**Test IV and Final Exam Resources (copied from my last Math 187 class page)**

Solutions to the worksheets now appear under week 15.

The hour exams are of course study resources for this test. Here is a set of solutions to your Test I (speaking in Fall 2013). Here is Test II with solutions (in fall 2013). Here is Test III with solutions (Fall 2013). All of the old hour exams are study resources as well (see below).

Here are detailed solutions to two of the problems on the RSA worksheet.

Here is the Spring 2013 final exam.

Here is the Spring 2012 final exam. You will not have a proof question about graph theory.

Comments on the following final papers were written in Spring 2012. Their coverage was not identical to yours but they should be useful.

M187Su06final.pdf A good study resource for you – everything here is something I might ask. This particular test has some irritating typos in it.

M187F06final.pdf we didnt do bubble sort; otherwise this is quite similar in coverage.

M187S07final.pdf This is a nice study resource for you, too. Since it is a summer exam, it has quite a lot of the later material (it is also test 4, as it were).

M187F08final.pdf Disregard the question about groups. This was from the semester where we were trying to save paper and made the students use blue books, so it doesn't have a separate page for each question.

M187S10final.pdf Also a good test. Notice that problems in the same area are not always phrased the same in these exams; these tests are not clones (some questions are pretty stereotyped, of course).

Review material for Test IV and the final will start to appear next week.

- Friday Dec 9: this will be a day of review. You should bring questions.
- Wednesday, Dec 7 (a date which will live in infamy): Lectured on section 39 (last lecture of the year).
Last homework assignment, due at the final: 39.1, 39.2, 39.9, 39.10 (to do this you need to read theorem 39.5, which describes the usual way of computing gcds using prime factorization), 39.11 (prove this using results in this section), 39.22 (yes, you really can prove this using the results of this section). If you have questions about these, I'll be quite willing to discuss them on Friday.

- Monday Dec 5: Talked about the RSA algorithm.
Homework (due Friday) is this worksheet. Notice that the worksheet gives answers: the point is to find the detailed work that gives the answers. Sometime tomorrow I am likely to add some example problems which are shorter; there are some such examples in the posted sample tests.

This gives fully worked solutions to the first two problems on the RSA worksheet.

This gives solutions to the modular exponentiation worksheet and the rest of the RSA worksheet (except problem 5, which is too large to be a test question).

**NEW**: Here is a worked example of the RSA algorithm in parts which may suggest how a test question would be structured. Notice that an actual question might have only some of these parts. You are welcome to turn in work for parts 5 and 6 of this handout as an additional optional homework set.

- Wednesday Nov 30th: gave more section 38 examples and did examples of the repeated squaring method of computing powers in modular arithmetic.
Here is a worksheet on modular exponentiation. I'll tell you tomorrow what "Fermat's little theorem" is; I am mostly interested in the repeated squaring calculations at this point. The worksheet is due Wednesday (so that you

**will**have complete information about Fermat's little theorem before you need to complete it) but**do**start computing those powers by repeated squaring now. You will want to be able to do this on test 4, and it requires practice. - Monday Nov 28th: discussed section 38 on solving linear equations in modular arithmetic and the Chinese remainder theorem.
Homework due on Friday: 38.1, 38.3, 38.4 (a bit of fun), 38.8 (for this you need to read and apply the result in problem 38.7, but you do not have to do problem 38.7; you should check that the answers you get in 38.8 are correct).

- Friday, Nov 18: covered section 37 on modular arithmetic.
Homework, due Wed after the break: 37.1 acegikmoq (the funny symbols stand for addition, subtraction, multiplication and division in modular arithmetic), 37.2 (Z

_{n}refers to mod n arithmetic), 37.3, 37.4, 37.5 (optional but has a very interesting answer), 37.14 (we will have much more to say about exponentiation in modular arithmetic later). - Wednesday, Nov 16: covered section 36.
Homework: 36.2, 36.5, 36.11, 36.12, 36.13, 36.14, 36.21, due Monday after the break.

- Monday, Nov 14: covered section 35 and started section 36.
Homework: please remind me to collect section 24 on Wednesday...the new homework due on Friday is 35.1/2 (really the same question), 35.3, 35.5 (the answer to both parts is no; the point is to give counterexamples), 35.6 (I stated the result incorrectly for the case b<0; see if you can identify my error and state it sensibly), 35.9, 36.1.

- Monday November 7: covered section 26 on composition of functions. This is the last section covered on the test.
Homework (due Monday after the exam): 26.1 acegi, 26.2, 26.10, 26.11. There will be some more section 26 problems assigned (proof exercises) in a future assignment not covered on this test.

