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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