The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and how they work. The book offers techniques for both reading and writing proofs, discusses techniques in proving if/then statements by contrapositive and proofing by contradiction, includes the negation statement, and/or, examines various theorems, such as the if and only-if, equivalence theorems, existence theorems, and the uniqueness theorems. In addition, the use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are also covered. The book also provides mathematical topics for practicing proof techniques.
Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book accessible as well as invaluable.
1. Cover image - p. i 2. Title page - p. ii 3. Table of Contents - p. iii 4. Copyright - p. iv 5. Dedication - p. v 6. Preface - p. ix 7. List of Symbols - p. xi 8. Chapter 1. Getting Started - p. xiv 9. Chapter 2. Basic Techniques to Prove If/Then Statements - p. 6 10. What Does “If/Then” Mean? - p. 6 11. Direct Proof - p. 11 12. The Negation of a Statement; AND/OR - p. 20 13. Proof by Contrapositive - p. 26 14. Proof by Contradiction - p. 32 15. Chapter 3. Special Kinds of Theorems - p. 42 16. “If and Only If” or Equivalence Theorems - p. 42 17. Use of Counterexamples - p. 53 18. Mathematical Induction - p. 56 19. Existence Theorems - p. 72 20. Uniqueness Theorems - p. 75 21. Composite Statements - p. 82 22. Equality of Numbers - p. 92 23. Chapter 4. Some Mathematical Topics on Which to Practice Proof Techniques - p. 95 24. Basic Set Theory and Indexed Families - p. 95 25. Power Sets or Parts of a set - p. 110 26. Cartesian Product of Sets - p. 114 27. Indexed Families of Sets - p. 117 28. About Functions - p. 121 29. Composition of Functions - p. 129 30. A Little More About Functions and Sets - p. 134 31. Relations - p. 139 32. Most Common Properties of Relations - p. 143 33. More About Equivalence Relations - p. 147 34. A Special Relation and More Facts About Equivalence Classes - p. 150 35. The Basics of Groups - p. 160 36. Some Properties of Binary Operations - p. 164 37. Special Elements - p. 171 38. When the Properties Fit Together - p. 176 39. Group Sizes and Structures - p. 184 40. Groups (mod m) and Arithmetic (mod m) - p. 188 41. Permutations and Symmetric Groups - p. 196 42. Isomorphism and Subgroups - p. 200 43. An Important Theorem - p. 206 44. Limits - p. 209 45. Getting Closer - p. 214 46. Functions and Limits - p. 223 47. Cardinality and Sizes of Infinity - p. 236 48. Chapter 5. Review Exercises - p. 247 49. Exercises Without Solutions - p. 252 50. Collection of “Proofs” - p. 262 51. Solutions for the Exercises at the End of the Sections and the Review Exercises - p. 268 52. Chapter 2 Basic Techniques to Prove If/Then Statements - p. 268 53. Chapter 3 Special Kinds of Theorems - p. 280 54. Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques - p. 309 55. Solutions for the Review Exercises - p. 374 56. Index - p. 397
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and how they work. The book offers techniques for both reading and writing proofs, discusses techniques in proving if/then statements by contrapositive and proofing by contradiction, includes the negation statement, and/or, examines various theorems, such as the if and only-if, equivalence theorems, existence theorems, and the uniqueness theorems. In addition, the use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are also covered. The book also provides mathematical topics for practicing proof techniques.
Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book accessible as well as invaluable.
1. Cover image - p. i 2. Title page - p. ii 3. Table of Contents - p. iii 4. Copyright - p. iv 5. Dedication - p. v 6. Preface - p. ix 7. List of Symbols - p. xi 8. Chapter 1. Getting Started - p. xiv 9. Chapter 2. Basic Techniques to Prove If/Then Statements - p. 6 10. What Does “If/Then” Mean? - p. 6 11. Direct Proof - p. 11 12. The Negation of a Statement; AND/OR - p. 20 13. Proof by Contrapositive - p. 26 14. Proof by Contradiction - p. 32 15. Chapter 3. Special Kinds of Theorems - p. 42 16. “If and Only If” or Equivalence Theorems - p. 42 17. Use of Counterexamples - p. 53 18. Mathematical Induction - p. 56 19. Existence Theorems - p. 72 20. Uniqueness Theorems - p. 75 21. Composite Statements - p. 82 22. Equality of Numbers - p. 92 23. Chapter 4. Some Mathematical Topics on Which to Practice Proof Techniques - p. 95 24. Basic Set Theory and Indexed Families - p. 95 25. Power Sets or Parts of a set - p. 110 26. Cartesian Product of Sets - p. 114 27. Indexed Families of Sets - p. 117 28. About Functions - p. 121 29. Composition of Functions - p. 129 30. A Little More About Functions and Sets - p. 134 31. Relations - p. 139 32. Most Common Properties of Relations - p. 143 33. More About Equivalence Relations - p. 147 34. A Special Relation and More Facts About Equivalence Classes - p. 150 35. The Basics of Groups - p. 160 36. Some Properties of Binary Operations - p. 164 37. Special Elements - p. 171 38. When the Properties Fit Together - p. 176 39. Group Sizes and Structures - p. 184 40. Groups (mod m) and Arithmetic (mod m) - p. 188 41. Permutations and Symmetric Groups - p. 196 42. Isomorphism and Subgroups - p. 200 43. An Important Theorem - p. 206 44. Limits - p. 209 45. Getting Closer - p. 214 46. Functions and Limits - p. 223 47. Cardinality and Sizes of Infinity - p. 236 48. Chapter 5. Review Exercises - p. 247 49. Exercises Without Solutions - p. 252 50. Collection of “Proofs” - p. 262 51. Solutions for the Exercises at the End of the Sections and the Review Exercises - p. 268 52. Chapter 2 Basic Techniques to Prove If/Then Statements - p. 268 53. Chapter 3 Special Kinds of Theorems - p. 280 54. Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques - p. 309 55. Solutions for the Review Exercises - p. 374 56. Index - p. 397
