Math Camp Day 1. Tomohiro Kusano. University of Tokyo September 4,

Transcription

1 Math Camp 2015 Day 1 Tomohiro Kusano University of Tokyo September 4,

2 Outline I. Logic Basic Logic, T-F table, logical equivalence II. Set Theory Quantifiers, set operations, De Morgan s Law III. Basic Topology Distance, open set, compact set, convex set IV. Sequence Convergence, Squeeze theorem, Cauchy sequence 1

3 Logic 2

4 Why Logic? In theoretical economics, you will encounter a number of mathematical statements like the following: An allocation is Pareto efficient if holds. or equivalently, What is the meaning of the symbols, We use them because they are rigorous in the sense that every reader understands the statement in exactly the same way. Although this example is intentionally made more complicated than necessary, you have to be able to understand what this kind of statement is getting at. 3

5 Proposition Definition A proposition is a declaration that can be either true or false, but not both. Exercise Determine whether the following statements are propositions is divisible by It s Sunday today. 4. Microeconomics is more interesting than macroeconomics appears in the decimal representation of π. 4

6 Propositional Function Proposition must be judged if it is true or false in objective way. 1 and 5 are propositions because it can be judged by calculation ( appears at the 523,551,502nd decimal digit!). 4 is not a proposition because there is no objective manner to check whether this is true or not. We cannot say anything for 2 and 3 unless x and Today is specified. Once they are specified, we can judge. This kind of statements are called propositional function. We denote it by P x, P(today), 5

7 Basic Logic Let P and Q be propositions. Negation: Not P Logical product: P and Q Logical sum: P or Q Implication: If P, then Q Example P: 36 is divisible by 2, Q: 36 is divisible by is not divisible by 2 36 is divisible by 2 and 3 36 is divisible by 2 or 3 If 36 is divisible by 2, then also divisible by 3. 6

8 Basic Logic If P and Q are unrelated, we don t say If P, then Q in real life, but in math, we do. The following are equivalent statements of P implies Q. P only if Q. P is a sufficient condition for Q. Q is a necessary condition for P. We write as which we read P if and only if Q, or P is a necessary and sufficient condition for Q. 7

9 True-False Table Note T T F T T T T F F T F F F T T T F T F F T F F T is false only if P is true and Q is false. Note that is always true when P is false. If I had studied hard, I would have pass the exam! We can contradict him only if he studied hard and failed the exam. 8

10 Logical Equivalence Definition is the converse of is the inverse of is the contrapositive of Definition If two statements have the same truth value for all possible truth values of component variables, they are said to be logically equivalent. (Notation) P is logically equivalent to Q 9

11 Proof by Contrapositive T T T T T T T F F T T F F T T F F T F F T T T T Note that We may thus proof to show if it is difficult to show it directly (Proof by contrapositive). 10

12 Note on Contraposition (1/2) Consider the following propositions. P A mother doesn t scold the child. A child doesn t get down to study. In this case, is If a mother doesn t scold the child, he/she doesn t get down to study., which seems to be plausible. For however, If a child gets down to study, his/her mother scolds., seems to be strange. There is something wrong with the argument, but what is it? Q 11

13 Note on Contraposition (2/2) Rewrite the propositions as the following. P A mother doesn t scold the child at time t. Q A child doesn t get down to study at time t + 1. is not affected by this modification. But the meaning of changes to If a child gets down to study at time t + 1, it implies that his/her mother scolded at time t.. We have to take time into account if there exists a time lag between the premise and the conclusion of a proposition. 12

14 Useful Relationships The following are some useful relationships Exercise Show This is what we call proof by contradiction. When you prove a proposition using proof by contradiction, you have to specify the subject and the object of contradiction, like A contradicts B.. DO NOT write This is a contradiction., There is a contradiction., etc 13

15 Proof by Induction Proof by induction is used to establish a given statement for all natural numbers. Let P(n) denotes a propositional function with a natural number n. The (basic) procedure is the following: 1. Show that P(1) holds. 2. Show that if P(k) holds, then also P(k + 1) holds. 14

