Problems, Conjectures and Proofs
MATH 1200 3.00
Lecturer: Paul Szeptycki
Office: TEL 2031
Mondays 5:30-7:00 in SL107
Mondays 7:00-8:30 in either VH 1016 (tutorial 1) or VH 1018 (tutorial 2). You are enrolled in one of these.
Office hours Tuesday and Thursday 1:00-2:00
Course Description From the undergraduate course calendar: Extended exploration of elementary problems leading to conjectures, partial solutions, revisions, and convincing reasoning, and hence to proofs. Emphasis on problem solving, reasoning, and proving. Regular participation is required.
Main Objectives: This course is designed for first year mathematics majors to develop basic skills essential for more advanced courses in mathematics. An emphasis will be placed on writing proofs. There are two main aspects to proofs that we will focus on, problem solving and exposition. One of the challenges in upper division mathematics courses is to come up with proofs to problems that are not familiar or similar to problems presented in lectures or in the textbook. This requires a variety of problems solving techniques and lots of practice to develop the confidence to attack a problem without yet knowing how to solve it. Once one has solved a problem, or come up with a proof, one must then clearly communicate that solution in a clear and concise manner. Learning how to present convincing reasoning (aka a proof) is the main objective of this course. We will be learning the language of mathematical exposition.
Of course, this means a lot of hard work on your part. You will not succeed in this course if you do not actively engage in solving problems, writing proofs, and working through examples each and every week. If you fall behind one week, it can be very difficult to get caught up. E.g., attend the tutorials and start work on the homework assignments early!!
Prerequisite 12U Advanced Functions (MHF4U) or Advanced Functions and Introductory Calculus (MCB4U). Course credit exclusion 2200 3.00. NCR note: Not open to any student who is taking or has passed a MATH course at the 3000 level or higher. (or equivalent).
A Concise Introduction to Pure Mathematics by Martin Liebeck, CRC Press ISBN 978-1584881933
We will cover, time permitting, chapters 1-6, 8, 10, 11, 13, 17-19, 21, and 22 of the text.
- Assignments 40%.
- Fall exam 20%
- Final exam 40%
- Homework will be assigned more or less weekly. Each homework assignment will consist of 1 or 2 questions based on the current material being covered in class -- the assignment will be due the following week.
- The fall exam will be given in the December final exam period.
- Assignment 1. Due date: September 23
- FALL EXAM: Wednesday December 11, 7-9pm in TC Rexall.
- LAST LECTURE OF FALL TERM: Monday, December 2
- LAST LECTURE OF THE YEAR: Monday, March 31.
- FINAL EXAM: TBA
Any form of academic dishonesty (e.g., cheating on a test, quiz or final exam etc...) will not be tolerated. If I suspect a student of cheating, I will deal with the case in accordance with the Senate Policy on Academic Honesty . Penalties for cheating on a test, quiz or final exam may range from failing the course, suspension from the university or even expulsion from the university. DO NOT CHEAT, it is not worth it.
- September 10 The first homework assignment: Write out proofs for problems 6a, 6b and 10 from the problems at the end of chapter 1. Due in two weeks (September 23).
- September 16 Please make sure to have read chapter 1 by this week. We will start learning how to write proofs.
- September 23
- Homework 1 is due. Please hand it in at your tutorial after the lecture today. I will post a solution shortly.
- We will finish up a couple of loose ends from Chapter 1 and start discussing Chapter 2.
- Chapter 2 is the beginning of our study of the number systems: The real numbers, the rational numbers and the integers. Here are some short notes about the basic rules (aka axioms ) that we may assume and use in our proofs about these number systems: axioms for the integers and axioms for the real numbers
- If we cover enough of chapter 2 I will assign the next homework assignment in class.
- September 24
- Assignment 2 was assigned yesterday: Do problem 4(a) from the Chapter 2 exercises. This assignment is due on Monday September 30!!
