| Week 1: | ||
| Wednesday, September 5th | Introduction. Sets: Venn
diagrams; set identities. |
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| Friday, September 7th | Logic: propositional calculus; logical operators. | |
Week 2: |
Reading: Notes on logic by Jacques Verstraete are posted here. Also compare with Rosen. |
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| Monday September 10th |
Logic: The conditional and biconditional (equivalence) statements. | |
| Wednesday, September 12th | Logic: tautologies and contradictions. Examples of Direct Proofs and Proof by Contradiction. | Assignment
1 assigned. |
| Friday, September 14th | Logic: combinatorial circuits; circuit complexity. | |
Week 3: |
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| Monday September 17th | Asymptotic Bounds. Polynomial and Exponential Time Algorithms. | |
| For more background on the topic of this and the next lecture see these notes (especially Section 5). | ||
| Wednesday, September 19th | Example: P and NP.
Notes on the definition of NP, by Bruce Shepherd: pages 29-31 of this document. |
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| Friday, September 21st | Arrow's impossibility
theorem. See these
notes. |
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Week 4: |
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| Monday September 24th | Logic: predicate calculus.
|
Assignment 1 due. |
| Wednesday, September 26th | Logic: proofs by induction. | Assignment 2 assigned. |
| Friday, September
28th |
Number Theory: prime factorisation. (Reading:"Why isn't the fundamental theorem of arithmetic obvious".) | |
Week 5: |
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| Monday October 1st | Number Theory: applications
of unique prime factorizations. |
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| Wednesday, October 3rd | Number Theory: Greatest
common divisiors. |
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| Friday, October 5th | Number Theory: Euclid's Algorithm. | |
Week 6: |
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| Monday October 8th | Thanksgiving. No lecture. | |
| Wednesday, October 10th | Number Theory: Modular
arithmetic. Supplemental notes on modular arithmetic, by Jacques Verstraete. (NOTE: there are two errors in these notes. In the first example on page 4, the 3 should be 13. In the second example, the computation has a mistake; the correct answer to 121x=23(mod 501) is 83, not 151.) |
Assignment 2 due. |
| Friday, October 12th | Number Theory: Modular
arithmetic and prime fields. |
. |
Week 7: |
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| Monday, October 15th | Number Theory: Modular Equations. | |
| Wednesday, October 17th | Number Theory: Fermat
Test, Miller-Rabin Test, Recognizing Primes. |
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| Friday, October 19th | Review for midterm. | |
Week 8: |
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| Monday, October 22th |
Midterm.
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| Wednesday, October 24th | Example: cryptography. | |
| Wednesday, October 24th | ||
| Friday, October 26th | Combinatorics: functions, counting. | |
Week 9: |
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| Monday, October 29th | Combinatorics: subsets; the Binomial theorem. | |
| Wednesday, October 31st | Combinatorics: Pascal's triangle. | |
| Friday, Nov 2nd |
Combinatorics: solving recurrence equations. | |
Week 10: |
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| Monday, November 5th | Combinatorics: inclusion-exclusion. | |
| Wednesday, November 7th | Combinatorics: inclusion-exclusion; the derangement problem. | |
| Friday, November 9th | Combinatorics: inclusion-exclusion; Euler's function. | |
Week 11: |
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| Monday, November 12th. | Combinatorics: the Multinomial theorem. | |
| Wednesday, November 14th | Graph Theory: the Handshaking lemma. Euler circuits. (Supplemental material: Early writings on graph theory: Euler circuits and the Königsberg bridge problem.) | |
| Friday, November 16th | Graph Theory: Hamiltonian cycles. | |
Week 12: |
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| Monday, November 19th. | Graph Theory: trees. | |
| Wednesday, November 21st | Graph Theory: representing spanning trees. | |
| Friday, November 23rd | Graph Theory: counting spanning trees. | |
Week 13: |
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| Monday November 26th | Graph Theory: planar graphs. | |
| Wednesday, November 28th | Graph Theory: graph coloring; characterising bipartite graphs. | |
| Friday, November 30th |
Graph Theory: the 4-color theorem; the 5-color theorem. | |
| Week 14: | ||
| Monday, December 3nd | Game Theory |
|
| Wednesday, December 5th |
Review |