Math 4190, Summer I 2019
Math 4190, Summer I 2019
"Mathematics, rightly viewed, possesses not only truth, but supreme
beauty." --Bertrand Russell
An Open Textbook is a textbook published under a Creative Commons license, and is usually freely available online. Sometimes, print copies can be purchased at production costs. In this course, we will exclusively use Open resources.
- Radiolab story on Russell's paradox and undecidable problems (36:45--50:30).
Section 1: Sets and counting.
Section 2: Logic.
- Lecture 1.1: Basic set theory. [YouTube (60:20) | Slides]
- Lecture 1.2: Inclusion-exclusion. [YouTube (36:42) | Slides]
- Lecture 1.3: Permutations and combinations. [YouTube (41:41) | Slides]
- Lecture 1.4: Binomial and multinomial coefficients. [YouTube (38:43) | Slides]
- Lecture 1.5: Multisets and multichoosing. [YouTube (47:04) Slides]
- Lecture 1.6: Combinatorial proofs. [YouTube (47:51) | Slides]
Section 3: Basic number theory.
- Lecture 2.1: Propositions and logical operators. [YouTube (42:31) | Slides]
- Lecture 2.2: Tautology and contradiction. [YouTube (26:51) | Slides]
- Lecture 2.3: Equivalence and implication. [YouTube (42:59) | Slides]
- Lecture 2.4: Axiomatic systems. [YouTube (32:03) | Slides]
- Lecture 2.5: Proofs in propositional calculus. [YouTube (36:50) | Slides]
- Lecture 2.6: Proofs over a universe. [YouTube (37:45) | Slides]
- Lecture 2.7: Quantifiers. [YouTube (40:04) | Slides]
- Lecture 2.8: Set-theoretic proofs. [YouTube (47:30) | Slides]
- Lecture 2.9: Russell's paradox and the halting problem. [YouTube (41:26) | Slides]
Section 4: Relations and functions.
- Lecture 3.1: The pigeonhole principle. [YouTube (34:41) | Slides]
- Lecture 3.2: Parity, and proving existential statements. [YouTube (27:13) |
- Lecture 3.3: Proving universal statements. [YouTube (31:00) |
- Lecture 3.4: Divisibility and primes. YouTube (29:45) | Slides]
- Lecture 3.5: Rational and irrational numbers. [YouTube (31:44) | Slides]
- Lecture 3.6: Quotient, remainder, ceiling and floor. [YouTube (31:42) |
- Lecture 3.7: The Euclidean algorithm. [YouTube (41:24) | Slides]
Section 5: Cyptography.
- Lecture 4.1: Binary relations on a set. [YouTube (41:30) |
- Lecture 4.2: Equivalence relations and equivalence classes. [YouTube (52:14) | Slides]
- Lecture 4.3: Partially ordered sets. [YouTube (49:55) | Slides]
- Lecture 4.4: Functions. [YouTube (58:17) | Slides]
- Lecture 4.5: Cardinalities and infinite sets. [YouTube (57:09) | Slides]
- Lecture 5.1: Symmetric cryptographic
ciphers. [YouTube (34:15)
- Lecture 5.2: Public-key cryptosystems and RSA. [YouTube (44:09) | Slides]
- Lecture 5.3: Why RSA works. [YouTube (40:38) | Slides]
- Lecture 5.4: The Diffie-Hellman key exchange.
- Lecture 5.5: Error-correcting codes.
Most of the homework will be done on WeBWork, a free online homework system. One pedagogical advantage of this is that it gives you feedback right away on whether you got the answer right or wrong.
I will post the pdf of the WeBWork assignments here. There will also be some written homework assignments for problems involving written proofs.
- Homework 1: Basic set theory. pdf
- Homework 2: Venn diagrams. pdf
- Homework 3: Counting and Permutations. pdf
- Homework 4: Combinations and multisets. pdf
- Homework 5: Propositional logic (statements). pdf
- Homework 6: Propositional logic (truth tables). pdf
- Homework 7: Propositional logic (equivalence, proofs). pdf
- Homework 8: Quantifiers. pdf
- Homework 9: Quantifiers, pigeonhole principle. pdf
- Homework 10: Divisibility. pdf
- Homework 11: Quotient, remainder and the Euclidean algorithm. pdf
- Homework 12: Binary relations. pdf
- Homework 13: Equivalence relations and functions. pdf
- Homework 14: Cryptography. pdf