Passage to Abstract Mathematics / Edition 1

by Mark E. Watkins, Jeffrey L. Meyer

Overview

Passage to Abstract Mathematics helps students progress from a facility with computational procedures to an understanding of abstract mathematical concepts. Students develop their ability in mathematical communication through reading proofs, constructing proofs, and writing proofs in correct mathematical language.

Concise, practical, and highly valuable, the text is ideal for students who have taken lower-division mathematics courses and need the tools requisite to study more advanced, abstract mathematics.

The text features material that instructors of upper-level courses in set theory, analysis, topology, and modern algebra presume students have already learned by the time they enter advanced courses. It places emphasis on complete and correct definitions, as well as expressing mathematics in correct syntax. The core material consists of the first five closely knit chapters: Logic, Numbers, Sets, Functions, and Induction.

To support active and continuous learning, exercises are embedded within the text material immediately following a definition or theorem. The explanatory comments, hints to solutions, and thought-provoking questions that appear within brackets throughout the text all serve to deepen the student's understanding of the material.

In the second edition, the chapter entitled Functions precedes the chapter entitled Induction, and select material has been clarified or corrected. Number theoretic digressions such as Euclid's Algorithm and the Chinese Remainder Theorem have been deleted.

Product Details

ISBN-13:	9780321738639
Publisher:	Pearson
Publication date:	01/20/2011
Series:	Featured Titles for Transition to Advanced Mathematics Series
Pages:	256
Product dimensions:	7.00(w) x 9.20(h) x 0.70(d)

About the Author

Mark E. Watkins holds a Ph.D. in mathematics from Yale University, where he was a student of Øystein Ore. He is a professor emeritus of mathematics at Syracuse University. Dr. Watkins has published over 70 research articles, mainly in algebraic and topological graph theory, and has coauthored the books Combinatorics with Emphasis on the Theory of Graphs and Locally Finite, Planar, Edge-Transitive Graphs.

Jeffrey L. Meyer holds a Ph.D. in number theory from the University of Illinois, where he was the tenth Ph.D. student of Bruce C. Berndt. He is an associate teaching professor of mathematics at Syracuse University. Dr. Meyer has published both research and expository papers in analytic number theory, especially Dedekind Sums.

Table of Contents

Preface for the Instructor vii

Preface for the Student xiii

1. Logic and Proof

1.1 Proofs, what and why?

1.2 Statements and Non-statements

1.3 Logical Operations and Logical Equivalence

1.4 Conditionals, Tautologies and Contradictions

1.5 Methods of Proof

1.6 Quantifiers

1.7 Further Exercises

2. Numbers

2.1 Basic Ideas of Sets

2.2 Sets of Numbers

2.3 Some properties of N and Z

2.4 Prime Numbers

2.5 gcd’s and lcm’s

2.6 Euclid’s Algorithm

2.7 Rational Numbers and Algebraic Numbers

2.8 Further Exercises

3. Sets

3.1 Subsets

3.2 Operations with Sets

3.3 The Complement of a Set

3.4 The Cartesian Product

3.5 Families of Sets

3.6 Further Exercises

4. Induction

4.1 An Inductive Example

4.2 The Principle of Mathematical Induction

4.3 The Principle of Strong Induction

4.4 The Binomial Theorem

4.5 Further Exercises

5. Functions

5.1 Functional Notation

5.2 Operations on Functions

5.3 Induced Set Functions

5.4 Surjections, Injections, and Bijections

5.5 Identity Functions, Cancellation, Inverse Functions, and Restrictions

5.6 Further Exercises

6. Binary Relations

6.1 Partitions

6.2 Equivalence Relations

6.3 Order Relations

6.4 Bounds and Extremal Elements

6.5 Applications to Calculus

6.6 Functions Revisited

6.7 Further Exercises

7. Infinite Sets and Cardinality

7.1 Counting

7.2 Properties of Countable Sets

7.3 Counting Countable Sets

7.4 Binary Relations on Cardinal Numbers

7.5 Uncountable Sets

7.6 Further Exercises

8. Algebraic Systems

8.1 Binary Operations

8.2 Modular Arithmetic

8.3 Numbers Revisited

8.4 Complex Numbers

8.5 Further Exercises

Index of Symbols and Notation

Index