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Summary: The Art Of Proof
Basic Training For Deeper
Mathematics
Matthias Beck & Ross Geoghegan
c 2010 by the authors. All rights reserved. This work may not be translated
or copied in whole or in part without the permission of the authors. This is a
beta version of a book that will be published by Springer by the end of 2010.
The most current version of the beta version of this book is available at the
website http://math.sfsu.edu/beck/aop.html.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Notes for Students. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Notes for Instructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Part I: The Discrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 First Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Philosophical Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Natural Numbers and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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