- Solution to Assignment 1
- October 1
- Solution to Assignment 2
- I gave a rather impromptu homework assignment 3 yesterday in class. You should write up solutions to each of the problems, but, please, hand in only one to be graded.
- Next week we will start to discuss chapter 3. Please read chapter 3 and look over the suggested homework.
- October 7
- Homework 3 is due this evening, please hand in at the beginning of the tutorials. I'll post a solution by tomorrow afternoon.
- Here is homework assignment 4 . It is a bit longer than usual but since we have no classes next week you have two weeks to work on it. I'll hand out hard copies in class later this afternoon.
- October 8 Update on Assignment 4. Please do only problems 1 and 2 to hand in on October 21.
- October 21
- I just noticed I never posted a solution to assignment 3. Here it is.
- Assignment 4 is due today. Please hand in problems 1 and 2 at the beginning of your tutorial tonight.
- We will cover Chapter 4 today. Please read Chapter 4 and do the suggested problems from the text. Next week we will start Chapter 5, so you can start reading that as well.
- Here is homework assignment 5 . It is due next week, Monday October 28.
- October 28
- Homework 5 is due at the beginning of this weeks tutorial.
- We will finish chapter 4 and 5 (chapter 5 is very short) in class this week. Please see the suggested homework for chapter 5 below.
- homework assignment 6 . It is due next week, Monday November 4.
- November 2
- Here are the solutions to Homework 4 and Homework 5.
- We will finish chapter 5 and start chapter 6 on Monday, November 4.
- Homework 7 will consist of problem 4 from the end of chapter 6 exercises. It will be due either in one week or two, depending on how much we cover this Monday.
- November 11
- We'll finish chapter 6 today. Homework 7 is due next Monday, November 18. PLEASE NOTE CHANGE IN HOMEWORK 7: IT ONLY CONSISTS OF ONE EXERCISE, PROBLEM 4 FROM THE END OF CHAPTER 6.
- Here is the solution to homework 6 .
- November 18
- Homework 7 is due today, in the tutorials. I will post a solution tomorrow
- We will finish Chapter 6 this afternoon and then skip to the next topic: Induction, covered in Chapter 8.
- Please note the date, time and location of the midterm: Wednesday December 11, 7-9pm in TC Rexall.
- November 25
- homework assignment 8 . It is due next week, Monday November 4.
- We will finish up the material on induction this afternoon and next week (last class) we will spend reviewing for the midterm.
- Here is some information about the the midterm.
- December 2
- LAST LECTURE OF THE TERM. Woooohoooo.
- I will post the solution to homework 8 by tomorrow or late tonight.
- New information regarding the midterm: there are a total of 10 questions, 10 points each for a 100 point exam.
- Here is the solution to homework 7 .
- Here is the solution to homework 8 .
- December 4 Please note that you will be allowed 3 hours to take the midterm exam.
- December 9 I've made a couple of office hour appointments for Tuesday but none for today. So I will not be coming to the office today but if you want to meet with me on Tuesday, I will be in my office 2-4pm.
- December 28 Hope you are all having a pleasant break!
- You should have all received your midterm grades by email as well as a scanned and graded copy of your test.
- If you are feeling bored and needing something to think about to pass the time, here is a diversion inspired by a small party favour found in a Christmas Cracker.
- January 5
- Tomorrow we will spend a bit of time reviewing, discussing the problem I posted earlier and taking up questions.
- Tutorials will be held tomorrow -- the tutors will be going over the midterm test so if you had problems understanding any of the problems please attend!!
- We will also start covering Chapter 10. So please start reading it.
- January 7 If you are still feeling a bit confused by problems 2 and 3 from the Test or if you just want some more examples to think about, please look at the following list of practice problems on quantifiers that was written up by one of the other
MATH 1200 instructors.
- January 13
- We will finish chapter 10 today (hopefully!!).
- homework assignment 9 . It is due next week, Monday January 20.
- January 27
- Here is a solution to assignment 9. It will be handed back in tutorials later tonight.
