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Math 220
Spring 2017 Journal

Thursday, May 4
The final examination was given, and solutions are available.
Thursday, April 27
We followed up on the exercise from the end of last class, summarized the most memorable experience from the past semester, and reviewed for the final exam by listing all the main concepts. The notes from class are available.
Tuesday, April 25
We discussed unique factorization of integers and an application to the greatest common divisor. The notes from class are available. Then we worked in groups on Exercises 8, 17, and 18 in Section 5.4 (pages 186–187).
Saturday, April 22
Here are some review exercises from Chapter 5 that you can use to help prepare for the comprehensive final exam to be held on Thursday, May 4. All of these exercises have either answers or hints in the back of the book.
Section 5.1
5(a)(d), 11(a), 19(a)
Section 5.2
4(a), 8(d), 16(a)
Section 5.3
1(a), 4(a), 11, 16(a)
Section 5.4
4, 7, 14, 18
Thursday, April 20
We worked on proofs of two propositions: namely, if a prime \(p\) divides a product \(ab\), then either \(p \mid a\) or \(p\mid b\); and the irrationality of the square root of \(2\).
Reminders: The final draft of the paper is due on April 25, and the final examination takes place on the afternoon of Thursday, May 4 from 3:00 to 5:00.
Tuesday, April 18
We discussed the Euclidean algorithm and worked examples. The notes from class are available.
The assignment to hand in next time is Exercise 10(b) on page 180 in Section 5.3.
Thursday, April 13
We worked on some more induction problems, including Exercises 5, 6, 7, and 11 in Section 5.2. The notes from class are available.
There is no assignment to hand in next class.
Tuesday, April 11
We worked on the method of mathematical induction, including Exercises 1(a) and 4(b) in Section 5.2. The notes from class are available.
The assignment to hand in next time is Exercise 1(b) in Section 5.2.
Thursday, April 6
We expanded the game by including order axioms in the set of axioms for the integers, and we discussed the notion of well ordering. The notes from class are available.
The assignment to hand in next time is Exercise 5(c) on page 157 in Section 5.1, which is Q7 on page 154. Again, the goal is to justify each step by citing either one of the axioms A1–A10 or one of the previously proved propositions P1–P8 or Q1–Q6.
Tuesday, April 4
We worked in groups on parts (a)–(d) of Exercise 1 on page 157. The assignment to hand in next class is part (e) of this exercise, which is P8 on page 153. The goal is to justify each step by citing either one of the axioms A1–A8 or one of the previously proved propositions P1–P7.
Thursday, March 30
I returned the graded exams. We began a new chapter, concerning properties of the integers, and wrote our first mathematical poem. The notes from class are available.
The assignment is to finish the peer editing task, which is due in class on Tuesday, April 4.
Tuesday, March 28
The second exam was given, and solutions are available.
The due date for the peer editing assignment is hereby changed from March 30 to April 4. Bring your edits to class on April 4 to give to the author, and also post a digital copy of your comments to eCampus.
Friday, March 24
I posted a solution to Exercise 15 that we worked on in class yesterday.
If you left something important behind in the classroom yesterday, then please get in touch with me—I might have what you lost.
Thursday, March 23
We reviewed for the examination to be given on Tuesday, March 28. The notes from class are available.
Wednesday, March 22
I have posted an anonymous feedback form for you to write comments about what you would like me to change or keep the same during the rest of the semester. Please answer the three questions to let me know your thoughts.
Here are some review exercises that you can use to help prepare for the upcoming exam to be held on Tuesday, March 28. All of these exercises have answers in the back of the book, so you can confirm your solutions.
Section 3.1
3(a)(d)(g), 10(a), 18(a)(d)(g), 21(a)
Section 3.2
2(a)(d), 13(b)(d), 24(a)
Section 3.3
1(a), 3(a), 7(a), 10(a)(d) (you did this before), 13(a)
Section 4.1
2(a)(d), 7(a)(d), 10(a)(d), 23(a), 25(a)
Section 4.2
3(a)(d), 4(a), 5(a), 12(a), 16(a)
Tuesday, March 21
We discussed the key words in Section 4.2: relation, reflexive, symmetric, antisymmetric, transitive, equivalence relation, equivalence class, partial ordering, linear ordering. And we swapped around the draft papers for peer review. The notes from class are available.
The assignment for next time (not to hand in) is to make a list of the main concepts from Chapters 3 and 4 (in preparation for the exam to be given on Tuesday, March 28).
Thursday, March 9
Reminder: The draft paper is due in class (two hard-copy printouts) on Tuesday, March 21.
In class today, we followed up on the concept of binary operations. The notes from class are available.
The assignment for Spring Break is to travel safely.
Tuesday, March 7
The due date for the first draft of the paper is hereby changed from March 9 to Tuesday, March 21 (after Spring Break). In class, we discussed the notion of a binary operation on a set. The notes from class are available.
The assignment due next class is to type up a solution to Exercise 2 on page 134 (which we worked on in groups during class).
Thursday, March 2
We discussed composite functions and inverse functions. The notes from class are available. I also posted the notes from last class, which I forgot to post on Tuesday.
The assignment due next class is to type up a solution to Exercise 10 on pages 119–120 in Section 3.3, which we started in class. Notice that the problem wraps over to page 120.
Tuesday, February 28
The Blocker building closed around the beginning of class due to a plumbing issue, so anybody who was running late could not enter the building; but the rest of us were able to meet. I returned the graded exams. We discussed notions of injective functions, surjective functions, bijective functions, and permutations. Then we worked in groups on Exercises 1 and 12 on pages 105–107 in Section 3.2.
