Class Journal

We will keep track of what we cover in class each day through this journal. It won't contain any of the nitty-gritty details of what we have done in a given class, but will instead serve to summarize the key theorems we proved or discussions we had.

Week 1

• Monday, 1.31.11: Dana briefly summarized the structure of the course. We plan to discuss the course in more detail on Wednesday. We discussed the meaning of theorem and proof; many interesting points were made by various students. Dana discussed Definitions 1.1 and 1.2, which introduce even and odd, respectively. With Dana as scribe, the class produced a proof of Theorem 1.3. AS bravely volunteered to present Theorem 1.4 on the spot. AS did an excellent job, but we ran out of time and the last line needed some fixing. Great first go!
• Wednesday, 2.2.11: No class...snow day!
• Friday, 2.4.11: Excellent first day of presentations! After briefly discussing the syllabus, we jumped right into student presentations. PC was first up and dispensed with Theorem 1.5. Next, CQ provided a counterexample to 1.6. Right before the end of class, JD quickly jotted down a proof of Theorem 1.7. Great discussions on all the problems.

Week 2

• Monday, 2.7.11: We began by taking a quick look at the first couple items on the Elements of Style of Proofs handout. Next, we briefly discussed the difference between "$n$ divides $m$" and "$\frac{m}{n}$". We then jumped right into student presentations. ZG made short work of Theorem 1.10. MS quickly dispensed with Theorem 1.11. AW and RC2 simultaneously wrote up solutions to Problem 1.14 and Theorem 1.15, respectively. Each presentation provided the backdrop for great class discussions.
• Wednesday, 2.9.11: As a class, we discussed 1.17, 1.21, and 1.25. RC1 and BC came to the board and hammered out a couple truth tables for 1.23. We left 1.24, 1.28, and 1.29 to discuss on Friday.
• Friday, 2.11.11: We had students simultaneiously write-up solutions on the board to 1.24, 1.28, 1.29, 1.32, 1.33, 1.34, and 1.35. The following students wrote-up solutions: PC, AD, PH, SJ, JL, AO, IR, and AS. With the time we had left, students worked in small groups on the next couple problems in the course notes.

Week 3

• Monday, 2.14.11: Another good class. AS presented 1.36, SJ provided a counterexample for 1.37, PW put 1.38 on the board, and RC2 and CQ presented proof by contrapostion for 1.40 and 1.41, respectively.
• Wednesday, 2.16.11: No class (Winter Carnival).
• Friday, 2.18.11: CQ presented 1.43. RC1 proved 1.45 and RC2 provided an alternate and shorter proof. TB showed us 1.49 and SB went above and beyond the call of duty on 1.50. As a class, we discussed 1.44, 1.46, and 1.47.

Week 4

• Monday, 2.21.11: Dana did a quick summary of (i) direct proof, (ii) proof via the contrapositive, and (iii) proof by contradiction for propositions of the form $A\implies B$. AO proved 1.52 via the contrapositive while RC2 tacked the same theorem via a proof by contradiction. AW presented 1.53 and used a proof by contradiction.
• Wednesday, 2.23.11: Dana reviewed the basic concepts of free variables, predicates, and quantification. As a group, we discussed most of the exercises. We also had NG, MS, and BC present 1.62, 1.63, and 1.65, respectively.
• Friday, 2.25.11: Due to the weather and closing of campus, we treated class as an optional review session.

Week 5

• Monday, 2.28.11: We had people put up solutions to all the problems at once. MS, TV, PC, ZG, and AW wrote up solutions to 1.68, 1.71, 1.74, 1.75, and 1.76, respectively. NG and RC2 each presented half of 1.78.
• Wednesday, 3.2.11: As a class, we discussed 1.88, 1.89, and 1.90. CQ and RC1 each tackled half of 1.85 and BP and SJ each presented parts of 1.92.
• Friday, 3.4.11: We had the in-class portion of Exam 1.

Week 6

• Monday, 3.7.11: We worked in small groups on the problems assigned for the first half of Section 2.1, which deals with an introduction to set theory.
• Wednesday, 3.9.11: SB presented 2.10, MS proved 2.11 (Transitivity of Subset), PC and BP tag teamed 2.20, and CQ presented a few parts of 2.21.
• Friday, 3.11.11: AW did a great job presenting 2.22 and RC2 had a clever approach involving the use of a table to tackle 2.26.

Week 7

• Monday, 3.14.11: $\pi$ day! AS presented one direction of 2.36.
• Wednesday, 3.16.11: Dr. Maatta covered from me while I was out of town. Students presented 2.38 and 2.39. The class discussed 2.40, 2.41, 2.42, and 2.43, which are all related to Russell's Paradox.
• Friday, 3.11.11: No class today.

