Fundamental Concepts of Mathematics

Table of Contents

CONTENTS CONTENTS PREFACE 1. SET NOTATION Objects. Sets. Subsets. Rule of specification. Complements. Intersection. Union. Ordered pairs and Cartesian product. One-to-one correspondence. Mappings. v vii 2. MATHEMATICAL INDUCTION 37 A proof by induction. The well-ordering principle. The principle of mathematical induction. Miscellaneous remarks. Another version of the principle of mathematical induction. Recursive formulae. 3. COMBINATORICS 61 Boxes and balls. Remarks. Permutations. Permutations in which repetitions are allowed. Permutations of objects some of which are alike. Circular permutations. Combinations. Combinations with repetitions. Binomial theorem. 4. ARITHMETIC 94 Absolute value. Divisibility. Euclidean algorithm. The greatest common divisor. The least common multiple. An effective division algorithm for the evaluation of gcd. Prime numbers. The fundamental theorem of arithmetic. The infinity of prime numbers. Congruence. Chinese remainder theorem. 5. THE REAL NUMBERS 125 The number line. Some basic assumptions. Some wellknown inequalities. Denseness of the rational numbers. Postulate of continuity. Powers and roots. Existence of roots. Powers and logarithm. 6. LIMIT AND CONVERGENCE 157 Null sequence. Convergent sequence. Divergent sequence. Sum, product and quotient of convergent sequences. The sandwich theorem. Monotone vi Contents sequence. Cauchy's convergence test. Series. Geometric series and harmonic series. Some useful rules. Test of convergence. Appendix. 7. COMPLEX NUMBERS 193 Equations and number systems. One-dimensional number system. Two-dimensional number system. Complex numbers. Standard notations. Complex conjugate. Equations with real coefficients. De Moivre's theorem. The n-th roots. Geometry of complex numbers. Circles. Straight lines. Appendix. ANSWERS TO EXERCISES INDEX 243 253 ...