Please see below for lecture summaries, homework and other study material. For your grades, please see Canvas.
| Date | Topics |
| Jan.10. | Basic definitions and notation about sets. Notes. |
| Jan.12. | Cartesian products of sets. Power sets.
Notes. Homework 1 (due Wed., Jan. 19): 1.1: 3, 12, 18, 29, 31; 1.2: 9, 15; 1.3: 13, 14, 15; 1.4: 14, 15. |
| Jan.14. | Cardinalities of power sets. Set unions, intersections, differences and complements. Notes. |
| Jan.17. | Martin Luther King Holiday, no class. |
| Jan.19. | Using Venn diagrams. Proofs of set equalities. Notes.
Homework 2 (due Wed., Jan. 26): 1.5: 2, 6, 10; 1.6: 2, 4, 6; 1.7: 4, 10; 1.8: 2, 4, 8, 14. |
| Jan.21. | Indexed sets. Statements. "And", "or", and "not" operators. Notes. |
| Jan.24. | Conditional statements. How to prove various types of statements. Logical equivalences. Notes. |
| Jan.26. | DeMorgan's Law for logic. From logic to sets. Open sentences and
quantifiers. Notes. Homework 3 (due Wed., Feb. 2): 2.1: 2, 4, 10; 2.4: 1, 4, 5; 2.6: 2, 3, 12. 2.7: 4, 8. |
| Jan.28. | The multiplication, addition, and subtraction principles for counting. Example problems. Notes. |
| Jan.31. | Practice problems. Factorials. Permutations vs combinations. Notes. |
| Feb.02. | Class cancelled due to inclement weather. |
| Feb.04. | Binomial coefficients. Combinatorial identities. Pascal's triangle.
Notes. Homework 4 (due Wed., Feb. 9): 3.3: 2, 4, 8, 10; 3.4: 3, 10, 12; 3.5: 4, 6, 10, 12, 18. |
| Feb.07. | The binomial theorem and its applications. Another combinatorial identity. Notes. |
| Feb.09. |
The inclusion-exclusion principle.
Notes. Homework 5 (due Wed., Feb. 16): 3.6: 3, 7, 12; 3.7: 2, 4, 6, 8; 3.8: 2, 4, 6, 8. 10. |
| Feb.11. | Worksheet on permutations and combinations.. (Solutions) |
| Feb.14. | Multisets. The bars-and-stars method. Notes |
| Feb.16. | More bars-and-stars problems. The word problem (permutations of multisets).
Notes. Homework 6 (due Wed., Feb. 23): 3.8: 5, 7, 9, 11, 13, 14; 3.9: 2, 4; 3.10: 5, 6. |
| Feb.18. | Summary of counting problem types. The pigeonhole principle. Notes. |
| Feb.21. | The division principle. Combinatorial proofs.
Notes. Worksheet on counting problems involving multisets (Solutions). Topics for Midterm I. |
| Feb.23. | More on combinatorial proofs. Homework discussion. List of midterm topics.
Notes. No homework due next week. |
| Feb.25. | Midterm 1. |
| Feb.28. | Direct proofs of conditional statements. Case discussions. Notes. |
| Mar.02. | More direct proofs. Worksheet on direct
proofs. Notes. Homework 7 (due Wed., Mar. 09): Chapter 4: 4, 10, 13, 16, 20, 26; Chapter 5: 6, 10, 15, 20. |
| Mar.04. | Congruence of integers. Contrapositive proofs. Notes. |
| Mar.07. | Proof by contradiction. Recommendations for mathematical writing. Notes. |
| Mar.09. | The proof that there are infinitely many primes. Notes. Second worksheet on proofs.
Homework 8 (due Wed., Mar. 16): Chapter 5: 4, 9, 12, 28; Chapter 6: 8, 9, 10, 14, 15. |
| Mar.11. | Proofs of equivalences and some non-conditional statements. Notes. |
| Mar.16. | GCDs and the Euclidean algorithm. Constructive vs. non-constructive proofs. Notes. Homework 9 (due Wed., Mar. 30): Chapter 7: 6, 12, 13, 23, 26, 31, 32, 33. Chapter 8: 2, 19, 28. |
| Mar.18. | More proof examples from Chapter 7. Proofs of set containments. Notes. |
| Mar.28. | Proofs of set equalities. Disproofs and counter-examples. Notes. |
| Mar.30. | More disproofs. Notes.
Third worksheet on proofs.
No homework due next week. |
| Apr.01. | Midterm 2. |
| Apr.04. | Proof by mathematical inductions: strategy and examples. Notes. |
| Apr.06. | More proofs by induction. Introduction to strong mathematical induction. Notes. Homework 10 (due Wed. Apr. 13): Chapter 10: 2, 8, 15, 16, 18, 19, 20. |
| Apr.08. | More strong induction proofs. Graphs and trees. Notes. |
| Apr.11. | More recursions and inductions: the fundamental theorem of arithmetic, Fibonacci numbers. Notes. |
| Apr.13. | More on the Fibonacci sequence. Notes.
Worksheet on (strong) mathematical induction. Homework 11 (due Wed. Apr. 20): Chapter 10: 6, 21, 23, 24, 26, 30, 42. |
| Apr.15. | Discussion for the induction worksheet. Notes. |
| Apr.18. | Relations. Reflexivity, symmetry and transitivity. Equivalence relations. Notes. |
| Apr.20. | Proofs verifying a relation is an equivalence relation. Homework discussion. Notes. Homework 12 (due Wed. Apr. 27): 11.1: 2, 4, 8; 11.2: 4, 8, 15; 11.3: 4, 8. |
| Apr.22. | Partitions from equivalence relations. Notes. |
| Apr.25. | Basic notions and proofs regarding functions. Notes. |
| Apr.27. | Review for final exam. Notes. |