Week 8

• Spring Break!

Week 9

• Monday, 3.28.11: $\pi$ day! AS presented one direction of 2.36.
• Wednesday, 3.30.11: IR, PH, ZG, BP, BC, and AS each presented 2.48-2.53, respectively. As a class we discussed 2.55 and 2.56. CQ did an excellent proof of 2.58(1).
• Friday, 4.1.11: MS, RC1, BP, and CQ each represented 2.69, 2.70, 2.72, and 2.73, respectively. As a class we discussed 2.67 and 2.71. Great class!

Week 10

• Monday, 4.4.11: No class.
• Wednesday, 4.6.11: Lots of good discussion! RC1 quickly dispatched with 2.74, NG and CQ tag-teamed 2.75 with a little help from the class, and RC2 and AS simultaneously presented 2.76 and 2.77.
• Friday, 4.1.11: RC2 simultaneously took care of 2.78(1) and 2.81(2). BP with KD's help, quickly dispatched 2.80. From his seat, MS described a perfect solution to 2.81. RC1 presented 2.83(1) and Dana quickly sketched a proof for the rest of 2.83.

Week 11

• Monday, 4.11.11: PC attempted to prove 2.85. After some class discussion, we wrote down a valid proof. For 2.86, 2.87, 2.88, and 2.91, we discussed them as a class with Dana as scribe. We didn't end up discussing 2.90.
• Wednesday, 4.13.11: TV made an attempt at 2.92, but there were some flaws that we discussed. BP (with CQ as scribe) presented one direction of 2.93. BP also provided a proof for 2.83. SJ made short work of 2.94 and 2.96. PH showed us an example for 2.98. Right at the end of class, MS quickly discussed 2.95.
• Friday, 4.15.11: We had the in-class portion of Exam 1.

Week 12

• Monday, 4.18.11: We went over the in-class portion of Exam 2.
• Wednesday, 4.20.11: Dana lectured over the first half of the notes for Section 3.1: Relations.
• Friday, 4.22.11: First, Dana discussed the relationship between digraphs and reflexive, symmetric, and transitive, which answered 3.25. AD explained how to do 3.20, ZG presented 3.22, and then BC talked us through 3.26. We finished class by having AW, AJ, RC1, and RC2 explain the answers to 3.27 (each did two parts).

Week 13

• Monday, 4.25.11: Great class. All of the problems and discussion centered around equivalence relations. PC presented 3.30, TV showed gave an example for 3.31, IR presented 3.32, AS tackled 3.35, and Dana discussed 3.36. CQ, AW, ZG, AS, RC2, and RC1 each presented two parts of 3.33.
• Wednesday, 4.27.11: We continued our discussion of equivalence relations. MS realized he made a mistake, but pushed on and nailed down 3.40. RC2 and CQ each presented one half of 3.41.
• Friday, 4.29.11: We went outside! We wheeled a portable chalk board outside and sat in a circle on the bricks by Center Lodge. BC discussed 3.51, both RC1 and CQ provided alternate solutions to 3.52, AS tackled 3.53, RC2 presented a painless proof to 3.54, RC1 came up a second time and discussed 3.55, and lastly CQ came up a second time and presented 3.58.

Week 14

• Monday, 5.2.11: Dana spent the first half of class discussing the big picture surrounding relations and partitions. After that RC2 presented 3.60 and 3.62 presented 3.62. We finished class with Dana proving 3.63 and 3.64.
• Wednesday, 5.4.11: NG, PH, AO, and BC each presented parts of 3.72/3.74, AW did an excellent job on 3.76, MS attacked 3.77, TB showed up 3.78, and RC2 proved 3.81.
• Friday, 5.6.11: Lots of presentations today! AD and IR each presented two parts of 3.83, JL presented 3.84, MS and AS each presented two part of 3.85, PW proved both directions of 3.86, and JL, RC1, IR, and CQ each presented parts of 3.89.

Week 15

• Monday, 5.9.11: Class went fast today! Dana discussed some of the underlying concepts related to function composition. PW and AS each presented two parts of 3.93. MS proved 3.94 (a classic) and then Dana talked about how to streamline the proof.
• Wednesday, 5.11.11: Where does the time go? I think we bit off more than we could chew today. SB presented one direction (the more difficult direction) of 3.102. BC made a great attempt at 3.106, but didn't handle the situation in his proof by contradiction where $f$ was not onto.
• Friday, 5.13.11: coming soon...