16 Exercise Exercise Show the following propositions using proof by induction Suppose Then This is a special case of Bernoulli's inequality. 15

17 Set Theory 16

18 Basic of Sets Definition A set is a collection of elements. Examples The set of countries: {China, Japan, Sweden, }. The set of actions: {listen, sleep, leave, }. We will only focus on the sets of numbers: The set of natural numbers: The set of integers: The set of rational numbers: The set of real numbers: 17

19 Defining Sets Two major ways to define a set: The extensional definition lists all the elements: The intensional definition takes the form like More generally, means the set A consists of elements for which is true. A set may be empty, which we denote If x is an element of the set A then we write The negation is 18

20 Quantifiers Let denote a propositional function whose truth value depends on the value of x. Example is larger than 5. Now we introduce (universal quantifier) and (existential quantifier). for all there exists such that is true if every element of X satisfies is true if at least one of the elements of X satisfies 19

21 Negation and Quantifiers Negation of and : Example Let Then, means x 2 is greater than 0 for any real number x.. Of course we know this is false. To show this, what we have to do is to find a real number x such that P x doesn t hold. That is, the negation of is In this case, x = 0 is the counterexample. 20

22 The order of Quantifiers If we mix two quantifiers, the order does matter. Example I: set of cooks (I = {A, B, C}). C: set of colors (C = {Green, Red, Blue}). 1. i wears c-colored cooking hat. 2. i wears c-colored cooking hat. We all wear a cooking hat with the same color. We all wear a cooking hat but its colors are different. 21

23 Exercise Exercise Choose all the formulae which are equivalent to (1) (2) (3) (4) 22

24 Definition Let U be the universal set. Subset Union Intersection Set difference Cartesian product Complement Set operations U B A U A B 23

25 includes the case Tips If then is denoted by Example The relationship on sets of numbers: This is called n-dimensional Euclidean space. 24

26 Examples Example Let within the universe of 25

27 Examples Example y B A 2 3 x 26

28 De Morgan s Laws Consider sets A and B. Then, This generalize to any number of sets. De Morgan s Laws We often abbreviate to 27

29 Basic Topology 28

30 Euclidean Spaces We will focus on as the universal set. We shall introduce the concept of distance to measure the relationship between each element. Definition The Euclidean distance (norm) between and is defined by More generally, Universal set + Distance = Metric space. 29

31 Example Example Keep in mind that how to define the distance is not so obvious at all. The concept of metric space treats the distance in much more general way. 30

32 ε-neighborhood Definition The ε-neighborhood (open ball) of is define by Example The case with 31

33 Open and Closed Sets Definition A set is an open set if Definition A set is an closed set if its complement is an open set. A set may be open and closed at the same time. A set may neither be open nor closed. 32

34 Examples Open set Closed set Roughly speaking, a closed set includes the boundaries, whereas an open set doesn t. is open. is closed. 33

35 Examples Examples In the space of Open Closed

36 Exercise Exercise Discuss if the following sets are open and/or closed. (1) (2) 35

37 Properties Theorem 1. is open and closed. 2. is open and closed. 3. are open is open. 4. are open is open. 5. are closed is closed. 6. are closed is closed. 3 and 6 holds also with while 4 and 5 don t. 36

38 Exercise Exercise Prove the followings. (1) If is a sequence of open sets, then is open. (2) If is a sequence of closed sets, then is closed. Disprove the followings. In particular, find counterexamples. (3) If is a sequence of open sets, then is open. (4) If is a sequence of closed sets, then is closed. 37

39 Technical Examples Exercise Show that ε-neighborhood of x is open. 38

40 Max and Min Consider the set As you all know, Formally, How about or Neither B nor C has max or min. We thus seek to generalize the concept of maximum and minimum

