# Welcome to Math 287, section 001, Fall 2015

Welcome to the class. New material will be added at the top (just below this introductory paragraph) You should also look at the syllabus.

Notice that my office hours are now posted on my main page.

## Manual of Logical Style

Here is my "manual of logical style". It now contains sample proofs in a final section.

## Presentations

Here is the handout for the presentation assignment. It is a good idea to team up with a partner and send me your choice of problem (and the name of your partner) by email as soon as possible. It is first-come first served by order of arrival in my mailbox.

Watch this space. I will supply a list of proofs which students can present -- they will be assigned to pairs of students or individual students on a first-come first-served basis. The assignment involves preparing a slide presentation of the proof of the theorem you are assigned (you can run these past me before presenting them -- I do not want to critique proof presentations which are incorrect, as I am interested in communication more than in testing of specific content). The presenters should each take a role in the actual presentation, and should both be expected to answer questions from the audience or from me.

I remind you that presentations may run into the final exam period. If they do, not only presenters but the entire class will be expected to appear. Attendance will be taken if this happens.

## Last Logic Lab

The Lab2.sagews file in my Lab2 project now contains the full worked out example of quantifier reasoning that I did in class, and I may add more examples. Your assignment is to complete the exact proof that I did as an example in class. I will provide a file with details of the argument in English that I was working from. You can turn it in any time before the end of the class.

I am hoping that we will have one more lab day where we can complete all lab work that is still dangling. If you have handed in the first logic lab and/or the LaTeX assignment you should expect to get an email about it. I am still accepting those assignments.

## Homework 10:

Here is a file with the proofs of the triangle inequality and the addition property of limits I did in class, with promised exercises, due on Wednesday November 11.

On Monday remember that we are working on the constant multiple property of limits, and also I may do something to encourage people to continue computer lab activities.

## Homework 9:

Here is a handout with the limit proofs I did in class (read them) and exercises at the end for you to write out, due on the Wednesday after Halloween. The third one is optional: the first two should have basically the same outline as my examples, and so are mostly writing exercises. You might try duplicating the handout with your own LaTeX skills for practice.

## LaTeX Lab exercise

Typeset the contents of page 123 of the book as best you can in LaTeX. You can turn this in to me by sending me a .tex file or by sharing a Sage project in which you have files implementing this. You may do this in a group of two or three people. Get it to me soon and I will reply with (one hopes) helpful comments!

## Homework 8

Due Wednesday 28 Oct.

These are proofs to be written in mathematical English. You are welcome to structure them with goal statements, indentation, line numbers, and so forth as I do in class. You are also welcome to write plain paragraphs of English, as long as they contain what is needed to communicate the proof. Try to be explicit about where you are using or proving quantified statements, as I was in my board examples. Other logical issues also arise in some of these problems.

4.1, 4.2, 4.5, 5.10 (define largest element of a set in your work), 5.26, 5.46

Having constructed this assignment, I comment that the book does not give me perfect material for what I have in mind. These are interesting proof problems, but not really a systematic quantifier reasoning workout. Please work on them and ask questions about them if you need to...

I am thinking of posting notes on the examples I did in class Wednesday. If I do, they will appear here.

## Homework 7

This homework is due 10/26/2015.

Section 2.10 problems: 2.68, 2.70, 2.71, 2.72, 2.75, 2.76, 2.78, 2.79

Here are the grades on Test I for students who took the test at the regular time and place, posted by the ID number on the test paper.

Here are the solutions. They are not completely typeset yet; I will update them fully later.

## Solution Set for Homework 3 and 4

Here are solutions to Homework 3 and 4. Other study materials for the test may appear here.

## Imminent Events

On Sept. 28th, I am planning to lecture further on Chapter 3, and review topics for a pending test.

On Sept. 30th, I am hoping to have a computer lab on propositional logic. It might be on the 5th, depending on issues I may encounter in setting it up.

On Wednesday Oct. 7 we will have Test I, covering material we will have done up to the Sept 28th lecture; I changed my mind and am leaving Oct. 5 free for review or other activities. The coverage of the test is still up to the topics covered on Sept. 28th.

## Homework Assignment VI

This assignment is due on Monday Oct. 5. Do problems 3.16. 3.18, 3.19, 3.20, 3.21. I will have a few words to say on Wednesday about appropriate use of lemmas. Again, a nice clear write-up is important.

## Homework Assignment V

This assignment is due on Wed. the 30th.

3.5, 3.6, 3.7, 3.8, 3.9, 3.10

It's quite brief and I know that Dr. Kaiser covered a couple of these. What I hope to see on these papers is a nice clear writeup (this class is about communication!). I'll talk more about the kind of style I want to see in my lecture on Monday.

## A Visit from Dr. Kaiser

Dr. Uwe Kaiser will be teaching the class on Wednesday the 23rd. He will cover material from chapter 3 in the book, and I will post a homework assignment from the book on Thursday after I read his communications about what he covered. It won't be propositional logic -- these will be proofs in arithmetic and algebra. But you should recognize the same approach to proving an if-then statement that I am teaching in the propositional logic material.

## Homework Assignment IV

This assignment is due on Monday, September 21st, revised to Wednesday, Sept. 23d.

Prove the following using the rules in the style manual, and not using any kind of replacement of statements with other statements of logically equivalent form (for example, do not use de Morgan's laws).

• Prove ~(P v Q) <-> ~P & ~Q (Notice that this is the one of de Morgan's laws that is not proved as an example).

• Prove (P v ~Q) & (Q v R) -> P v R (Hint: this is a proof by cases)

• Prove (P -> Q) <-> (~Q -> ~P) (the well known equivalence of an implication with its contrapositive; we did this in class, so it is low effort if you have perfect notes, but you might do better to work it out yourself. Do not use the rule of modus tollens or the alternative strategy for proving an implication; this makes it too easy)

• Verify the rule of destructive dilemma (you have the verification of constructive dilemma in class and in the style manual as a model). The rule says that if P implies R, and Q implies S, and either R is false or S is false, then either P or Q must be false. Start, of course, by expressing the premises and the conclusion in symbols. This part had a typo in it -- two letters were exchanged

## Homework Assignment III

This will be due Wednesday September 16th. This is really a warmup.

I use -> for implies, ~ for not and & for and here because they are on my keyboard...

Prove the statement ((A -> B) & (B -> C)) -> (A -> C) using the rules described in the style manual. This justifies the well-known logical rule "hypothetical syllogism".

Prove the statement (P -> Q) <-> ~(P & ~Q) using the rules described in the style manual. This verifies a perhaps unexpected logical equivalent of implication.

## Homework Assignment II

This assignment is due Wednesday Sept 9. (Monday is the Labor Day holiday).

Exercises 2.1, 2.2, 2.9, 2.10, 2.13. 2.14, 2.15, 2.16, 2.20, 2.25, 2.27, 2.31, 2.34, 2.39, 2.47, 2.48, 2.49

I do expect to hear some questions on Wednesday!

## Homework Assignment I

This assignment is due Wednesday Sept 2.

Exercises 1.3, 1.4, 1.5, 1.6, 1.10 (these are different examples for each part), 1.11, 1.14, 1.19, 1.21, 1.22 (I was writing Ac in lecture for what they write as A: that is, the complement of A or U-A), 1.28, 1.30, 1.32, 1.33, 1.35.

Every now and then a homework problem may turn into a class example, particularly as I am likely to mark only selected problems from each set; be ready on Monday with your questions about these!