- Assignment 10 consists of 1 problem: Problem 6 from the end of Chapter 11.
- February 10
- Here is a solution to assignment 10 including the bonus problem.
- Last week I assigned problem 4 from the end of Chapter 13. It is due tonight.
- February 11
- First off, I had the leap year calculation flipped around: Leap years occur every 4 years with the exception that a century year is only a leap year if divisible by 400. So the year 2000 was a leap year, but the years 2100, 2200 and 2300 will not be (I believe that in class I reversed this stating that 2000 was not, but 2100, 2200 and 2300 would be). So what is the correct solution to Problem 11 in Chapter 13?
- Here is a solution to assignment 11.
- Now a short announcement from our sponsor: If you have completed at least 18 credits towards your mathematics major, you might have received a request to fill out a survey about your program of study. Here is a message regarding that survey from our undergraduate program director: Some of you will have already been sent a request to fill out a survey concerning undergraduate programs in Mathematics and Statistics. The purpose of the survey is to solicit student input into a review of our undergraduate programs to help assess programs' strengths and weaknesses and ultimately to make them more coherent and effective. The survey is detailed and will take approximately 20 minutes to complete. In case you have been contacted, we ask that you please take the time to fill out the survey, as your opinions are an essential ingredient to our assessment and planning. Your cooperation is very much appreciated.
- ANNOUNCEMENT: Club Infinity (the undergraduate mathematics club) is hosting a presentation on the history of prime numbers by one of our faculty members. Here are the details:
Speaker: Prof. Youness Lamzouri
Time and Location: Tuesday, February 25, 4pm. Stong College, Rm 216
Title: The Mysteries of Prime Numbers
Abstract: There are infinitely many primes, as was first proved by Euclid more than two thousand years ago. Since that time mathematicians have been fascinated with these mysterious numbers.
Although the primes are simply defined, many problems about them remain open, such as the famous twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2) and Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes). One of seven millennium problems: the Riemann hypothesis (worth 1M$), is deeply connected with the distribution of prime numbers.
In this lecture, I will describe the history of this fascinating subject, and present some of its recent development, including some of my work on the distribution of primes in arithmetic progressions and the so-called prime number races.
Note: Refreshments will be served.
- February 24
- March 3
- We will finish chapter 17 and start 18 today.
- Homework 12 is due in the tutorial after class.
- Homework 13 is due next week, Monday March 10.
- March 12
- Here are solutions to assignment 12 and assignment 13.
- Assignment 14 consists of problem 4 from the end of Chapter 19 exercises on functions. This assignment is due on Monday March 24 (the monday after next).
- March 30
- The final exam will be comprehensive. We covered chapters 1-6, 8, 10, 11, 13, 17-19, and 21.
- Most of the test, approximately 2/3, will be devoted to the chapters not covered on the first term test (starting with chapter 8 on induction).
- The exam will be common with the other sections. To help you prepare for the Final, each of the instructors came up with a list of problems, that we have compiled together into a list of FINAL EXAM REVIEW QUESTIONS. We have agreed that a large portion of the final exam will be constructed from questions either from this list or very similar to questions on this list.
- Final Exam period OFFICE HOURS:
- Tuesday April 8 1-2pm (or later if there is a demand)
- Tuesday April 15, 11-1pm
- Tuesday April 22, 11-1pm
I will give a list of suggested homework problems from the end of each section and perhaps from other sources. Do as many of these as you can. They might appear on future tests so if you have trouble with any of them, bring your questions to class and your tutorial.
- Chapter 1: All of them!
- Chapter 2: 1-5, 7, 8.
- Chapter 3: All of them!
- Chapter 4: 1-8.
- Chapter 5: all of them.
- Chapter 6: all of them.
- Chapter 8: all of them.
- Chapter 10: 1-9.
- Chapter 11: 1-9
- Chapter 13 All of them
- Chapter 17 1-8, 10.
- Chapter 18 1-8.
- Chapter 19. All of them.
- Chapter 21. 1, 2b, 3, 4.