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This is the eBook of the printed book and may not include any media, website access codes, or print supplements that may come packaged with the bound book. For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number theory, combinatorics, and calculus. The exercises receive consistent praise from users for their thoughtfulness and creativity. They help students progress from understanding and analyzing proofs and techniques to producing well-constructed proofs independently. This book is also an excellent reference for students to use in future courses when writing or reading proofs. 0134746759 / 9780134746753 Chartrand/Polimeni/Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, 4/e Cover 1 Title Page 2 Copyright Page 3 Dedication 4 Contents 5 Chapter 0: Communicating Mathematics 18 0.1 Learning Mathematics 19 0.2 What Others Have Said About Writing 20 0.3 Mathematical Writing 22 0.4 Using Symbols 23 0.5 Writing Mathematical Expressions 25 0.6 Common Words and Phrases in Mathematics 27 0.7 Some Closing Comments About Writing 29 Chapter 1: Sets 31 1.1 Describing a Set 31 1.2 Subsets 35 1.3 Set Operations 40 1.4 Indexed Collections of Sets 44 1.5 Partitions of Sets 48 1.6 Cartesian Products of Sets 50 Chapter 1 Supplemental Exercises 52 Chapter 2: Logic 55 2.1 Statements 55 2.2 Negations 58 2.3 Disjunctions and Conjunctions 60 2.4 Implications 62 2.5 More on Implications 66 2.6 Biconditionals 70 2.7 Tautologies and Contradictions 74 2.8 Logical Equivalence 77 2.9 Some Fundamental Properties of Logical Equivalence 79 2.10 Quantified Statements 82 2.11 Characterizations 93 Chapter 2 Supplemental Exercises 95 Chapter 3: Direct Proof and Proof by Contrapositive 98 3.1 Trivial and Vacuous Proofs 99 3.2 Direct Proofs 102 3.3 Proof by Contrapositive 106 3.4 Proof by Cases 111 3.5 Proof Evaluations 115 Chapter 3 Supplemental Exercises 119 Chapter 4: More on Direct Proof and Proof by Contrapositive 122 4.1 Proofs Involving Divisibility of Integers 122 4.2 Proofs Involving Congruence of Integers 127 4.3 Proofs Involving Real Numbers 130 4.4 Proofs Involving Sets 134 4.5 Fundamental Properties of Set Operations 137 4.6 Proofs Involving Cartesian Products of Sets 139 Chapter 4 Supplemental Exercises 140 Chapter 5: Existence and Proof by Contradiction 144 5.1 Counterexamples 144 5.2 Proof by Contradiction 148 5.3 A Review of Three Proof Techniques 155 5.4 Existence Proofs 158 5.5 Disproving Existence Statements 163 Chapter 5 Supplemental Exercises 166 Chapter 6: Mathematical Induction 169 6.1 The Principle of Mathematical Induction 169 6.2 A More General Principle of Mathematical Induction 179 6.3 The Strong Principle of Mathematical Induction 187 6.4 Proof by Minimum Counterexample 191 Chapter 6 Supplemental Exercises 195 Chapter 7: Reviewing Proof Techniques 198 7.1 Reviewing Direct Proof and Proof by Contrapositive 199 7.2 Reviewing Proof by Contradiction and Existence Proofs 202 7.3 Reviewing Induction Proofs 205 7.4 Reviewing Evaluations of Proposed Proofs 206 Exercises for Chapter 7 210 Chapter 8: Prove or Disprove 217 8.1 Conjectures in Mathematics 217 8.2 Revisiting Quantified Statements 222 8.3 Testing Statements 228 Chapter 8 Supplemental Exercises 237 Chapter 9: Equivalence Relations 241 9.1 Relations 241 9.2 Properties of Relations 