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Normal 0 false false false Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition , prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses. Cover......Page 1 Title Page......Page 2 Copyright Page......Page 3 Contents......Page 5 0 Communicating Mathematics......Page 16 Learning Mathematics......Page 17 What Others Have Said About Writing......Page 19 Mathematical Writing......Page 20 Using Symbols......Page 21 Writing Mathematical Expressions......Page 23 Common Words and Phrases in Mathematics......Page 25 Some Closing Comments About Writing......Page 27 1.1 Describing a Set......Page 29 1.2 Subsets......Page 33 1.3 Set Operations......Page 36 1.4 Indexed Collections of Sets......Page 39 1.5 Partitions of Sets......Page 42 1.6 Cartesian Products of Sets......Page 43 Exercises for Chapter 1......Page 44 2.1 Statements......Page 52 2.2 The Negation of a Statement......Page 54 2.3 The Disjunction and Conjunction of Statements......Page 56 2.4 The Implication......Page 57 2.5 More on Implications......Page 59 2.6 The Biconditional......Page 62 2.7 Tautologies and Contradictions......Page 64 2.8 Logical Equivalence......Page 66 2.9 Some Fundamental Properties of Logical Equivalence......Page 68 2.10 Quantified Statements......Page 70 2.11 Characterizations of Statements......Page 78 Exercises for Chapter 2......Page 79 3 Direct Proof and Proof by Contrapositive......Page 92 3.1 Trivial and Vacuous Proofs......Page 93 3.2 Direct Proofs......Page 95 3.3 Proof by Contrapositive......Page 99 3.4 Proof by Cases......Page 104 3.5 Proof Evaluations......Page 107 Exercises for Chapter 3......Page 108 4.1 Proofs Involving Divisibility of Integers......Page 114 4.2 Proofs Involving Congruence of Integers......Page 118 4.3 Proofs Involving Real Numbers......Page 120 4.4 Proofs Involving Sets......Page 123 4.5 Fundamental Properties of Set Operations......Page 126 4.6 Proofs Involving Cartesian Products of Sets......Page 128 Exercises for Chapter 4......Page 129 5.1 Counterexamples......Page 135 5.2 Proof by Contradiction......Page 139 5.3 A Review of Three Proof Techniques......Page 145 5.4 Existence Proofs......Page 147 5.5 Disproving Existence Statements......Page 151 Exercises for Chapter 5......Page 152 6.1 The Principle of Mathematical Induction......Page 157 6.2 A More General Principle of Mathematical Induction......Page 166 6.3 Proof by Minimum Counterexample......Page 173 6.4 The Strong Principle of Mathematical Induction......Page 176 Exercises for Chapter 6......Page 180 7.1 Conjectures in Mathematics......Page 185 7.2 Revisiting Quantified Statements......Page 188 7.3 Testing Statements......Page 193 Exercises for Chapter 7......Page 200 8.1 Relations......Page 207 8.2 Properties of Relations......Page 208 8.3 Equivalence Relations......Page 211 8.4 Properties of Equivalence Classes......Page 213 8.5 Congruence Modulo n......Page 217 8.6 The Integers Modulo n......Page 222 Exercises for Chapter 8......Page 225 9.1 The Definition of Function......Page 231 9.2 The Set of All Functions from A to B......Page 234 9.3 One-to-One and Onto Functions......Page 235 9.4 Bijective Functions......Page 237 9.5 Composition of Functions......Page 240 9.6 Inverse Functions......Page 244 9.7 Permutations......Page 247 Exercises for Chapter 9......Page 249 10 Cardinalities of Sets......Page 257 10.1 Numerically Equivalent Sets......Page 258 10.2 Denumerable Sets......Page 259 10.3 Uncountable Sets......Page 265 10.4 Comparing Cardinalities of Sets......Page 270 10.5 The Schröder–Bernstein Theorem......Page 273 Exercises