Mathematics 201: Introduction to Proofs
Textbooks:
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Syllabus:Sets, logic, proofs in algebra and calculus.
Learning objectives:This class is designed to guide students through the transition from example-based calculus courses to advanced proof-based classes in mathematics. Emphasis is placed on learning how to recognize and handle valid mathematical statements, to create proofs of true statements, and to disprove false statements. A second objective is to learn how to communicate mathematics effectively, both in written and spoken form. Thus it should be appreciated, for example, that a string of disconnected statements cannot constitute a valid argument. Proof-writing should be recognized as an art, where the level of detail to be included in the proof of a statement has to be matched to the level of that statement and to the intended audience.Sample Assignments:
Read (BP) Sections 1.3 to 1.7; do (BP) Exercises 1.3: 2, 4, 10 (p.14); 1.4: 4, 5 (p.16); 1.5: 2(a)-(c), 4(f)(g)(h) (p.18); 1.6: 2(a),(f),(g) (p.20); 1.7: 10, 12 (p.23). Read (BP) Section 1.8; do (BP) Exercises 1.8: 2, 6, 9, 14 (p.27). Read (BP) Sections 2.1, 2.2; do (BP) Exercises 2.2: 1, 3-5, 7-10. (p.39). Read (BP) Sections 2.3, 2.4; do (BP) Exercises 2.3: 2, 4, 8, 10 (p.42); 2.4: 2, 4 (p.44). Read (BP) Sections 2.5, 2.6; do (BP) Exercises 2.5: 10 (p.46); 2.6: 6, 8, 12, 14 (p.49). Read (BP) Sections 2.7 to 2.9; do (BP) Exercises 2.7: 4, 8, 10 (p.51); 2.9: 2, 4, 6, 10 (p.55). Read (BP) Section 2.10; do (BP) Exercises 2.10: 2, 4, 6, 8 (pp.58-9). Read (BP) Sections 4.1 to 4.3; do (BP) Exercises 4: 2, 4, 6, 10, 12 (p.98). Read (BP) Sections 4.4, 4.5; do (BP) Exercises 4: 14, 16, 18, 20, 26 (pp.98-9). Read (BP) Sections 5.1, 5.3; do (BP) Exercises 5: 2, 4, 6, 12, 16 (p.108). First Graded Homework Read (BP) Sections 6.1, 6.2; do (BP) Exercises 6: 2, 6, 8, 10 (p.116). Read (BP) Chapter 7; do (BP) Exercises 7: 2, 4, 12, 18, 20 (p.127). Read (BP) Sections 8.1 to 8.3; do (BP) Exercises 8: 4, 6, 8, 20, 22 (p.143). Read (BP) Chapter 9; do (BP) Exercises 9: 4, 12, 16, 28, 30, 34 (p.151). Test #1. Read (BP) Section 3.4; do (BP) Exercises 3.4: 2, 4, 8, 10, 12 (p.78). Read (BP) Chapter 10 through p.158; do (BP) Exercises 10: 2, 4, 6, 8, 16 (pp.167-8), using induction or otherwise. Read (BP) Section 10.1; do (BP) Exercises 10: 10, 18, 22, 24 (pp.167-8), using induction or otherwise. Read (BP) Sections 11.0, 12.1; do (BP) Exercises 11.0: 2, 4, 12, 14 (p.176); 12.1: 2, 4, 8, 10 (pp.198-199). Read (BP) Sections 12.2, 12.4, 12.6; do (BP) Exercises 12.2: 2, 4 (p.202); 12.4: 2, 6 (p.208); 12.6: 2, 6 (p.214). Read (BP) Sections 12.5, 14.1 (through Theorem 14.1); do (BP) Exercises 12.2: 9, 10 (p.202); 12.5: 2, 6 (p.212); 14.1: 4, 6, 8 (p.274). Read (BP) Sections 14.1, 14.2; do (BP) Exercises 14.2: 2, 4, 6, 8 (p.280). Second Graded Homework. Read (BA) Section 1.1; do (BA) Exercises 1.1: 1, 2, 4, 6, 9 (p.24). Read (BA) Sections 1.2.1 (not proof of Ex. 1.2.3), 1.2.3, 1.2.4; do (BA) Exercises 1.2: 4, 7, 9 (p.30). Read (BA) Section 1.2.2; do (BA) Exercises 1.2: 1, 2, 8 (p.30). Read (BA) Section 1.3; do (BA) Exercises 1.3: 1, 2, 3, 5 (p.34). Read (BA) Section 2.1 through Section 2.1.1; do (BA) Exercises 2.1: 1, 2, 9, 10 (pp.49-50). Test #2. Complete reading of (BA) Section 2.1; do (BA) Exercises 2.1: 3, 4, 13, 14, 15, 22 (pp.49-50). Read (BA) Section 2.2.1 and Prop. 2.2.5; do (BA) Exercises 2.2: 1, 2, 3, 5, 7 (p.60). Read (BA) Sections 3.2.1-3; do (BA) Exercises 3.2: 1, 3, 5 (prove continuity at x = 0), (p.108). Read (BA) Section 2.4; do (BA) Exercises 2.4: 1, 4, 5 (p.71). Read (BA) Sections 2.5.1-3; do (BA) Exercises 2.5: 1, 2, 3(b)(d) (p.81). Third Graded Homework. Read (BA) Prop. 2.5.14. Final. |