243 9.3 Equivalence Relations 247 9.4 Properties of Equivalence Classes 252 9.5 Congruence Modulo n 256 9.6 The Integers Modulo n 262 Chapter 9 Supplemental Exercises 265 Chapter 10: Functions 268 10.1 The Definition of Function 268 10.2 One-to-one and Onto Functions 273 10.3 Bijective Functions 276 10.4 Composition of Functions 280 10.5 Inverse Functions 284 Chapter 10 Supplemental Exercises 291 Chapter 11: Cardinalities of Sets 295 11.1 Numerically Equivalent Sets 296 11.2 Denumerable Sets 297 11.3 Uncountable Sets 305 11.4 Comparing Cardinalities of Sets 310 11.5 The Schr ̈oder-Bernstein Theorem 313 Chapter 11 Supplemental Exercises 318 Chapter 12: Proofs in Number Theory 320 12.1 Divisibility Properties of Integers 320 12.2 The Division Algorithm 322 12.3 Greatest Common Divisors 327 12.4 The Euclidean Algorithm 329 12.5 Relatively Prime Integers 332 12.6 The Fundamental Theorem of Arithmetic 335 12.7 Concepts Involving Sums of Divisors 339 Chapter 12 Supplemental Exercises 341 Chapter 13: Proofs in Combinatorics 344 13.1 The Multiplication and Addition Principles 344 13.2 The Principle of Inclusion-Exclusion 350 13.3 The Pigeonhole Principle 353 13.4 Permutations and Combinations 357 13.5 The Pascal Triangle 365 13.6 The Binomial Theorem 369 13.7 Permutations and Combinations with Repetition 374 Chapter 13 Supplemental Exercises 380 Chapter 14: Proofs in Calculus 382 14.1 Limits of Sequences 382 14.2 Infinite Series 390 14.3 Limits of Functions 395 14.4 Fundamental Properties of Limits of Functions 403 14.5 Continuity 409 14.6 Differentiability 412 Chapter 14 Supplemental Exercises 414 Chapter 15: Proofs in Group Theory 417 15.1 Binary Operations 417 15.2 Groups 422 15.3 Permutation Groups 428 15.4 Fundamental Properties of Groups 431 15.5 Subgroups 435 15.6 Isomorphic Groups 440 Chapter 15 Supplemental Exercises 445 Chapter 16: Proofs in Ring Theory 447 16.1 Rings 448 16.2 Elementary Properties of Rings 453 16.3 Subrings 456 16.4 Integral Domains 458 16.5 Fields 461 Exercises for Chapter 16 463 Chapter 17: Proofs in Linear Algebra 466 17.1 Properties of Vectors in 3-Space 467 17.2 Vector Spaces 468 17.3 Matrices 471 17.4 Some Properties of Vector Spaces 474 17.5 Subspaces 476 17.6 Spans of Vectors 479 17.7 Linear Dependence and Independence 482 17.8 Linear Transformations 485 17.9 Properties of Linear Transformations 490 Exercises for Chapter 17 492 Chapter 18: Proofs with Real and Complex Numbers 497 18.1 The Real Numbers as an Ordered Field 497 18.2 The Real Numbers and the Completeness Axiom 499 18.3 Open and Closed Sets of Real Numbers 503 18.4 Compact Sets of Real Numbers 508 18.5 Complex Numbers 510 18.6 De Moivre’s Theorem and Euler’s Formula 515 Exercises for Chapter 18 519 Chapter 19: Proofs in Topology 527 19.1 Metric Spaces 527 19.2 Open Sets in Metric Spaces 531 19.3 Continuity in Metric Spaces 536 19.4 Topological Spaces 539 19.5 Continuity in Topological Spaces 542 Exercises for Chapter 19 544 Answers and Hints to Selected Odd-NumberedExercises in Chapters 16–19 601 Answers to Odd-Numbered Section Exercises 548 References 614 Credits 615 Index of Symbols 617 Index 618 Gary Chartrand, Western Michigan University ; Albert D. Polimeni, State University Of New York At Fredonia ; Ping Zhang, Western Michigan University. Includes Bibliographical References And Indexes.