Welcome to MAT 137! Course website:

Transcription

1 Welcome to MAT 137! Course website: Read the course outline Office hours to be posted here Online forum: Piazza Precalculus review: If you haven t gotten an from Alfonso, there is a problem with your U of T address. Lecture videos on YouTube. Before Wednesday s lecture, watch: 2 videos on Sets and Notation First video on Quantifiers

2 How do I get an A in this class? Let s look at what happened in # of submitted problem sets A A+B F 10 35% 58% 4% 9 19% 41% 9% 8 5% 22% 22% 5 to 7 1% 9% 45% less than 5 2% 2% 79% The moral: train like you re at the gym. ( kpmann/getana.pdf)

3 A warmup problem Pick 4 points at random on a circle. Draw lines between every pair of points. How many regions have we divided the circle into? # of points # of regions

4 A warmup problem Pick 4 points at random on a circle. Draw lines between every pair of points. How many regions have we divided the circle into? # of points # of regions ?? N??

5 Fire Which of the following statements are equivalent to the statement, No two students in this class are not on fire.? Which are equivalent to its negation? 1 All student in this class, except at most one, are on fire. 2 Two students in this class are on fire. 3 For any pair of students in this class, one of them is on fire. 4 At least two students in this class are not on fire. 5 If I choose two students in this class and one of them is not on fire, then the other one is on fire.

6 MAT 137, day 2 Course website: probably more reliable than individual instructors! Today: Sets and notation Basics of quantifiers Lecture videos on YouTube: Before Friday s class, watch the remaining 3 videos on quantifiers.

7 Warmup What are the following sets? 1 [2, 4] (3, 5) 2 [2, 4] (3, 5) 3 [π, e] 4 [0, 0] 5 (0, 0)

8 Describing a set An irrational number is one which is real, but not rational. B is the set of positive rational and negative irrational numbers. Write a definition for B using only mathematical notation.

9 True or false? Is the following statement true or false? Prove it. Claim Let A, B, and C be subsets of R. Then (A B) C = A (B C).

10 Some sets What are the following sets? 1 A = {x R : y [0, 1], x < y} 2 B = {x R : y [0, 1] s.t. x < y} 3 C = {x [0, 1] : y [0, 1], x < y} 4 D = {x [0, 1] : y [0, 1] s.t. x < y} 5 E = {x [0, 1] : y R s.t. x < y} 6 E = {x [0, 1] : y R, x < y}

11 MAT 137, day 3 Course website: Today: Quantifiers Negation Lecture videos on YouTube: Before Monday s class, watch the 2 videos on conditionals. Practice problems: see Logic section of the precalculus website

12 Negation Write the negation of these statements without using any negative words ( no, not, none, etc.): 1 My favourite integer is greater than 7. 2 Every student at the U of T likes English more than any other language. 3 There is a country in the European Union with fewer than 1000 inhabitants. Write the negation of these statements as simply as possible: 1 Every student at the U of T has a cellphone. 2 I like pistachios and walnuts.

13 More negation Write the negation of this statement without using any negative words ( no, not, none, etc.): Every page in this book contains at least one word whose first and last letters both come alphabetically before M.

14 True or false? There is a pink elephant in this room. All elephants in this room are pink.

15 Domination, part 1 Given two sets A and B of real numbers, we say that B dominates A when the following statement is true: For every a A, there exists b B such that a < b. Here are some questions: 1 Find a set that dominates itself. 2 Find a set that doesn t dominate itself. 3 Does the empty set dominate itself?

16 MAT 137, day 4 Course website: Today: conditional statements Lecture videos on YouTube: Before Wednesday s class, watch the first 3 videos on definitions and proofs. Office hours and Math Aid Centre hours begin this week! Tutorials begin next week you have to sign up for one.

17 Domination Given two sets A and B of real numbers, we say that B dominates A when the following statement is true: For every a A, there exists b B such that a < b. Find two non-empty sets A and B such that the following three properties hold: 1 A B is empty; 2 A dominates B; 3 B dominates A.

18 Cards Four cards lie on the table in front of you. You know that each card has a letter on one side and a number on the other. At the moment, you can read the symbols E, P, 3, and 8 on the sides that are up. I tell you: If a card has a vowel on one side, then it has an odd number on the other side. Which cards do you need to turn over in order to verify whether I am telling the truth or not?

19 MAT 137, day 5 Course website: Today: conditional statements, definitions, proofs Lecture videos on YouTube: By Friday, watch remaining 3 videos on definitions and proofs Check out new video on negating conditionals!

20 Intervals We define three sets: A = {x Q : x 2 < 2} B = [ 2, 2) C = ( 1, 0) Which of the following statements are true? 1 If x A, then x B 4 If x C, then x B 2 If x B, then x A 5 If x C Q, then x A 3 If x B, then x C 6 If x A, then x C Q

21 Periodic functions Let f be a function with domain R. We want to define what it means for f to be periodic. It is one (or more) of the following: 1 For every x R and for every T > 0, f(x + T ) = f(x). 2 For every x R there exists T > 0 such that f(x + T ) = f(x). 3 There exists T > 0 such that x R = f(x + T ) = f(x). 4 There exists T > 0 such that f(x + T ) = f(x) = x R. 5 There exists T > 0 such that for every x R, f(x + T ) = f(x). 6 For every T > 0 there exists x R such that f(x + T ) = f(x). Which ones are correct? What do the wrong ones mean?

22 MAT 137, day 6 Course website: Today: definitions and proofs, continued For Monday, read 1.3, Review of Inequalities, in the book Tutorials start next week

23 Periodic functions Let f be a function with domain R. We want to define what it means for f to be periodic. It is one (or more) of the following: 1 For every x R and for every T > 0, f(x + T ) = f(x). 2 For every x R there exists T > 0 such that f(x + T ) = f(x). 3 There exists T > 0 such that x R = f(x + T ) = f(x). 4 There exists T > 0 such that f(x + T ) = f(x) = x R. 5 There exists T > 0 such that for every x R, f(x + T ) = f(x). 6 For every T > 0 there exists x R such that f(x + T ) = f(x). Which ones are correct? What do the wrong ones mean?

24 Bad proof #1 The following proof is wrong. Why? Claim For every n, n + 1 = n. Proof. We proceed by induction on n. Suppose the claim is true for n = k. We will prove that it s also true for n = k + 1. By induction, we know that k + 1 = k. Adding one to both sides, we get that (k + 1) + 1 = k + 1.

25 Bad proof #2 The following proof is wrong. Why? Claim On a certain island, there are n 2 cities, some of which are connected by roads. If each city is connected by a road to at least one other city, then you can travel from any city to any other city along the roads. Proof. We proceed by induction on n. The claim is clearly true for n = 1. Now suppose the claim is true for an island with n = k cities. To prove that it s also true for n = k + 1, we add another city to this island. This new city is connected by a road to at least one of the old cities, from which you can get to any other old city by the inductive hypothesis. Thus you can travel from the new city to any other city, as well as between any two of the old cities. This proves that the claim holds for n = k + 1, so by induction it holds for all n.