A Discrete Transition to Advanced Mathematics (Pure and Applied Undergraduate Texts) (Pure and Applied Undergraduate Texts, 3)

As the title indicates, this book is intended for courses aimed at bridging the gap between lower-level mathematics and advanced mathematics. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. While including many traditional topics, the text offers innovative material throughout. Surprising results are used to motivate the reader. The last three chapters address topics such as continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio, and may be used for independent reading assignments. The treatment of sequences may be used to introduce epsilon-delta proofs. The selection of topics provides flexibility for the instructor in a course designed to spark the interest of students through exciting material while preparing them for subsequent proof-based courses. The book includes a large number of problems of varying difficulty. A student manual with solutions to selected problems is available. For more information regarding the student manual, please contact AMS Member and Customer Services at cust-serv@ams.org. An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org. Cover Title Copyright Preface Contents 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 2 Proofs 2.1 Proof Techniques 2.2 Mathematical Induction 2.3 The Pigeonhole Principle 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 4 Combinatorics 4.1 Getting from Point A to Points 4.2 The Fundamental Principle of Counting 4.3 A Formula for the Binomial Coefficients 4.4 Combinatorics with Indistinguishable Objects 4.5 Probability 5 Relations 5.1 Relations 5.2 Equivalence Relations 5.3 Partial Orders 5.4 Quotient Spaces 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 7 Graph Theory 7.1 Graphs 7.2 Matrices, Digraphs, and Relations 7.3 Shortest Paths in Weighted Graphs 7.4 Trees 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 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 10 Continued Fractions 10.1 Finite Continued Fractions 10.2 Convergents of a Continued Fraction 10.3 Infinite Continued Fractions 10.4 Applications of Continued Fractions Answers or Hints for Selected Exercises Bibliography Index A B C D E F G H I K L M N O P Q R S T U V W Z Back Cover "As the title indicates, this book is intended for courses aimed at bridging the gap between lower-level mathematics and advanced mathematics. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. While including many traditional topics, the text offers innovative material throughout. Surprising results are used to motivate the reader. The last three chapters address topics such as continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio, and may be used for independent reading assignments. The treatment of sequences may be used to introduce epsilon-delta proofs. The selection of topics provides flexibility for the instructor in a course designed to spark the interest of students through exciting material while preparing them for subsequent proof-based courses."--Publisher's description

offering Many Elegant And Surprising Results, An Engaging Writing Style, And More Than 650 Exercises, This Mathematics Text Is Designed To Bridge The Gap Between Computational Lower-level Courses, And More Theoretical Upper-level Courses. The Text Is Appropriate For Discrete Mathematics Courses, As Well As Introductory Proofs Courses, And Is Accessible To Students Having Two Or Three Semesters Of Calculus, Or Introductory Linear Algebra. Topics Include Proofs, Number Theory, Combinatorics, Graph Theory, Divisibility Tests, Binomial Coefficients, And Fibonacci Numbers And Pascal's Triangle. Annotation ©2003 Book News, Inc., Portland, Or

Offers an introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. This book addresses topics such as continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio.