUCTION 122 ROOTED TREES 123 SORTING 124 SEARCHING STRATEGIES 125 WEIGHTED GRAPHS 126 THE SHORTEST PATH AND TRAVELLING SALESMAN PROBLEMS 127 NETWORKS AND FLOWS REFERENCES AND FURTHER READING HINTS AND SOLUTIONS TO SELECTED EXERCISES