41 Boundedness Definition 1. is bounded above if and M is called an upper bound of A. 2. Similarly, is bounded below if and M is called an lower bound of A. If the set is bounded from above and below, the set is said to be bounded

42 sup and inf The least upper bound of A is called the supremum of A, which we denote The greatest lower bound of A is called the infimum of A, which we denote Formally, let U(A) and L(A) denote the sets of upper and lower bounds of A, defined by Then, If then we set If then we set 41

43 Examples Examples max min sup inf There always exist sup and inf in whereas max and min sometimes don t. 42

44 Exercise Exercise Let (1) Find the upper bound and the lower bound of A. (2) Examine if the supremum and the infimum of A exists. (3) Examine if the maximum and the minimum of A exists. 43

45 Compact Set Definition A set is bounded if Definition A set is a compact set if it is closed and bounded. In general space, a compact set is defined in more sophisticated way. However, this is the definition of compact set that you will encounter in the coursework. 44

46 Examples Examples The case with compact bounded, not closed closed, not bounded Compact set is an important concept for the existence of the solution to the optimization problem and also for proving the existence of the equilibrium. 45

47 Convex Set Definition A set is convex if Intuitively, a set A is convex if the straight line joining any two points in A is itself contained in A. convex convex convex nonconvex nonconvex nonconvex 46

48 Sequence 47

49 Sequence A sequence in is a specification of a point for each integer We denote them Examples (Fibonacci sequence) 48

50 Definition Convergence A sequence in converges to if We often denote them or A sequence which converges to some limit is called convergent sequence. A sequence is called divergent otherwise. If we write Switch inequality for 49

51 Definition Properties of Sequence A sequence in is bounded if Definition A sequence in is increasing if A sequence in is decreasing if The sequence is said to be monotone if it is either increasing or decreasing. 50

52 Examples Examples 1. If then It is bounded and decreasing. 2. If then it diverges to, unbounded and increasing. 3. If then it converges to 1, bounded and not monotone. 4. If then it is divergent, bounded and not monotone. 51

53 Convergence and Boundedness Theorem Every convergent sequence is bounded. Theorem (Monotone Convergence Theorem) Let be monotone sequence in Then converges (to x) if and only if it is bounded. Moreover, if it is increasing, if it is decreasing. 52

54 Some Properties of Convergence Theorem Consider two sequences and such that Then, For any constant 4. provided and 5. If then 53

55 Squeeze Theorem Theorem (Squeeze Theorem) If and then Example Let Then, 54

56 Cauchy Sequence Definition A sequence in is a Cauchy sequence if Intuitively, if we choose sufficiently large n and m, the distance between two points and can be arbitrarily small. 55

57 Complete Metric Space Theorem A sequence in is a convergent sequence if and only if it is a Cauchy sequence. We can examine the convergence of a sequence in without knowledge of the limiting value (Cauchy criterion for convergence). A metric space in which every Cauchy sequence converges is said to be complete. Thus is a complete metric space. But, not all the metric spaces are complete ( is not complete). Intuitively, complete no missing point. 56

58 Examples Examples 1. is a Cauchy sequence. For any ε, if we set then for any 2. Consider sequence in such that It is a Cauchy sequence, but, so is not complete. 57

59 Closed Set Revisited The following is another characterization of closed set. Theorem is closed if and only if every convergent sequence in X is convergent in X. Example Consider and Then

60 References 1. Rudin, W. (1976), Principles of Mathematical Analysis, McGraw-Hill, 3 rd edition. 2. Sundaram, R.K. (1996), A First Course in Optimization Theory, Cambridge University Press. 3. Chiang, A. C. (2005), Fundamental Methods in Mathematical Economics, McGraw-Hill. 4. Carter, M. (2001), Foundations in Mathematical Economics, MIT Press. 5. 神谷 浦井 (1996), 経済学のための数学入門, 東大出版会. 6. 鈴木 (2003), 集合と位相への入門 ユークリッド空間の位相, サイエンス社. 59