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Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers

ISBN: 978-0-470-55138-7
338 pages
August 2010
Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers (0470551380) cover image
An accessible introduction to abstract mathematics with an emphasis on proof writing

Addressing the importance of constructing and understanding mathematical proofs, Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The author supplies a seamless, hands-on presentation of number systems, utilizing key elements of logic and set theory and encouraging readers to abide by the fundamental rule that you are not allowed to use any results that you have not proved yet.

The book begins with a focus on the elements of logic used in everyday mathematical language, exposing readers to standard proof methods and Russell's Paradox. Once this foundation is established, subsequent chapters explore more rigorous mathematical exposition that outlines the requisite elements of Zermelo-Fraenkel set theory and constructs the natural numbers and integers as well as rational, real, and complex numbers in a rigorous, yet accessible manner. Abstraction is introduced as a tool, and special focus is dedicated to concrete, accessible applications, such as public key encryption, that are made possible by abstract ideas. The book concludes with a self-contained proof of Abel's Theorem and an investigation of deeper set theory by introducing the Axiom of Choice, ordinal numbers, and cardinal numbers.

Throughout each chapter, proofs are written in much detail with explicit indications that emphasize the main ideas and techniques of proof writing. Exercises at varied levels of mathematical development allow readers to test their understanding of the material, and a related Web site features video presentations for each topic, which can be used along with the book or independently for self-study.

Classroom-tested to ensure a fluid and accessible presentation, Fundamentals of Mathematics is an excellent book for mathematics courses on proofs, logic, and set theory at the upper-undergraduate level as well as a supplement for transition courses that prepare students for the rigorous mathematical reasoning of advanced calculus, real analysis, and modern algebra. The book is also a suitable reference for professionals in all areas of mathematics education who are interested in mathematical proofs and the foundation upon which all mathematics is built.

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Preface.

Questions.

1 Logic.

1.1 Statements.

1.2 Implications.

1.3 Conjunction, Disjunction and Negation.

1.4 Special Focus on Negation.

1.5 Variables and Quantifiers.

1.6 Proofs.

1.7 Using Tautologies to Analyze Arguments.

1.8 Russell's Paradox.

2 Set Theory.

2.1 Sets and Objects.

2.2 The Axiom of Specification.

2.3 The Axiom of Extension.

2.4 The Axiom of Unions.

2.5 The Axiom of Powers, Relations and Functions.

2.6 The Axiom of Infinity and the Natural Numbers.

3 Number Systems I: Natural Numbers.

3.1 Arithmetic With Natural Numbers.

3.2 Ordering the Natural Numbers.

3.3 A More Abstract Viewpoint: Binary Operations.

3.4 Induction.

3.5 Sums and Products.

3.6 Divisibility.

3.7 Equivalence Relations.

3.8 Arithmetic Modulo m.

3.9 Public Key Encryption.

4 Number Systems II: Integers.

4.1 Arithmetic With Integers.

4.2 Groups and Rings.

4.3 Finding the Natural Numbers in the Integers.

4.4 Ordered Rings.

4.5 Division in Rings.

4.6 Countable Sets.

5 Number Systems III: Fields.

5.1 Arithmetic With Rational Numbers.

5.2 Fields.

5.3 Ordered Fields.

5.4 A Problem With the Rational Numbers.

5.5 The Real Numbers.

5.6 Uncountable Sets.

5.7 The Complex Numbers.

5.8 Solving Polynomial Equations.

5.9 Beyond Fields: Vector Spaces and Algebras.

6 Unsolvability of the Quintic by Radicals.

6.1 Irreducible Polynomials.

6.2 Field Extensions and Splitting Fields.

6.3 Uniqueness of the Splitting Field.

6.4 Field Automorphisms and Galois Groups.

6.5 Normal Field Extensions.

6.6 The Groups Sn

6.7 The Fundamental Theorem of Galois Theory and Normal Subgroups.

6.8 Consequences of Solvability by Radicals.

6.9 Abel's Theorem.

7 More Axioms.

7.1 The Axiom of Choice, Zorn's Lemma and the Well-Ordering Theorem.

7.2 Ordinal Numbers and the Axiom of Replacement.

7.3 Cardinal Numbers and the Continuum Hypothesis.

A Historical Overview and Commentary.

A.1 Ancient Times: Greece and Rome.

A.2 The Dark Ages and First New Developments.

A.3 There is No Quintic Formula: Abel and Galois.

A.4 Understanding Irrational Numbers: Set Theory.

Conclusion and Outlook.

Bibliography.

Index.

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BERND S.W. SCHRÖDER, 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. Schröder is the author of Mathematical Analysis: A Concise Introduction and A Workbook for Differential Equations, both published by Wiley.
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  • Enforces the fundamental rule that you are not allowed to use any results that you have not proved yet, and consequently starts with an axiomatic system and builds from there
  • Introduces proof methods in a separate section of the Logic chapter to facilitate the development of proof writing skills.  In the early chapters, proofs are written in much detail with explicit indications, and the level of detail is gradually reduced until the writing style is typical for a standard mathematics text.
  • Provides results, applications, and open problems that are typically covered in the high school curriculum in an effort to draw upon as many connections as possible from a reader's existing knowledge base 
  • Features many types of exercises at varied levels of mathematical development and successfully guides readers through the presented mathematical fundamentals
  • Extensively classroom-tested to prepare readers for their first "targeted" proof class, whether it is analysis, algebra, linear algebra, or another subject altogether
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"This is a lovely introduction to mathematics. The book gives an elegant construction of the familiar number systems while at the same time introducing the student to mathematical logic, set theory and rigorous mathematical proof." (Zentralblatt MATH, 2011)
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