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Softcover ISBN:  9781470472047 
Product Code:  AMSTEXT/63 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Sale Price:  $57.85 
eBook ISBN:  9781470474270 
Product Code:  AMSTEXT/63.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Sale Price:  $55.25 
Softcover ISBN:  9781470472047 
eBook ISBN:  9781470474270 
Product Code:  AMSTEXT/63.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 
Sale Price:  $113.10 $85.48 

Book DetailsPure and Applied Undergraduate TextsVolume: 63; 2023; 523 ppMSC: Primary 00
This is a Revised Edition of: AMSTEXT/3
This textbook bridges the gap between lowerdivision mathematics courses and advanced mathematical thinking. Featuring clear writing and appealing topics, the book introduces techniques for writing proofs in the context of discrete mathematics. By illuminating the concepts behind techniques, the authors create opportunities for readers to sharpen critical thinking skills and develop mathematical maturity.
Beginning with an introduction to sets and logic, the book goes on to establish the basics of proof techniques. From here, chapters explore proofs in the context of number theory, combinatorics, functions and cardinality, and graph theory. A selection of extension topics concludes the book, including continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio.
A Discrete Transition to Advanced Mathematics is suitable for an introduction to proof course or a course in discrete mathematics. Abundant examples and exercises invite readers to get involved, and the wealth of topics allows for course customization and further reading. This new edition has been expanded and modernized throughout, featuring:
 A new chapter on combinatorial geometry;
 An expanded treatment of the combinatorics of indistinguishable objects;
 New sections on the inclusionexclusion principle and circular permutations;
 Over 365 new exercises.
ReadershipUndergraduate students who need a strong conceptual foundation for higher mathematical thinking.

