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数学的基本概念(Elias Zakon)Basic Concepts of Mathematics (Elias Zakon)
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208页
这本书帮助学生完成从纯粹的操纵性到严格的数学的过渡,其主题涵盖了基本集合论,领域(着重于实数),对三维几何的回顾以及线性空间的性质。
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The Zakon Series on Mathematical Analysis
Basic Concepts of
Mathematics
Elias Zakon
University of Windsor
The Trillia Group West Lafayette, IN
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Copyright Notice
Basic Concepts of Mathematics
c
1973 Elias Zakon
c
2001 Bradley J. Lucier and Tamara Zakon
Distributed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0
International
(CC BY-NC-ND 4.0) Public License. Informally, this license allows you to:
Share: copy and redistribute the material in any medium or format
under the following conditions:
Attribution: You must give appropriate credit, provide a link to the license, and
indicate if changes were made. You may do so in any reasonable manner, but not in
any way that suggests the licensor endorses you or your use.
NonCommercial: You may not use the material for commercial purposes.
NoDerivatives: If you remix, transform, or build upon the material, you may not
distribute the modified material.
No additional restrictions: You may not apply legal terms or technological mea-
sures that legally restrict others from doing anything t he license permits.
Full license terms are at:
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode.
If you would like more rights to this work (e.g., commercial publication rights or including
parts of this work in another work, etc.), please contact the publisher.
Published by The Trillia Group, West Lafayette, Indiana, USA
ISBN 978-1-931705-00-3
First published: May 26, 2001. This version released: May 19, 2017.
Technical Typist: Judy Mitchell. Copy Editor: John Spiegelman. Logo: Miriam Bogdanic.
The phrase “The Trillia Group” and The Trillia Group logo are trademarks of The Trillia
Group and may not be used without permission.
This book was prepared by Bradley J. Lucier and Tamara Zakon from a manuscript
prepared by Elias Zakon. We intend to correct and update this work as needed. If you notice
any mistakes in this work, please send e-mail to Bradley Lucier (lucier@math.purdue.edu)
and they will be corrected in a later version.
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Contents
∗
Preface
vii
About the Author ix
Chapter 1. Some Set Theoretical Notions 1
1. Introduction. Sets and the ir Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Problems in Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. Log ic al Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4. Relation s (Correspondences) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Problems in the Theory of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5. Map pings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Problems on Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
∗
6. Composition of Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Problems on the Composition o f Re la tion s. . . . . . . . . . . . . . . . . . . . . . . . .30
∗
7. Equivalenc e Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Problems on Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8. Sequ ences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Problems on Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
∗
9. Some Theorems on Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Problems on Countable and Uncountable Sets . . . . . . . . . . . . . . . . . . . . . 48
Chapter 2. The Real Number System 51
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2. Axioms of an Ordered Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3. Arithmetic O perations in a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4. Inequalities in an Ordered Field. Absolu te Values . . . . . . . . . . . . . . . . . . . .58
Problems on Arithmetic Operations and Inequalities in a Field . . . . 62
5. Natural Numbe rs. Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3
6. Indu ction (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Problems on Natural Numbers a nd Induction . . . . . . . . . . . . . . . . . . . . . . 71
7. Integers and R a tionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Problems on Integers and Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8. Bounded Sets in an Ordered Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
∗
“Starred” sections may be omitted by beginners.
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