- Friday, November 4: covered section 24 on functions.
Homework due Wednesday: 24.1 acegi (leaving others as examples to ask about), 24.2, 24.3, 24.4, 24.8, 24.11, 24.14, 24.21

- Monday and Wednesday were devoted to recurrence relations, and that assignment is due Monday.

- Friday Oct 28 : recurrence relations started. I'm planning to say more about section 23 on Monday.
Homework, due next Friday (not Wed): 23.1ace, 23.2egikmno, 23.3ac (Monday lecture needed), 23.7 (Monday lecture needed), 23.11

- Wednesday Oct 26: discussion of Monday continued.
Homework due next Monday: 20.1 ace (I'll review contrapositives); 20.4 aceg, 20.9, 20.13, 20.16* (optional challenge!), 21.2, 21.7, 21.8 (I am likely to give this one away as a class exercise, but try it), 22.7* (an optional challenge), 22.9 (induction on the length of the line, I think), 22.10 (you may use basic facts about sums of angles of a triangle), 22.16 acd (other parts may be requested as demonstrations; this is prep for the next section), 22.17. You may use section 22 methods on section 21 problems and vice versa.

- Monday Oct 24: discussion of induction, least counterexample, proofs of negative statements.

- Friday Oct 21: induction proofs started. Homework: 22.4 acde, 22.5 ace. I'm making this due Friday the 28th partly because I am posting it late (Sunday afternoon, sorry) and partly because you genuinely will need more time to work on these. There will be more assignments on this material.
- Monday test review, Wednesday was the test.

- Friday Oct 14th: A day of counting word problems.
Homework from the xerox sheet, due on Monday (not Friday) after the exam. This is rather long so I am giving extra time to work through it. The various counting principles will all appear on the exam, but the counting problems on the exam will be simpler as a rule. p. 381 probs 1cdef (e and f are on p. 382), p. 383 problem 10 (look at problem 9), 13bd, 14, p. 384 problem 16, p. 385 probs 20, 22, p. 395 probs 1,5,6,7,9, p. 396 probs 13, 15, 21, 24, p. 397, 25, 39, p. 406 problem 5, p. 407 23, 24, 25, 27, p. 417 problems 10, 16, p. 418 18, 20, 23, 28.

- Wednesday Oct 12: Dr. Champion will be covering section 19.
Homework: 19.1, 19.2, 19.3, 19.7.

- Monday Oct 10: Covered section 18. No homework as yet -- I'll make up a sheet of counting problems, watch this space.

- Fri Oct 7: finished section 17, though I may have a little more to say on Monday.
Homework due Friday the 14th (I will be out again on the 12th and I see no reason for Dr. Champion to have to manage the papers; use your extra time wisely): 17.1, 17.2, 17.3, 17.5, 17.6, 17.8, 17.10, 17.11 (this is a trick question -- there is a special case you need to think of!), 17.14, 17.18, 17.25, 17.33

- Wed Oct 5: finished section 16, started section 17.
- Mon Oct 3. Dr Champion will be teaching the class. Section 15 to be finished, and perhaps section 16.
Homework: 15.1, 15.3, 15.7, 15.11, 15:14/15 (view this as one problem), 15.16, 16.2, 16.7/8 (view this as one problem), 16.11, 16.15. Appropriate modifications will be made if coverage isn't sufficient for these problems. Due date of the section 16 problems deferred to next Monday, since we are not done discussing section 16; the section 15 problems are due Friday (along with the section 14 problems).

- Fri Sept 30th: started section 15. Homework vacation, I'm not done with the section.
- Wed Sept 28th: discussed section 14 on relations.
Homework due next Wednesday (due date deferred to Friday): 14.1bd, 14.2, 14.3, 14.4, 14.6, 14.7, 14.13, 14.14, 14.16, 14.17ace

- Monday Sept 26: Test post-mortem; discussed section 13.
Homework, due Friday the 30th: 12.1, 12.17, 12.30ac (I'll do one of the other parts on request to give you a model), 13.2 (this is interesting algebra rather than combinatorial proof), 13.3 (hint: think about sequences of the digits 0,1,2 of length n: there are 3

^{n}of these), 13.5, 13.6.

- Wednesday, September 21st: Reviewed for test. Discussed last items in section 12.
- Monday, September 19th: Covered section 12. 12.1, 12.3, 12.9, 12.24, 12.12, 12.21 (for proofs in 12.21, supply Venn diagrams), 12.25, 12.28. These are not all models for questions on Test I: those will be Venn diagram proofs and simple counting questions.

