# Fundamental Concepts of Mathematics

Kam-Tim Leung, P.H. Cheung, "Fundamental Concepts of Mathematics"

Hong Kong University Press | 1991 | ISBN: 9622091814 | 260 pages | File type: PDF | 3,3 mb

CONTENTS

1. SET NOTATION

Objects. Sets. Subsets. Rule of specification.

Complements. Intersection. Union. Ordered pairs and

Cartesian product. One-to-one correspondence.

Mappings.

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

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

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