for Chapter 10......Page 277 11.1 Divisibility Properties of Integers......Page 281 11.2 The Division Algorithm......Page 282 11.3 Greatest Common Divisors......Page 286 11.4 The Euclidean Algorithm......Page 287 11.5 Relatively Prime Integers......Page 290 11.6 The Fundamental Theorem of Arithmetic......Page 292 11.7 Concepts Involving Sums of Divisors......Page 295 Exercises for Chapter 11......Page 296 12.1 Limits of Sequences......Page 303 12.2 Infinite Series......Page 310 12.3 Limits of Functions......Page 315 12.4 Fundamental Properties of Limits of Functions......Page 322 12.5 Continuity......Page 327 12.6 Differentiability......Page 329 Exercises for Chapter 12......Page 332 13.1 Binary Operations......Page 337 13.2 Groups......Page 341 13.3 Permutation Groups......Page 345 13.4 Fundamental Properties of Groups......Page 348 13.5 Subgroups......Page 351 13.6 Isomorphic Groups......Page 355 Exercises for Chapter 13......Page 359 14.1 Rings......Page 366 14.2 Elementary Properties of Rings......Page 371 14.3 Subrings......Page 374 14.4 Integral Domains......Page 376 14.5 Fields......Page 378 Exercises for Chapter 14......Page 381 15.1 Properties of Vectors in 3-Space......Page 384 15.2 Vector Spaces......Page 386 15.3 Matrices......Page 389 15.4 Some Properties of Vector Spaces......Page 392 15.5 Subspaces......Page 394 15.6 Spans of Vectors......Page 397 15.7 Linear Dependence and Independence......Page 399 15.8 Linear Transformations......Page 403 15.9 Properties of Linear Transformations......Page 407 Exercises for Chapter 15......Page 410 16.1 Metric Spaces......Page 415 16.2 Open Sets in Metric Spaces......Page 418 16.3 Continuity in Metric Spaces......Page 423 16.4 Topological Spaces......Page 426 16.5 Continuity in Topological Spaces......Page 428 Exercises for Chapter 16......Page 430 Answers and Hints to Selected Odd-Numbered Exercises in Chapters 14–16 (online)......Page 9 Answers and Hints to Odd-Numbered Section Exercises......Page 440 References......Page 483 Index of Symbols......Page 484 C......Page 485 F......Page 486 N......Page 487 S......Page 488 Z......Page 489 Cover 1 Title Page 2 Copyright Page 3 Contents 5 0 Communicating Mathematics 16 Learning Mathematics 17 What Others Have Said About Writing 19 Mathematical Writing 20 Using Symbols 21 Writing Mathematical Expressions 23 Common Words and Phrases in Mathematics 25 Some Closing Comments About Writing 27 1 Sets 29 1.1 Describing a Set 29 1.2 Subsets 33 1.3 Set Operations 36 1.4 Indexed Collections of Sets 39 1.5 Partitions of Sets 42 1.6 Cartesian Products of Sets 43 Exercises for Chapter 1 44 2 Logic 52 2.1 Statements 52 2.2 The Negation of a Statement 54 2.3 The Disjunction and Conjunction of Statements 56 2.4 The Implication 57 2.5 More on Implications 59 2.6 The Biconditional 62 2.7 Tautologies and Contradictions 64 2.8 Logical Equivalence 66 2.9 Some Fundamental Properties of Logical Equivalence 68 2.10 Quantified Statements 70 2.11 Characterizations of Statements 78 Exercises for Chapter 2 79 3 Direct Proof and Proof by Contrapositive 92 3.1 Trivial and Vacuous Proofs 93 3.2 Direct Proofs 95 3.3 Proof by Contrapositive 99 3.4 Proof by Cases 104 3.5 Proof Evaluations 107 Exercises for Chapter 3 108 4 More on Direct Proof and Proof by Contrapositive 114 4.1 Proofs Involving Divisibility of Integers 114 4.2 Proofs Involving Congruence of Integers 118 4.3 Proofs Involving Real Numbers 120 4.4 Proofs Involving Sets 123 4.5 Fundamental Properties of Set Operations 126 4.6 Proofs Involving Cartesian Products of Sets 128 Exercises for Chapter 4 129 5 Existence and Proof by