NEW: Here are solutions to the modular exponential worksheet. If you turned in your section 37 and worksheet in class, you can pick up your paper with solutions attached (to selected 37 problems as well) at my office door.

The first batch of posted stuff is copied from the Fall 2016 class page. The comments are from Fall 2016, too: I will probably
update them as I look over these sample exams (and I will add the Fall 2016 final if I can find it. Please note that we do not cover permutations and we do not cover graph theory. We could ** almost ** do the Chinese Remainder Theorem, and I might actually show you how to do these problems, but it would be a stretch to introduce this at the very last minute. All of this is a function of the class having been reduced from 4 credits to 3 credits: I had a lot more time in those past years. I will also post your tests from this term as resources, and naturally you can use the old hour exams already posted as resources (they are still all on this page: I moved the review material for the old pages to the end).

The hour exams this term are of course study resources for this test. Here is your Test I (speaking in Spring 2018). Here is Test II (in spring 2018). Here is Test III (Spring 2018). All of the old hour exams are study resources as well (see below). I am planning to find or make solution sets for the spring 2018 exams and post them here some time this week. You may expect that final exam questions on a topic covered on the hour exams will in general terms be similar to what I asked on the hour exams: I am not going to be dreaming up new tricky approaches.

** below this point copied from Fall 2016 final review: comments may be outdated **

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 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).

- Friday April 27: this will be strictly test review.
I remind you that sample tests are posted above. Also, note that you can use your textbook and a standard sized sheet of notebook paper with whatever notes you want on it on the final. Bring your questions to the review! Also, please fill out course evaluations.

- Wednesday, April 25: discussed the RSA cryptosystem. You may try this optional worksheet on the subject, which you may turn in up until the morning of the final. Solutions will be posted after Friday's review session. If there is an RSA problem on the final, it will be for extra credit.
- Monday, April 23: section 37 homework, belatedly: 37.1, 37.2. Their notation is funny: circled operation symbols are the modular arithmetic operations; when they talk about Z
_{n}they are talking about the set of remainders mod n with the modular arithmetic operations. 37.3, 37.4: in these problems, we do not have methods of solution, but you can find solutions by plugging in all possible values, 37.14ab (which is on the topic of today's lecture!) You might have picked up a hint at the very end of the lecture about why computing 3^{64}in mod 100 arithmetic (or just computing 3^{64}mod 100) is actually rather easy -- and absolutely does not involve computing 3^{64}: the magic words are "square repeatedly" (and simplify using the modulus). In addition, do this worksheet. Everything is due on Friday.

- Wednesday, April 18: we covered the Euclidean Algorithm for computing greatest common divisors. Homework: section 36: 36.1 and 36.2: carry out the algorithm taught in class in the table format I used, with positive numbers, then make appropriate adjustments to handle sign changes; 36.12 (you can actually carry out the algorithm symbolically, more or less), 36.13, 36.20, 36.21.
Here is my Euclidean algorithm spreadsheet calculator.

- Monday, April 16: I discussed sizes of infinite sets, then changed gears and discussed the division algorithm. Homework: section 35: 35.1, 35.2, 35.3, 35.7.

- Monday, April 9: Section 25: 25.1, 25.3, 25.5, 25.6, 25.8, 25.9, 25.16 (a proof that a function f:A -> B is a bijection from A to B has two parts: show that it is one to one, and show that it is onto B), 25.17. Take a look at the lattice point example in the section, too.

This is a demanding problem set, but notice that 26.1 is full of routine stuff you can do straightforwardly.

- Wednesday, March 21: More section 23. Additional problems from section 23, due Wednesday after the break (along with the problems assigned on the 19th): 23.3, 23.7, 23.9, 23.l1 (I don't remember which kind of section 23 question 23.11 is, but it is an interesting question).
- Monday, March 19: we talked about section 23 on recurrence relations. I'll have more to say about this section on Wednesday: here is a brief
homework assignment. I do hope people will have questions about the induction assignment on Wed!
Homework: 23.1 bdf (leaving other parts as classroom examples); 23.2 bfijk: on at least two of the parts, please write induction proofs that the formula works. DUE Wed after break along with the problems assigned on the 21st (changed due date for this but not for the induction assignment).