- Friday, Sept. 16th: Discussed proof strategies. Do problem 1,2,3,4,6 in the proof strategies handout (problem 5 is harder, I believe).
Due Wednesday the 27th, after the test (there was a typo here earlier): this material will not be on Test I.
- Wednesday Sept. 14th: we discussed chapter 11.
11.1 acegik; 11.2 acegik (this leaves others to ask me about in class); 11.4 all parts; 11.5 aceg. If you can write a correct answer to 11.8 I will award extra credit. Due next Monday.

- Monday Sept. 12: I delayed collecting section 8 to Wed because I did not have section 6 marked.
I'll have section 6 back on Wed. I lectured section 10 on sets.
Homework, due Friday, 10.1, 10.3 (I didn't use the word "cardinality" in my lecture: the cardinality of a set, written |A|, is the number of elements in the set A.), 10.4 (the notation 2

^{A}denotes the power set of A, the set of all subsets of A), 10.5, 10.8, 10.10.

- Friday, Sept. 9: we talked about more counting problems. I'll post a handout with some additional counting problems for homework later today (I decided to delay this).
- Wednesday, Sept.7: lectured section 8 and a little of section 9. Homework due next Monday, 8.2, 8.3, 8.6, 8.8, 8.9, 8.10, 8.12, 8.15, 8.18, 9.2, 9.5, 9.10.

- Friday, Sept. 2: I am a goofball, and forgot to post your homework! Here is a brief assignment
on section 6, now due on Friday. I will add additional material on proof strategies in due course.
Problems 6.2, 6.3, 6.4, 6.5, 6.9 (try to find the first counterexample to 6.9: it is amusing).
- Wednesday, August 31: Continued discussion of section 5 and Appendix D: touched on the basic idea of section 6. Do the following worksheet.
- Monday, August 29: Section 5 lectured. Homework (due Friday): 5.1, 5.6, 5.9, 5.13, 5.22. Notice that I added one more problem.

- Monday August 22nd: Please read the section "A Bit of Help" on pp. 2 and 3 in the book. I will lecture sections 2 and 3 (mostly section 3). Exercises (due on the 26th): 2.1 (follow instructions, and clearly write out instructions to solve the problem which you believe a fellow student could follow), 3.1, 3.2, 3.3, 3.4 (when they say "use the concept of natural numbers to create definitions..." they are being a bit too brief: you also need to use the notions of addition and/or subtraction), 3.6, 3.9, 3.12.
A positive integer M is

*special*iff there is a unique integer x such that`1 < x < M`and`x | M`(x is a factor of M). We saw in class that 4 is special and 25 is special. For Wednesday, give a brief description of the special integers using words you already know. - Wednesday August 24th: I will lecture section 4 and some additional material about propositional logic and truth tables.
Exercises: 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.10, 4.12acd (you are very welcome to do the other parts!). This is due on Monday the 29th.
- Friday August 26th: I will continue to lecture section 4 and some additional material about propositional logic and truth tables.
The homework is a worksheet on truth tables and logical notation posted here. This is due on Wednesday the 31st.

Here are solutions to Test 3 from Fall 2013.

Here are solutions to Test 3 from Spring 2013.

Other sample Test III papers: Fall 2006; Fall 2008; Spring 2007; Summer 2006; Summer 2007. Spring 2012 Some of these have different coverage from your test in ways which should be obvious.

Sample tests will appear sometime this evening or during the weekend.

Sample Test II papers: Fall 2006 Test II; Fall 08 Test II; Spring 2007 Test II; Spring '10 Test II; Summer 06 Test II; an old Test II review sheet Spring 2012 Test II paper

Here are the solutions to my Spring 2013 Test II.

Here is a little sheet of counting problems which I gave my spring 2013 class. Some of them are good practice for the test (the simpler ones!)

The tests do not all have identical coverage to ours. Questions that were on our Test I, or which mention things like Hasse diagrams and mathematical induction that we have not covered yet, are not going to be on this test. The counting word problemsOne more warning: a problem on the level of difficulty of the first two problems on the proof strategy handout is fair game.

Other announcements related to the test have been moved to the bottom of this file (such as the sample tests, which you may want to use when you study for the final as well).

Here is the test with solutions.

Fall 2006 Test I. Here is the Spring 2012 Test I paper.

Here is the Spring 2013 Test I paper with solutions. Here is the Fall 2013 Test I paper with solutions.