انتقال گسسته به ریاضیات پیشرفته، ویرایش دوم

This textbook bridges the gap between lower-division 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. New features include a chapter on combinatorial geometry, a more in-depth treatment of counting, and over 365 new exercises. Copyright 1 Contents 6 Preface 10 Preface to the Second Edition 14 Chapter 1. Sets and Logic 16 1.1. Sets 16 1.2. Set Operations 24 1.3. Partitions 38 1.4. Logic and Truth Tables 42 1.5. Quantifiers 52 1.6. Implications 58 Chapter 2. Proofs 68 2.1. Proof Techniques 68 2.2. Mathematical Induction 80 2.3. The Pigeonhole Principle 91 Chapter 3. Number Theory 100 3.1. Divisibility 100 3.2. The Euclidean Algorithm 111 3.3. The Fundamental Theorem of Arithmetic 120 3.4. Divisibility Tests 128 3.5. Number Patterns 139 Chapter 4. Combinatorics 150 4.1. Getting from Point A to Point B 150 4.2. The Fundamental Principle of Counting 160 4.3. A Formula for the Binomial Coefficients 170 4.4. Permutations with Indistinguishable Objects 179 4.5. Combinations with Indistinguishable Objects 185 4.6. The Inclusion-Exclusion Principle 192 4.7. Circular Permutations 199 4.8. Probability 204 Chapter 5. Relations 212 5.1. Relations 212 5.2. Equivalence Relations 220 5.3. Partial Orders 228 5.4. Quotient Spaces 238 Chapter 6. Functions and Cardinality 250 6.1. Functions 250 6.2. Inverse Relations and Inverse Functions 261 6.3. Cardinality of Infinite Sets 270 6.4. An Order Relation for Cardinal Numbers 277 Chapter 7. Graph Theory 286 7.1. Graphs 287 7.2. Matrices, Digraphs, and Relations 297 7.3. Shortest Paths in Weighted Graphs 310 7.4. Trees 319 Chapter 8. Sequences 328 8.1. Sequences 328 8.2. Finite Differences 335 8.3. Limits of Sequences of Real Numbers 344 8.4. Some Convergence Properties 352 8.5. Infinite Arithmetic 357 8.6. Recurrence Relations 371 Chapter 9. Fibonacci Numbers and Pascal’s Triangle 386 9.1. Pascal’s Triangle 387 9.2. The Fibonacci Numbers 400 9.3. The Golden Ratio 411 9.4. Fibonacci Numbers and the Golden Ratio 419 9.5. Pascal’s Triangle and the Fibonacci Numbers 427 Chapter 10. Combinatorial Geometry in the Plane 434 10.1. Polygons and Convex Sets 435 10.2. Pick’s Theorem 441 10.3. Irrational Approximations of π 447 10.4. Cotes’s Theorem (optional) 455 10.5. Tiling and Visibility 457 10.6. Covering Properties and Geometry of Point Sets 464 10.7. Linear Algebra and Packing the Plane 469 10.8. Helly’s Theorem 478 Chapter 11. Continued Fractions 484 11.1. Finite Continued Fractions 484 11.2. Convergents of a Continued Fraction 493 11.3. Infinite Continued Fractions 499 11.4. Applications of Continued Fractions 505 Answers or Hints for Selected Exercises 516 Bibliography 530 Index 532 As the title indicates, this text is intended for courses aimed at bridging the gap between lower level mathematics and advanced mathematics. The transition to advanced mathematics presented is discrete since continuous functions are not studied. 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. Including more topics than can be covered in one semester, the text offers innovative material throughout, particularly in the last three chapters (e.g. Fibonacci Numbers and Pascal's Triangle). This allows flexibility for the instructor and the ability to teach a deeper, richer course.

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