- Friday, March 16: problems may be done by induction, by strong induction, by least counterexample, as appropriate: 21.6, 21.7, 21.8, 22.4de, 22.5abde, 22.8, 22.9, 22.16 abe, 22.17. This assignment is due next Friday. The extended time is intended to support extra discussion; there may be another (shorter) assignment next week with the same due date.
- Wednesday, March 14 (well, writing on Thursday): talked some more about induction and its relations to the Well-Ordering Principle. I didn't assign anything more (there will be more homework assigned on Friday) but do notice that Monday's homework is due Friday!
- Monday, March 12: This is a rather miscellaneous set of exercises on things we have done recently: 17.3, 17.14 (a trick question about Pascal's Triangle: hint, think about the definition of binomial coefficients...what do they count?), chapter 3 self test questions 15, 20, 21, exercise 22.4abc.

- Friday, March 2: Started on section 17 on binomial coefficients. I expect still to be talking about it on Monday. Homework 16: 17.1, 17.2, 17.4, 17.6, 17.7, 17.8, 17.10, 17.32. Due next Wednesday.
- Wednesday Feb 28: Covered section 16 on partitions and counting of rearrangements: Homework 15, 16.1 (see if you can count the partitions of {1,2,3,4,5} as well), 16.2, 16.4, 16.5 (there had to be a proof!), 16.7, 16.8, 16.9 (this is a bit ambiguous; see if you can tell why I say this); 16.11. 16.13. 16.15. We are back in the land of counting word problems: I expect a lot of discussion of these on Friday, and I will give related examples.
- Monday, Feb 26: Covered section 15 on equivalence relations. Homework 14: 15.3, 15.4, 15.7, 15.8, 15.9 (this is a formal proof that two sets are equal), 15.11 (another exercise in formal proof strategy), 15.14 and 15.15 (together), 15.16.

- Friday, Feb 23: Section 14 on relations. Homework 13: 14.1ab, 14.2abc, 14.3, 14.4, 14.6, 14.14 (hard), 14.16, 14.17. Due next Wednesday.
- Wednesday, Feb 21: Section 12, 12.1aceg, 12.3, 12.4, 12.7, 12.9, 12.11, 12.17 (hard), 12.21 all parts, 12.27. Due next Monday.

- Friday, Feb 16: Here is the link to download pythonmarcel.py, the theorem prover program. Here is the lab manual for Marcel: the exercises are on page 55. Here are some of the starting commands set up in a text file.
- Wednesday, Feb 14: Started lecturing section 12: no assignment until I'm farther into the section. Bring your laptop to Friday's class!
- Monday, Feb 12 (writing on the 13th, sorry for lateness): Lectured section 11 on quantifiers. Please bring laptops to Wednesday and Friday class this week. On Wednesday it might be useful; on Friday it is needed (those without laptops can work with a partner in class and work in the math department computer lab to do the exercises).
Please notice that there is a new section on quantifiers in the manual of logical style above, which you might find helpful.

Homework 11: 11.1 acegi, 11.2 acegi, 11.3, 11.4bcfg, 11.5 aceg, 11.8 (hard).

- Monday, February 5: Finished lecturing section 10 on sets. Collected homework 7 and 8.
Homework 10 (due on Monday after the exam, but exam-relevant): 10.3, 10.4d-g, 10.5, 10.7a,d, 10.10, 10.14.

- Wednesday, February 7: Test review. I expect to spend time talking about sections 8 and 10, and problems from
sample exams. I'll take questions about anything on the exam, of course.
- Friday, February 9: Test I. Make sure you have a standard scientific calculator which is not a graphing calculator. It's not graphing capability which concerns me: I don't want a TI89 or similar gadget. It's easiest to check that you do not have a grapher; make sure you are not using one.

- Monday, January 29: nothing due (I'll accept Homework 5 as on time if you hand it in on this date).
I am expecting to lecture on formal logic (and discuss some issues which I expect to come up in the section 5 homework).

Homework 7 is on the last page of the Manual of Logical Style document. The new rules and examples we did in class are all there. Homework 7 is due next Monday (not Friday). A

**nasty typo**in the third question is fixed 1/31/2018. - Wednesday, January 31: Homework 6 due. I'm planning to have the computer lab on logic after Test I. Today I am going to do an example to assist you with Homework 7, then start lecturing section 8 on Lists.
Homework 8: 8.1, 8.2, 8.3, 8.5, 8.7, 8.9, 8.12, 8.13, 8.15. Due next Monday.