Table of Contents

Copyright

Contents

Preface

Preface to the Second Edition

Chapter 1. Sets and Logic

1.1. Sets

1.2. Set Operations

1.3. Partitions

1.4. Logic and Truth Tables

1.5. Quantifiers

1.6. Implications

Chapter 2. Proofs

2.1. Proof Techniques

2.2. Mathematical Induction

2.3. The Pigeonhole Principle

Chapter 3. Number Theory

3.1. Divisibility

3.2. The Euclidean Algorithm

3.3. The Fundamental Theorem of Arithmetic

3.4. Divisibility Tests

3.5. Number Patterns

Chapter 4. Combinatorics

4.1. Getting from Point A to Point B

4.2. The Fundamental Principle of Counting

4.3. A Formula for the Binomial Coefficients

4.4. Permutations with Indistinguishable Objects

4.5. Combinations with Indistinguishable Objects

4.6. The InclusionExclusion Principle

4.7. Circular Permutations

4.8. Probability

Chapter 5. Relations

5.1. Relations

5.2. Equivalence Relations

5.3. Partial Orders

5.4. Quotient Spaces

Chapter 6. Functions and Cardinality

6.1. Functions

6.2. Inverse Relations and Inverse Functions

6.3. Cardinality of Infinite Sets

6.4. An Order Relation for Cardinal Numbers

Chapter 7. Graph Theory

7.1. Graphs

7.2. Matrices, Digraphs, and Relations

7.3. Shortest Paths in Weighted Graphs

7.4. Trees

Chapter 8. Sequences

8.1. Sequences

8.2. Finite Differences

8.3. Limits of Sequences of Real Numbers

8.4. Some Convergence Properties

8.5. Infinite Arithmetic

8.6. Recurrence Relations

Chapter 9. Fibonacci Numbers and Pascal’s Triangle

9.1. Pascal’s Triangle

9.2. The Fibonacci Numbers

9.3. The Golden Ratio

9.4. Fibonacci Numbers and the Golden Ratio

9.5. Pascal’s Triangle and the Fibonacci Numbers

Chapter 10. Combinatorial Geometry in the Plane

10.1. Polygons and Convex Sets

10.2. Pick’s Theorem

10.3. Irrational Approximations of 𝜋

10.4. Cotes’s Theorem (optional)

10.5. Tiling and Visibility

10.6. Covering Properties and Geometry of Point Sets

10.7. Linear Algebra and Packing the Plane

10.8. Helly’s Theorem

Chapter 11. Continued Fractions

11.1. Finite Continued Fractions

11.2. Convergents of a Continued Fraction

11.3. Infinite Continued Fractions

11.4. Applications of Continued Fractions

Answers or Hints for Selected Exercises

Bibliography

Index


Additional Material

RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseInstructor's Solutions Manual – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
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This is a Revised Edition of: AMSTEXT/3
This textbook bridges the gap between lowerdivision mathematics courses and advanced mathematical thinking. Featuring clear writing and appealing topics, the book introduces techniques for writing proofs in the context of discrete mathematics. By illuminating the concepts behind techniques, the authors create opportunities for readers to sharpen critical thinking skills and develop mathematical maturity.
Beginning with an introduction to sets and logic, the book goes on to establish the basics of proof techniques. From here, chapters explore proofs in the context of number theory, combinatorics, functions and cardinality, and graph theory. A selection of extension topics concludes the book, including continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio.
A Discrete Transition to Advanced Mathematics is suitable for an introduction to proof course or a course in discrete mathematics. Abundant examples and exercises invite readers to get involved, and the wealth of topics allows for course customization and further reading. This new edition has been expanded and modernized throughout, featuring:
 A new chapter on combinatorial geometry;
 An expanded treatment of the combinatorics of indistinguishable objects;
 New sections on the inclusionexclusion principle and circular permutations;
 Over 365 new exercises.
Undergraduate students who need a strong conceptual foundation for higher mathematical thinking.

Copyright

Contents

Preface

Preface to the Second Edition

Chapter 1. Sets and Logic

1.1. Sets

1.2. Set Operations

1.3. Partitions

1.4. Logic and Truth Tables

1.5. Quantifiers

1.6. Implications

Chapter 2. Proofs

2.1. Proof Techniques

2.2. Mathematical Induction

2.3. The Pigeonhole Principle

Chapter 3. Number Theory

3.1. Divisibility

3.2. The Euclidean Algorithm

3.3. The Fundamental Theorem of Arithmetic

3.4. Divisibility Tests

3.5. Number Patterns

Chapter 4. Combinatorics

4.1. Getting from Point A to Point B

4.2. The Fundamental Principle of Counting

4.3. A Formula for the Binomial Coefficients

4.4. Permutations with Indistinguishable Objects

4.5. Combinations with Indistinguishable Objects

4.6. The InclusionExclusion Principle

4.7. Circular Permutations

4.8. Probability

Chapter 5. Relations

5.1. Relations

5.2. Equivalence Relations

5.3. Partial Orders

5.4. Quotient Spaces

Chapter 6. Functions and Cardinality

6.1. Functions

6.2. Inverse Relations and Inverse Functions

6.3. Cardinality of Infinite Sets

6.4. An Order Relation for Cardinal Numbers

Chapter 7. Graph Theory

7.1. Graphs

7.2. Matrices, Digraphs, and Relations

7.3. Shortest Paths in Weighted Graphs

7.4. Trees

Chapter 8. Sequences

8.1. Sequences

8.2. Finite Differences

8.3. Limits of Sequences of Real Numbers

8.4. Some Convergence Properties

8.5. Infinite Arithmetic

8.6. Recurrence Relations

Chapter 9. Fibonacci Numbers and Pascal’s Triangle

9.1. Pascal’s Triangle

9.2. The Fibonacci Numbers

9.3. The Golden Ratio

9.4. Fibonacci Numbers and the Golden Ratio

9.5. Pascal’s Triangle and the Fibonacci Numbers

Chapter 10. Combinatorial Geometry in the Plane

10.1. Polygons and Convex Sets

10.2. Pick’s Theorem

10.3. Irrational Approximations of 𝜋

10.4. Cotes’s Theorem (optional)

10.5. Tiling and Visibility

10.6. Covering Properties and Geometry of Point Sets

10.7. Linear Algebra and Packing the Plane

10.8. Helly’s Theorem

Chapter 11. Continued Fractions

11.1. Finite Continued Fractions

11.2. Convergents of a Continued Fraction

11.3. Infinite Continued Fractions

11.4. Applications of Continued Fractions

Answers or Hints for Selected Exercises

Bibliography

Index