Contents Acknowledgements Preface vi 1 Preliminaries 1 1.1 A Short Note on Proofs 1 1.2 Sets and Equivalence Relations... 3 1.3 Exercises 13 References and Suggested Readings 15 1.4 Sage 16 1.5 Sage Exercises 。。。 20 2 The Integers 22 2.1 Mathematical Induction.. 2.2 The Division Algorithm 的 2.3 Exercises 2.4 Programming Exercises 31 References and Suggested Readings.. 31 2.5Sage...············ 31 2.6 Sage Exercises 35 3 Groups 36 3.1 Integer Equivalence Classes and Symmetries..... 36 3.2 Definitions and Examples 40 3.3 Subgroups.·.················· 。 45 3.4 Exercises 47 3.5 Additional Exercises:Detecting Errors:········ 50 References and Suggested Readings..、..············· 51 52 3.7 Sage Exercises 57 4 Cyclic Groups 59 4.1 Cyclic Subgroups············· 59 4.2 Multiplicative Group of Complex Numbers........... 62 4.3 The Method of Repeated Squares...·············· 65 4.4 Exercises····························· 67 4.5 Programming Exercises·:··.···················: 70 References and Suggested Readings..................... 70 71 ⅸContents Acknowledgements v Preface vi 1 Preliminaries 1 1.1 A Short Note on Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 The Integers 22 2.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Programming Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Groups 36 3.1 Integer Equivalence Classes and Symmetries . . . . . . . . . . . . . . . . . . 36 3.2 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Additional Exercises: Detecting Errors . . . . . . . . . . . . . . . . . . . . . 50 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Cyclic Groups 59 4.1 Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Multiplicative Group of Complex Numbers . . . . . . . . . . . . . . . . . . 62 4.3 The Method of Repeated Squares . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Programming Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ix