**Solutions to Homework 6 are now found in the last section of the style manual.** - Friday, February 2: not a test date! Lectured section 9 and started section 10. We will still be lecturing section 10 on Monday.
Homework 9: 9.2, 9.5, 9.6, 9.8, 9.10, 9.11, 10.1, 10.2, 10.4a-c. Due next Wednesday.

- Monday, January 22: Lectured on section 6 and started on concepts of formal logic.
Homework 5: 6.2, 6.3, 6.4, 6.9 (there is an obvious first value of n to try which does give a composite value (not 1!): I suggest trying n=1, n=2, n=3 and so forth and finding out how long it takes to arrive at the first counterexample by this method); 6.11, 6.13

This was due on Friday the 26th, but I'll accept it on Monday. Your assignments are

**always**due two class sessions later, not counting dates of tests, unless I specifically say otherwise. If I do not state a due date, that is how you tell when it is due. - Wednesday, January 24: Started talking about formal logic. No homework yet. Look above for a link to my manual of logical style.
- Friday, January 26th: More formal logic. Homework 6: do the exercises on page 14 of the Manual of Logical Style document (link above). Don't hesitate to bother me about them! Due next Wednesday.
I just noticed that I had the assignment numbers wrong for the week three assignments, and corrected them (Sunday evening).

- Wednesday, January 17: Lecturing in section 5. We will still be discussing section 5 on Friday, and you will very possibly get more guidance re some problems I assign today in Friday's lecture. I do suggest
*reading*section 5.Homework 3: 5.2, 5.5, 5.6, 5.11, 5.12. There will be another section 5 assignment.

- Friday, January 19: Homework 2 due.
Homework 4: 5.13, 5.16, 5.20, 5.22, 5.24. Due next Wednesday.

challenge problem (not graded, but please turn it in if you have a solution): prove that x0 = 0 for any number x, using only the rules in Appendix D.

- Monday January 8th: Homework 1 from section 3: 3.1, 3.2 (I mentioned this in class), 3.4 (hint: the idea is to use the notion of natural numbers and the operation of addition to define less than, less than or equal to, greater than, greater than or equal to), 3.6, 3.10 (you can use the concept of distance between points in a plane in writing your definition), 3.12 (I'll talk about 3.12 a little on Wed), 3.13 (the computer part is optional).
A natural number m is

*composite*if and only if there is a natural number k such that 1 < k < m and m is divisible by k.A natural number m is

*special*if and only if there is a*unique*natural number k such that 1 < k < m and m is divisible by k. [this is a word I made up].Give an example of a number which is composite but not special. Find some special numbers, and give a brief description of what the special numbers are in familiar mathematical terms without using the word "special".

- Wednesday, January 10th: We started talking about section 4 (about logical operations on sentences): no homework was assigned.
- Friday, January 12th (deadline to add without permission number): Homework 1 due. We continued talking about operations of propositional logic, truth-table reasoning, the concepts of tautology, contradiction, logical equivalence, touched on inverses, converses and contrapositives of if-then sentences.
Homework 2: 4.1aceg, 4.2aeik, 4.4, 4.5, 4.6 (use truth tables for the last three questions), 4.12abd (an exploratory question: see what you discover), 7.7, 7.8, 7.11 beh, 7.17. Due next FRIDAY (I forgot about MLK day).

Samples of Test I papers from previous terms: Spring 2010 Test I Spring 2007 Test I Fall 2008 Test I

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. Two more test papers added Tuesday.Here is the Fall 2016 Test I paper with solutions.

Here are the Fall 2017 Test I and II papers with solutions. These tests are rather different from yours; we were using a different book. There are some relevant questions in Test 1, and there are two formal proof questions in Test II.

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 problems

The exam covers sections 18-26 (the last section we covered was 25, but we covered 26 first). We did not explicitly talk about sections 19 or 20.

Here I have copied my review material selection from Fall 2016:

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.

Test III has been marked and most papers have been returned. The class performance was very good.