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Foundations of Mathematics

Thomas Q. Sibley
Publisher: 
John Wiley
Publication Date: 
2010
Number of Pages: 
392
Format: 
Hardcover
Price: 
160.95
ISBN: 
9780470085011
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Bollman
, on
06/5/2010
]

Between graduate school, full-time teaching positions at three colleges, and a brief temporary gig as an adjunct, I have taught 86 different courses, with #87 and 88 on the fall schedule. I have never taught anything along the lines of “Transition to Advanced Mathematics,” the kind of course that is supposed to bridge the apparent gap between computational and theoretical mathematics classes. I’m not inherently opposed to the idea, I just have never been at a college where such a course is offered — my two tenure-track appointments have been at colleges where discrete mathematics fills that role.

Nonetheless, I am certainly aware of such courses, for which this book would be a fine choice as a textbook. My primary criterion concerns the far side of the transition: how well does a “transitions math” book give a good sense of where a math major taking the class is headed, in the form of a meaningful introduction to at least some of the ideas of analysis, algebra, and other advanced topics? Most of the competing books take on these important topics, but it’s easy to get caught up in the standard material on sets, logic, functions, and relations to an extent that the student doesn’t get a meaningful glimpse of what lies ahead.

On this score, Sibley has succeeded. The chapters on abstract algebra and real analysis provide a good overview of these important subjects and connect nicely to the preliminary material. A final chapter on the philosophy of mathematics ties the rest of the book together and provides some interesting questions to challenge the prospective mathematics major. Sibley’s book succeeds on two fronts: as a capstone to lower-division mathematics and as an introduction to the upper-division.


Mark Bollman (mbollman@albion.edu) is associate professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.


PART I

Chapter 1: LANGUAGE, LOGIC, AND SETS
1.1 Logic and Language
1.2 Implication
1.3 Quantifiers and Definitions
1.4 Introduction to Sets
1.5 Introduction to Number Theory
1.6 Additional Set Theory
Definitions from Chapter 1
Algebraic and Order Properties of Number Systems

Chapter 2: PROOFS
2.1 Proof Format I: Direct Proofs
2.2 Proof Format II: Contrapositive and Contradition
2.3 Proof Format III: Existence, Uniqueness, Or
2.4 Proof Format IV: Mathematical Induction
The Fundamental Theorem of Arithmetic
2.5 Further Advice and Practice in Proving
Proof Formats

Chapter 3: FUNCTIONS
3.1 Definitions
3.2 Composition, One-to-One, Onto, and Inverses
3.3 Images and Pre-Images of Sets
Definitions from Chapter 3

Chapter 4: RELATIONS
4.1 Relations
4.2 Equivalence Relations
4.3 Partitions and Equivalence Relations
4.4 Partial Orders
Definitions from Chapter 4

PART II

Chapter 5: INFINTE SETS
5.1 The Sizes of Sets
5.2 Countable Sets
5.3 Uncountable Sets
5.4 The Axiom of Choice and Its Equivalents
Definitions from Chapter 5

Chapter 6: INTRODUCTION TO DISCRETE MATHEMATICS
6.1 Graph Theory
6.2 Trees and Algorithms
6.3 Counting Principles I
6.4 Counting Principles II
Definitions from Chapter 6

Chapter 7: INTRODUCTION TO ABSTRACT ALGEBRA
7.1 Operations and Properties
7.2 Groups
Groups in Geometry
7.3 Rings and Fields
7.4 Lattices
7.5 Homomorphisms
Definitions from Chapter 7

Chapter 8: INTRODUCTION TO ANALYSIS
8.1 Real Numbers, Approximations, and Exact Values
Zeno’s Paradoxes
8.2 Limits of Functions
8.3 Continuous Functions and Counterexamples
Counterexamples in Rational Analysis
8.4 Sequences and Series
8.5 Discrete Dynamical Systems
The Intermediate Value Theorem
Definitions for Chapter 8

Chapter 9: METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS
9.1 Metamathematics
9.2 The Philosophy of Mathematics
Definitions for Chapter 9

Appendix: THE GREEK ALPHABET
Answers: SELECTED ANSWERS

Index
List of Symbols

Dummy View - NOT TO BE DELETED