The assignment due next class is to type up solutions to Exercises 8 and 20 on pages 106 and 108 (about the definitions of the words “surjective” and “injective”).
Thursday, February 23
We started Chapter 3, discussing the notions of function, domain, codomain, image, and inverse image. We worked on Exercises 7 and 17 on pages 94–95. A solution to Exercise 7 is that an element \(b\) of the codomain \(B\) is not in the image of the function \(f\) if there is no element \(a\) of the domain \(A\) for which \(f(a)=b\); alternatively, if \(f(a)\ne b\) for every element \(a\) of \(A\). The notes from class are available.
The outline for the paper is due today in eCampus. The assignment due next class is to type up a solution to Exercise 15 on page 95 in Section 3.1. Notice that this exercise is part 2 of Proposition 3.1.6 on page 89, which the author left to the reader.
Tuesday, February 21
The first exam was given, and solutions are available.
Thursday, February 16
We reviewed for the exam to be given next class and also worked in groups on the following exercises:
  • D1, page 46
  • D4, page 46
  • D1, page 60
  • D2, page 60
  • D2, page 80
The notes from class are available. The assignment is to review for the exam. Also, remember that the outline for the paper is due on Thursday, February 23.
Wednesday, February 15
Here are some review exercises that you can use to help prepare for the upcoming exam to be held on Tuesday, February 21. All of these exercises have answers in the back of the book, so you can confirm your solutions.
Section 1.1
2(a)(d)(g)(i), 3(a)(d)(g)(i), 9 (you did this one before)
Section 1.2
5(a)(d), 12(a), 15(a)
Section 1.3
3(a), 7(a)(c), 12, 17(a)(d)
Section 1.4
1, 12(a), 15(a)
Section 2.1
8(a)(e), 10(a), 20(a)(e), 21
Section 2.2
7(a), 24(a), 27
Section 2.3
11, 13, 20
Tuesday, February 14
We discussed the notions of Cartesian product, power set, partition, and the pigeonhole principle. We solved Exercise 5 on page 78 in groups. The notes from class are available.
The assignment for next time (not to hand in) is to make yourself a list of the main topics from Chapters 1 and 2 (in preparation for the first exam, which takes place on Tuesday, February 21).
Thursday, February 9
We discussed union, intersection, and De Morgan’s laws. Then we solved in groups Exercises 1, 3, and 26 on pages 68–71 in Section 2.2. The notes from class are available.
The assignment to hand in next time is Exercise 22 on page 70. One way to prove that two sets are equal is to show that each set is a subset of the other. An alternative method is to show the equality directly by applying the algebra of sets (the commutative, associative, and distributive laws and De Morgan’s laws: Proposition 2.2.2, Theorem 2.2.3, and Theorem 2.2.4).
Tuesday, February 7
We discussed various notions about sets: elements, subsets, complements, intervals in the real numbers, the empty set, and cardinality. The notes from class are available.
The assignment is to finish writing your termpaper proposal, due on February 9 before midnight (to be submitted at eCampus).
Thursday, February 2
We revisited various equivalent formulations of implication; solved together Exercise 4 on page 44; solved in groups an exercise on the algebra of the logical connectives \(\wedge\), \(\vee\), \(\implies\), and \(\iff\); made a list of the main concepts from Chapter 1; and solved in groups Exercise D5 on pages 46–47.
There is no assignment to hand in next class, but you should be working on your termpaper proposal, which is due on Thursday, February 9. I have posted a link in eCampus that you should use to submit the proposal.
Tuesday, January 31
We discussed the notions of necessary conditions, sufficient conditions, contrapositive, converse, and biconditional. We worked in groups on Exercise 17 on page 37 in Section 1.3. I have posted what the notes might have been if the classroom computer had been working.
The assignment to hand in next time is to type up in complete sentences a solution to Exercise 17 on page 37 in Section 1.3.
Thursday, January 26
In class, we discussed the exercises originally due today, the notion of logical equivalence, and the truth table for implication. The notes from class are available.
Here is the assignment to hand in next time.
  • Revise and correct the two problems originally due today.
  • Exercise D3 on page 28.
  • Exercise 12 on page 36.
Tuesday, January 24
In class, we discussed conjunction and disjunction, negations of conjunctions and disjunctions, and truth tables. We solved Exercise 4 on page 26 in Section 1.2 in groups. The notes from class are available.
The assignment to hand in next time is Exercises 8 and 14 on pages 26–27 in Section 1.2.
Thursday, January 19
We looked at some more examples of statements with quantifiers, and we discussed in groups the assignment originally due today and Exercise 9 on page 14 in Section 1.1. The notes from class are available.
I did not yet collect the assignment that was originally due today. You should revise your solution in light of the discussion during class and turn in the revision next class. Additionally, type up and turn in next class a solution to Exercise 9 on page 14. I have posted an annotated version of the exercise that can be opened on the web as well as a standalone pdf file.
If you have not yet signed up for a topic for the term paper, remember to pursue that task. I have updated the list of topics to show which ones have already been claimed.
Tuesday, January 17
In class, we discussed the notions of statements, open sentences, universal and existential quantifiers, and negation. This material is contained in Section 1.1 of the textbook, and the notes from class are available.
The assignment to hand in next time is available both in a form that can be opened on the web and as a standalone pdf file.
Monday, January 16, 2017
Welcome to Math 220. There will be regular updates here about assignments and the highlights of each class meeting.