Math 4190, Summer I 2020

# Math 4190, Summer I 2020

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

### Open Textbooks

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.

### Resources

• Radiolab story on Russell's paradox and undecidable problems (36:45--50:30).

### Lecture Notes

Section 1: Sets and counting.
• 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 2: Logic.
• 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 3: Basic number theory.
• Lecture 3.1: The pigeonhole principle. [YouTube (34:41) | Slides]
• Lecture 3.2: Parity, and proving existential statements. [YouTube (27:13) | Slides]
• Lecture 3.3: Proving universal statements. [YouTube (31:00) | Slides]
• 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) | Slides
• Lecture 3.7: The Euclidean algorithm. [YouTube (41:24) | Slides]
Section 4: Relations and functions.
• Lecture 4.1: Binary relations on a set. [YouTube (41:30) | Slides]
• 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]
Section 5: Cyptography.
• Lecture 5.1: Symmetric cryptographic ciphers. [YouTube (34:15) | Slides]
• 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.

### Homework

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, as they are available. It should be noted that many numerical values are randomized for different students, so most of these problems will be different from the ones that you have.
• 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