MAT137 Calculus! Welcome!

Transcription

1 MAT137 Calculus! Welcome! Beatriz Navarro-Lameda L0101 WF 1-4 MP202 office hours: Wednesday, May 17: 4-5 official website read course outline! remember to enrol in tutorial! Online Forum - Precalculus review:

2 Check your s regularly Mobile Access to your s

3 How did students do last year? Among students who wrote the final exam: # 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%

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5 MAT137 Calculus! Lecture 1 Today: - Sets and Notation - Quantifiers Homework before Friday s class: Watch all videos on - Conditionals (2) - Definitions and Proofs (6)

6 A warm-up problem Pick 4 points at random on a circle. Join every pair of points. In how many regions is the circle divided?

7 A warm-up problem # of points # of regions

8 A warm-up problem # of points # of regions

9 A warm-up problem # of points # of regions

10 A warm-up problem # of points # of regions

11 A warm-up problem # of points # of regions ??? N???

12 A warm-up problem # of points # of regions N???

13 A warm-up problem There are 31 regions!!!

14 A warm-up problem For N points, the number of regions is N 4 6N N 2 18N = 1 + ( ) N + 2 ( ) N 4

15 Fire Which of the following statements are equivalent to the statement, No two students in this class are not on fire.? 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. Which are equivalent to its negation?

16 Fire Which of the following statements are equivalent to the statement, No two students in this class are not on fire.? 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. Which are equivalent to its negation?

17 Negation I Write the negation of the statement without using any negative words ( no, not, none, etc.): Answer: My favourite integer number is greater than My favourite integer number is less than My favourite integer number is less than or equal to I don t like numbers greater than 20.

18 Negation II Write the negation of the following statement without using any negative words ( no, not, none, etc.): I like ice cream and smoothies

19 Sets and Notation Example (Warm up I) What are the following sets? 1 [0, 2] (1, 3] 2 [0, 2] (1, 3] 3 [1, 1] 4 (1, 1)

20 Sets and Notation Example (Warm up II ) What are the following sets? 1 A = {x R : x 2 = 2} 2 B = {x Z : x 2 = 2}

21 Sets and Notation Example (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}

22 Set Description Recall: Q is the set of all rational numbers i.e. numbers that can be written as p q for some p, q Z, q 0. Definition Irrational Number An irrational number is a number that is real but not rational. Example Let B be the set of positive rational numbers and negative irrational numbers. Write a definition for B using only mathematical notation.

23 Set Description Recall: Q is the set of all rational numbers i.e. numbers that can be written as p q for some p, q Z, q 0. Definition Irrational Number An irrational number is a number that is real but not rational. Example Let B be the set of positive rational numbers and negative irrational numbers. Write a definition for B using only mathematical notation. Notation: A \ B = {x A : x / B}

24 Quantifiers Negation I Write the negation of the following statement: Every prince has a white horse. Answer: 1 Every prince has everything but a white horse. 2 Every prince has a horse that is not white. 3 There is a prince who doesn t have a white horse.

25 Quantifiers Negation II 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. Negation: There is a page in this book for which every word has either its first or last letter coming alphabetically after L.

26 Negation III 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. Negation: There is a page in this book for which every word has either its first or last letter coming alphabetically after L.

27 True or False? s 1 There is a flying pig in this room. 2 All pigs in this room can fly.

28 Love Let M be the set of all men, and W be the set of all women. For x M and y W, let L(x, y) denote x loves y. Translate into English the following statements: 1 x M, y W s.t. L(x, y). 2 y W s.t. x M L(x, y).

29 True or False? If the statement is true, give a proof. If the statement is false, give a proof or a counterexample. 1 x R, y R s.t. x + y = 0 Meaning: Every real number has a negative. 2 y R s.t. x R, x + y = 0 Meaning: There is a real number that is the negative of every real number.

30 True or False? If the statement is true, give a proof. If the statement is false, give a proof or a counterexample. 1 x R, y R s.t. x + y = 0 Meaning: Every real number has a negative. TRUE 2 y R s.t. x R, x + y = 0 Meaning: There is a real number that is the negative of every real number. FALSE

31 Domination Definition: 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 are true: 1 A B is empty, 2 A dominates B 3 B dominates A