# MATH 220: Mathematical Proof, Winter term II, 2015/2016.

Text: Mathematical proofs, a transition to advanced mathematics, by G. Chartland, A. Polimeni, P. Zhang, Second or third edition.

### Course outline

The main aim of the course is to learn how write clear and correct mathematical proofs. In particular, this involves learning some of the language of mathematics, and also honing your precise reasoning skills. This course provides the gateway to more advanced mathematics. Tentatively, the course will cover the following topics: (chapter numbers refer to the textbook).
• Sets - definitions, set operations (chapter 1)
• Logic - logical connectives, quantifiers (chapter 2)
• Proofs - direct and contrapositive. (chapters 3 and 4)
• Proofs - existence and contradiction (chapter 5)
• Induction (chapter 6)
• Definition of a graph and some problems on graphs that use induction.
• Functions - injective, surjective, bijective, inverses and compositions (chapter 9)
• Cardinality of sets - finite sets and different types of infinite sets (chapter 10)
• Applications of what we have learned to combinatorics and graph theory (notes will be provided).

### Important announcement about the final exam.

The exam for all sections is in the gym OSBO A. When you enter, please follow instructions and SIT IN THE AREA DESIGNATED FOR YOUR SECTION. It is very important. You should not sit with another section, intentionally or by accident. For Section 203, YOUR EXAM BOOKLET SHOULD HAVE GREEN COVER PAGE. You can keep your bag at your table, with all the electronic devices turned off and in the bag. Please remember to bring your ID -- we will check.

### Review materials:

• Detailed list of topics. Please note that an earlier version of this list was missing congruences and composition of functions. It is now fixed (both of these topics are on the exam).
• Notes from J. Gordon's the review session on Thursday (Thanks to Isabella for taking the notes!)
• The format of the exam will be similar to the previous years.
Please see the old exams at Math Department site (scroll down to 220). Ignore all problems about sequences, limits, and the least upper bound, however (we had a bit more cardinality and graphs instead).
Some solutions for the old exams at Department wiki . Caution: some of them have big mistakes!
• There will be one or two questions on graphs. To review this topic, please look at all the posted notes and homeworks (see the individual sections' websites).
Especially helpful, Homework from Section 202/204. Solution .
• Workshop 6 on cardinality/review from Section 201.
• Online textbook by J. Farlow (I wish I heard about it earlier!). It has a nice section 3.4 on graphs (among other good things).
• Don't forget to look at all the review materials for the midterms below, and at all out workshop and homework solutions!
• If you are comfortable with all the things on this list, you are ready for the exam!
• ### Homework and quizzes

• You are strongly encouraged (and will receive a 2 points bonus at the end of the term if you do it) to type all your homework solutions using LaTeX. Here are some LaTeX resources .
• Every other week on the last class (that is, Thursday or Friday), the classes will be organized as workshops in which students in small groups will work problems. There will be a short quiz at the end.

### The Homework assignments:

• If you are using LaTeX, it is fine to draw pictures and diagrams by hand. A useful command \vskip1in produces a vertical space of specified length (in this example, 1 inch).
• Homework 1 . Due January 11 or 12. Solutions . (Marking: Q3 out of 9, Q5 out of 11; 20 total).
• Homework 2 . Due January 18 or 19. Solutions (Marking: Q2 out of 4, Q3 and Q4 out of 5 each, Q7 out of 3: 17 total).
• Homework 3 . Due January 28 or 29 (Thursday/Friday). Will be back to the usual schedule next week. Solutions . Marking: Question 2 (a, b -- 2 points each, c) -- 4 points (because it has 2 questions); Question 3 (1 point each), Question 8 (1 each), for the total of 20.
• Homework 4 . Due Monday February 1/Tuesday February 2. The LaTeX file with questions that you can download and use as a template for your solutions. Solutions . (Marking: Question 2: 1 each (3 total); Question 4 (1 points for interpreting the target statement correctly,and then 1 point each -- 5 total) Question 5 (2 each), Question 6(2 each). Total: 20 points.)
• Homework 5 . Due February 10th/11th. The LaTeX source .
Solutions.
• Homework 6 . Due Wednesday February 24/Thursday February 25 (Note the unusual date!) to allow time for office hours consultation after the break.
Solutions
.
• Homework 7 . Due Monday March7/Tuesday March 8.
Solutions.
• Homework 8 for Julia Gordon's and Maxime Bergeron's classes . The LaTeX file . Due Monday/Tuesday March 14/15. This is the assignment for Sections 201 and 203 only! If your Professor is Prof. Khosravi, then please check the individual section website for Sections 202 and 204 to see this and next week's homework and due date!
Solutions.
• For Homework 9 (both the assignment and its due dates), please see the individual sections' websites.

### Marking scheme and course policies

• Your mark will be based on Homework, workshop quizzes, two midterms, and the final exam:
• Weekly Homeworks (posted on Section webpage): 10%
• Workshop Quizzes (approximately once in two weeks, schedule to be announced): 5%
• Two midterms: 35%
• Final Exam: 50%.
• The midterms will be common between all sections, and are tentatively scheduled in the evenings (outside class time!) on Friday February 5 and Friday March 11, 5-7pm. This is to be confirmed but do not make any plans for these two evenings!
• Missed midterm policy: if you have a legitimate conflict with the midterm time please inform your instructor as early as possible, and at least 3 days in advance. One alternate sitting will be scheduled for each midterm; you need to have a valid documented conflict or a doctor's note to qualify to take the midterm at the alternate time. Not appearing for a midterm in the absence of such documented excuse will result in a score of zero.
• No late homework's will be accepted under any circumstances. It is likely that only a subset of the questions on each HW will be graded by the TA. Students who are unable to hand in a homework or quiz due to a medical or equivalent excuse may have that homework/quiz not count towards their final grade.
• Collaboration on homework: You are allowed to discuss homework with other students, but you have to first think about it yourself, and then write it alone in your own words. If you discussed homework solutions with someone, please acknowledge this on your submitted work, that is, put "discussed with Jane Smith" next to your name (and Jane Smith then has to put the similar acknowledgment of collaboration with you on her work). Please also see the notes on academic integrity .
• The final exam will be scheduled by the Registrar. Please do NOT make travel plans before the schedule is released later in the term.
• Other Resources: Drop-in tutorials begin during the second week of term. The schedule is available here .