Conjecture and Proof

Első borító
Cambridge University Press, 2001. jan. 1. - 118 oldal
0 Ismertetők
The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.
  

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Tartalomjegyzék

Proofs of Impossibility Proofs of Nonexistence
1
Proofs of Irrationality
3
The Elements of the Theory of Geometric Constructions
11
Constructible Regular Polygons
17
Some Basic Facts About Linear Spaces and Fields
21
Algebraic and Transcendental Numbers
27
Cauchys Functional Equation
33
Geometric Decompositions
39
Countable and Uncountable Sets
59
Isometries of R
67
The Problem of Invariant Measures
75
The BanachTarski Paradox
81
Open and Closed Sets in R The Cantor Set
85
The Peano Curve
93
Borel Sets
97
The Diagonal Method
103

Constructions Proofs of Existence
47
The Pigeonhole Principle
49
Liouville Numbers
55
References
107
Copyright

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Korlátozott előnézet - 2005

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