Fundamentals of Mathematics An Introduction

The foundation of mathematics is not found in a single discipline since it is a general way of thinking in a very rigorous logical fashion. This book was written especially for readers who are about to make their first contact with this very way of thinking.Chapters 1-5 provide a rigorous, self contained construction of the familiar number systems (natural numbers, integers, real, and complex numbers) from the axioms of set theory. This construction trains readers in many of the proof techniques that are ultimately used almost subconsciously. In addition to important applications, the author discusses the scientific method in general (which is the reason why civilization has advanced to todayrs"s highly technological state), the fundamental building blocks of digital processors (which make computers work), and public key encryption (which makes internet commerce secure). The book also includes examples and exercises on the mathematics typically learned in elementary and high school. Aside from serving education majors, this further connection of abstract content to familiar ideas explains why these ideas work so well.Chapter 6 provides a condensed introduction to abstract algebra, and it fits very naturally with the idea that number systems were expanded over and over to allow for the solution of certain types of equations.Finally, Chapter 7 puts the finishing touches on the excursion into set theory. The axioms presented there do not directly impact the elementary construction of the number systems, but once they are needed in an advanced class, readers will certainly appreciate them.Chapter coverage includes: Logic; Set Theory; Number Systems I: Natural Numbers; Number Systems II: Integers; Number Systems III: Fields; Unsolvability of the Quintic by Radicals; and More Axioms.

Bernd S.W. Schrder, PhD, is Edmundson/Crump Professor, Academic Director, and Program Chair of the Program of Mathematics and Statistics at Louisiana Tech University. He has authored more than thirty journal articles in his areas of research interest, which include ordered sets, probability theory, graph theory, harmonic analysis, computer science, and education. Dr. Schrder is the author of Mathematical Analysis: A Concise Introduction and A Workbook for Differential Equations, both published by Wiley.

Preface	p. vii
Questions	p. x
Logic	p. 1
Statements	p. 1
Implications	p. 5
Conjunction, Disjunction, and Negation	p. 11
Special Focus on Negation	p. 19
Variables and Quantifiers	p. 25
Proofs	p. 30
Using Tautologies to Analyze Arguments	p. 42
Russell's Paradox	p. 46
Set Theory	p. 51
Sets and Objects	p. 52
The Axiom of Specification	p. 56
The Axiom of Extension	p. 59
The Axiom of Unions	p. 67
The Axiom of Powers, Relations, and Functions	p. 73
The Axiom of Infinity and the Natural Numbers	p. 83
Number Systems I: Natural Numbers	p. 89
Arithmetic With Natural Numbers	p. 89
Ordering the Natural Numbers	p. 98
A More Abstract Viewpoint: Binary Operations	p. 103
Induction	p. 111
Sums and Products	p. 120
Divisibility	p. 133
Equivalence Relations	p. 142
Arithmetic Modulo m	p. 147
Public Key Encryption	p. 153
Number Systems II: Integers	p. 161
Arithmetic With Integers	p. 161
Groups and Rings	p. 167
Finding the Natural Numbers in the Integers	p. 175
Ordered Rings	p. 179
Division in Rings	p. 185
Countable Sets	p. 195
Number Systems III: Fields	p. 201
Arithmetic With Rational Numbers	p. 201
Fields	p. 205
Ordered Fields	p. 211
A Problem With the Rational Numbers	p. 213
The Real Numbers	p. 216
Uncountable Sets	p. 226
The Complex Numbers	p. 230
Solving Polynomial Equations	p. 233
Beyond Fields: Vector Spaces and Algebras	p. 243
Unsolvability of the Quintic by Radicals	p. 249
Irreducible Polynomials	p. 250
Field Extensions and Splitting Fields	p. 255
Uniqueness of the Splitting Field	p. 260
Field Automorphisms and Galois Groups	p. 269
Normal Field Extensions	p. 273
The Groups S_n	p. 276
The Fundamental Theorem of Galois Theory and Normal Subgroups	p. 281
Consequences of Solvability by Radicals	p. 292
Abel's Theorem	p. 298
More Axioms	p. 301
The Axiom of Choice, Zorn's Lemma, and the Well-Ordering Theorem	p. 301
Ordinal Numbers and the Axiom of Replacement	p. 308
Cardinal Numbers and the Continuum Hypothesis	p. 311
Historical Overview and Commentary	p. 317
Ancient Times: Greece and Rome	p. 318
The Dark Ages and First New Developments	p. 321
There is No Quintic Formula: Abel and Galois	p. 323
Understanding Irrational Numbers: Set Theory	p. 326
Conclusion and Outlook	p. 328
Bibliography	p. 329
Index	p. 333
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Fundamentals of Mathematics An Introduction to Proofs, Logic, Sets, and Numbers

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