A Discrete Transition to Advanced Mathematics

Description

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. 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