COMPREHENSIVE MATHEMATICS

By W. JAIND

CONTENTS I SETS, NUMBERS, AND GRAPHS FUNDAMENTALS—CONCEPTS AND LOGIC PROPOSITIONAL LOGIC ARCHITECTURE OF CONCEPTS AXIOMATIC SET THEORY THE AXIOMS BASIC CONCEPTS AND RESULTS BOOLEAN SET ALGEBRA THE BOOLEAN ALGEBRA OF SUBSETS FUNCTIONS AND RELATIONS GRAPHS AND FUNCTIONS RELATIONS ORDINAL AND NATURAL NUMBERS ORDINAL NUMBERS NATURAL NUMBERS RECURSION THEOREM AND UNIVERSAL PROPERTIES RECURSION THEOREM UNIVERSAL PROPERTIES UNIVERSAL PROPERTIES IN RELATIONAL DATABASE THEORY NATURAL ARITHMETIC NATURAL OPERATIONS EUCLID AND THE NORMAL FORMS IN?NITIES THE DIAGONALIZATION PROCEDURE THE CLASSICAL NUMBER DOMAINS Z, Q, R, AND C INTEGERS Z RATIONALS Q REAL NUMBERS R COMPLEX NUMBERS C 10 CATEGORIES OF GRAPHS 10 1 DIRECTED AND UNDIRECTED GRAPHS 10 2 MORPHISMS OF DIGRAPHS AND GRAPHS 10 3 CYCLES 11 CONSTRUCTION OF GRAPHS 12 SOME SPECIAL GRAPHS 12 1 N-ARY TREES 12 2 MOORE GRAPHS 13 PLANARITY 13 1 EULER’S FORMULA FOR POLYHEDRA 13 2 KURATOWSKI’S PLANARITY THEOREM 14 FIRST ADVANCED TOPIC 14 1 FLOATING POINT ARITHMETIC 14 2 EXAMPLE FOR AN ADDITION II ALGEBRA, FORMAL LOGIC, AND LINEAR GEOMETRY 15 MONOIDS, GROUPS, RINGS, AND FIELDS 15 1 MONOIDS 15 2 GROUPS 15 3 RINGS 15 4 FIELDS 16 PRIMES 16 1 PRIME FACTORIZATION 16 2 ROOTS OF POLYNOMIALS AND INTERPOLATION 17 FORMAL PROPOSITIONAL LOGIC 17 1 SYNTACTICS: THE LANGUAGE OF FORMAL PROPOSITIONAL LOGIC 17 2 SEMANTICS: LOGICAL ALGEBRAS 17 3 SIGNI?CATION: VALUATIONS 17 4 AXIOMATICS 18 FORMAL PREDICATE LOGIC 18 1 SYNTACTICS: FIRST-ORDER LANGUAGE 18 2 SEMANTICS: ?-STRUCTURES 18 3 SIGNI?CATION: MODELS 19 LANGUAGES, GRAMMARS, AND AUTOMATA 19 1 LANGUAGES 19 2 GRAMMARS 19 3 AUTOMATA AND ACCEPTORS 20 CATEGORIES OF MATRIXES 20 1 WHAT MATRIXES ARE 20 2 STANDARD OPERATIONS ON MATRIXES 20 3 SQUARE MATRIXES AND THEIR DETERMINANT 21 MODULES AND VECTOR SPACES 22 LINEAR DEPENDENCE, BASES, AND DIMENSION 22 1 BASES IN VECTOR SPACES 22 2 EQUATIONS 22 3 A?NE HOMOMORPHISMS 23 ALGORITHMS IN LINEAR ALGEBRA 23 1 GAUSS ELIMINATION 23 2 THE LUP DECOMPOSITION 24 LINEAR GEOMETRY 24 1 EUCLIDEAN VECTOR SPACES 24 2 TRIGONOMETRIC FUNCTIONS FROM TWO-DIMENSIONAL ROTATIONS 24 3 GRAM’S DETERMINANT AND THE SCHWARZ INEQUALITY 25 EIGENVALUES, THE VECTOR PRODUCT, AND QUATERNIONS 25 1 EIGENVALUES AND ROTATIONS 25 2 THE VECTOR PRODUCT 25 3 QUATERNIONS 26 SECOND ADVANCED TOPIC 26 1 GALOIS FIELDS 26 2 THE REED-SOLOMON (RS) ERROR CORRECTION CODE 26 3 THE RIVEST-SHAMIR-ADLEMAN (RSA) ENCRYPTION ALGORITHM A FURTHER READING