Contradiction 135 5.1 Counterexamples 135 5.2 Proof by Contradiction 139 5.3 A Review of Three Proof Techniques 145 5.4 Existence Proofs 147 5.5 Disproving Existence Statements 151 Exercises for Chapter 5 152 6 Mathematical Induction 157 6.1 The Principle of Mathematical Induction 157 6.2 A More General Principle of Mathematical Induction 166 6.3 Proof by Minimum Counterexample 173 6.4 The Strong Principle of Mathematical Induction 176 Exercises for Chapter 6 180 7 Prove or Disprove 185 7.1 Conjectures in Mathematics 185 7.2 Revisiting Quantified Statements 188 7.3 Testing Statements 193 Exercises for Chapter 7 200 8 Equivalence Relations 207 8.1 Relations 207 8.2 Properties of Relations 208 8.3 Equivalence Relations 211 8.4 Properties of Equivalence Classes 213 8.5 Congruence Modulo n 217 8.6 The Integers Modulo n 222 Exercises for Chapter 8 225 9 Functions 231 9.1 The Definition of Function 231 9.2 The Set of All Functions from A to B 234 9.3 One-to-One and Onto Functions 235 9.4 Bijective Functions 237 9.5 Composition of Functions 240 9.6 Inverse Functions 244 9.7 Permutations 247 Exercises for Chapter 9 249 10 Cardinalities of Sets 257 10.1 Numerically Equivalent Sets 258 10.2 Denumerable Sets 259 10.3 Uncountable Sets 265 10.4 Comparing Cardinalities of Sets 270 10.5 The Schröder–Bernstein Theorem 273 Exercises for Chapter 10 277 11 Proofs in Number Theory 281 11.1 Divisibility Properties of Integers 281 11.2 The Division Algorithm 282 11.3 Greatest Common Divisors 286 11.4 The Euclidean Algorithm 287 11.5 Relatively Prime Integers 290 11.6 The Fundamental Theorem of Arithmetic 292 11.7 Concepts Involving Sums of Divisors 295 Exercises for Chapter 11 296 12 Proofs in Calculus 303 12.1 Limits of Sequences 303 12.2 Infinite Series 310 12.3 Limits of Functions 315 12.4 Fundamental Properties of Limits of Functions 322 12.5 Continuity 327 12.6 Differentiability 329 Exercises for Chapter 12 332 13 Proofs in Group Theory 337 13.1 Binary Operations 337 13.2 Groups 341 13.3 Permutation Groups 345 13.4 Fundamental Properties of Groups 348 13.5 Subgroups 351 13.6 Isomorphic Groups 355 Exercises for Chapter 13 359 14 Proofs in Ring Theory (Online) 366 14.1 Rings 366 14.2 Elementary Properties of Rings 371 14.3 Subrings 374 14.4 Integral Domains 376 14.5 Fields 378 Exercises for Chapter 14 381 15 Proofs in Linear Algebra (Online) 384 15.1 Properties of Vectors in 3-Space 384 15.2 Vector Spaces 386 15.3 Matrices 389 15.4 Some Properties of Vector Spaces 392 15.5 Subspaces 394 15.6 Spans of Vectors 397 15.7 Linear Dependence and Independence 399 15.8 Linear Transformations 403 15.9 Properties of Linear Transformations 407 Exercises for Chapter 15 410 16 Proofs in Topology (Online) 415 16.1 Metric Spaces 415 16.2 Open Sets in Metric Spaces 418 16.3 Continuity in Metric Spaces 423 16.4 Topological Spaces 426 16.5 Continuity in Topological Spaces 428 Exercises for Chapter 16 430 Answers and Hints to Selected Odd-Numbered Exercises in Chapters 14–16 (online) 9 Answers and Hints to Odd-Numbered Section Exercises 440 References 483 Index of Symbols 484 Index 485 A 485 B 485 C 485 D 486 E 486 F 486 G 487 H 487 I 487 J 487 K 487 L 487 M 487 N 487 O 488 P 488 Q 488 R 488 S 488 T 489 U 489 V 489 W 489 Z 489 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.

Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

This book prepares students for the more abstract mathematics courses that follow calculus. The author introduces students to proof techniques, analyzing proofs, and writing proofs of their own. It also provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory