1、 1Lecture Note #1 Chapter 1 Set Theory and Properties of Rn 1.1 Set-Theoretic Notation and Concepts Aa Aa BA A= Xx | x satisfies P The set of real number R, the set of nonnegative real number R+, the set of positive real number R+. 1.1 Definitions. Suppose that a and b are real numbers, and that ab.
2、 Then we define: 1. a, b the closed interval from a to b by a, b= | bxaRx 2. a,b the open interval from a to b by a, b= | bxaRx :)( . With this notation, we can rewrite definition 1.3 as *)()(:),( aabfafAaa = and )(:)( afbAaBb = . 1.4 Definitions. If YXf : , and A and B are subsets of X and Y, respe
3、ctively, we define: 1. the image of A under f , denoted )(Af , by: )(:)(|)( xfyAxYyAf = . 2. the inverse image of B under f , denoted ),(1Bfby: .)(|)(1BxfXxBf =Examples: Figure 1.3, real world examples. We denote by P(A) the family of all subsets of A (the power set of A). If A and X are non-empty s
4、ets, and V is a set of subsets of X, then A is an index set for V iff there exists a function VAf : , which is both one-to-one and onto. (notice A can be exactly V) Example: Let X=R, A=N, and for each Nn , define nX by ./13,/1 nnXn= 41.6. Proposition. If A is a subset of S, and | = B is a family of
5、subsets of S, then: 1. =)(BA )(BAand =)(BA )(BA. 2. =)(BA )(BA. 3. =)(BA )(BA. 1.7 Theorem (DeMorgans Laws). Let S and A be non-empty sets, and let | AaXa= be a family of subsets of S. Then we have: 1. the complement of the union equals the intersection of the complements, that is: caAacaAaXX )()(=
6、2. the complement of the intersection equals the union of the complements, that : caAacaAaXX )()(= . 1.2 Properties of the Real Numbers (Review) A. Inequalities. If a, b, c, d R , then 1. if ab and cd, then a+cb+d, 2. if c0, then ab cacb 3. if c0, then ab cbca, 4. if 0. 1.15 Proposition. Suppose tha
7、t A and B are non-empty sets of real numbers satisfying: baBbAa :)( . Then infA and supB both exist, and we have infAsupB. Proof: In economic theory we often have occasion to consider sets which are scalar multiples or sums of other sets. In the case of subsets of the real numbers, we can define the
8、se as follows. 1.16 Definition. Let A and B be non-empty sets of real numbers, and let R . Then we define the sets A and A+B by: :)(| xxAxRxA = and :)(| baxBbAaRxBA +=+ . 1.17 Proposition. If A is a non-empty set of real numbers, which is bounded above, and is a nonnegative real number (+R ), then s
9、up( A ) exists, and we have sup( A )= supA. 1.18 Proposition. If A and B are non-empty sets of real numbers which are bounded above, then A+B is bounded above, and sup(A+B)=supA+supB. Homework: Page 10, problem 2, 4. Page 17, problem 5, 6. 1Lecture Note #2 1.3 Binary Relations A binary relation, R,
10、on a set X, is simply a rule such that for each x and y in X, we can determine whether sRy, yRx, or neither, or both. Ex: 1. yEx x=y. 2. Suppose RRf : , define yRx y )(xf 1.19. Definitions. Let G be a binary relation on a set X. We shall say that G is: 1. Total iff: xGyXyx :),( or yGx or yx = . 2. R
11、eflexive iff: xGxXx :)( . 3. Irreflexive iff xGxXx :)( . 4. Symmetric iff yGxxGyXyx :),( 5. asymmetric iff yGxxGyXyx :),( 6. antisymmetric iff: xGyXyx :),( Asymmetric relation is not the negation of symmetric relation; However, any asymmetric relation is also antisymmetric and irreflexive. Examples:
12、 Relation “” on R is total, reflexive, antisymmetric and transitive; but neither symmetric nor asymmetric. Relation “” on R is total, irreflesive, asymmetric and transitive. 1.21. Definition. Let G be a binary relation on a non-empty set, X. We shall say that G is a weak order (or that G is a weak o
13、rdering of X) iff G is total, reflexive, and transitive. Examples: 1. “”; 2. “at least as heavy as”; 3. Let X be any non-empty set, and let :f XR . Relation G defined on X: () ()xGy f x f y is a weak order on X. In working with orderings, it is often useful to divide the relation up by separating it
14、 into its so-called asymmetric and symmetric parts. 21.23 Proposition. Let G be a binary relation on a set X, and define P and I on X by: xPy xGy and where for 1(, , ) ,nx xxX=“ ix denotes the amount of the ith commodity available to the consumer, per unit of time. In this context, the consumer is s
15、upposed to choose according to her/his preference relation, G, defined over the consumption set. Let be a family of non-empty subsets of X. We then define :hX by: () |( ): hB x B y B xGy= . It is assumed that, if the consumer is restricted to a choice from B, then he or she will always choose an ele
16、ment of h(B). Typically one assumes that this preference relation is a weak order, and the asymmetric part of G, P, is called the consumers strict preference relation, while the symmetric part of G, I, is called the consumers indifference relation. 2. If G is a weak order on X, and P and I are the a
17、symmetric and symmetric parts of G, respectively, show that P and I satisfy the following conditions. For any w, x, y, and z in X: a. we have xPy yGx , (1.12) 3b. exactly one of the following conditions holds: xPy, yPx, or xIy. c. If wGx, xPy, then wPy. (Similarly, If wPx, xGy, then wPy) d. If xPz,
18、then either xPy or yPz. Proof: a. (xPy xGy and since xPz, it then follows from (c) that yPz. 3. As we have noted above, it is generally accepted practice to regard the economic theory of consumer behavior as being based upon the idea that given any two commodity bundles, x and y, a consumer can alwa
19、ys tell us whether x is at least as good as y, or whether y as good as (the binary relation, G) to be primitive (a basic, undefined, building block) of our theory, and usually assume that it is a weak order. The consumers strict preference relation, P, and indifference relation, I, are then derived
20、concepts of the theory; that is, they are defined from G. The foregoing considerations may raise a question in your mind; namely, what happens if we make (strict) preference the basic primitive of the theory? In other words, suppose we begin our theoretical considerations with the assumption that th
21、e consumer has a (strict) preference relation, P, which is asymmetric, irreflexive, and transitive; which are the properties which P has been shown to satisfy in Proposition 1.26. We can then define, G, the at-least-as-good-as relation, form P by (1.12). Do we now have an equivalent theory? The answ
22、er to this last question is No. Ex: 1xPy x y +. Thus 1xGy y x+. One can show that P is irreflexive, asymmetric and transitive. But the corresponding G is not transitive. 1.28 Definition. We shall say that a relation, P, on a set X is negatively transitive iff for all x, y, zX, we have: if xPz, then
23、either xPy or yPz If x is preferred to z, and y is any alternative not preferred to z, then x must be preferred to y. 41.29. Proposition. If G is a weak order on X, then P, its asymmetric part, is negatively transitive. Conversely, if P is a relation that is asymmetric and negatively transitive, and
24、 we define G on X by (1.12), above, then G is a weak order on X, and P is its asymmetric part. Proof: “” is by 1.27.2.d. “” we only show transitivity. Suppose xGy and yGz. If xGz , then zPx. By negative transitivity, we would have zPy or yPx. 1.30. Corollary. If P is a binary relation which is asymm
25、etric and negatively transitive, then P is also transitive. (Proposition 1.26) If we begin our development of consumer preference theory with strict preferences as the basic primitive of the theory, and we assume that the strict preference relation, P is asymmetric and negatively transitive, our the
26、ory is equivalent to one which takes the at-least-as-good-as relation, G, as the primitive, and assumes that G is a weak order. 1.31. Definition. If G is a binary relation on X, we shall say that a function :f XR represents G on X iff for all x and y in X, we have () (),xGy f x f y (1.19) 1.32. Prop
27、osition. Suppose that X is a finite set, and that G is a binary relation on X. Then there exists :f XR such that f represents G iff G is a weak order. Homework: Page 32, problem 1, 2, 3. 1Lecture Note 3 1.4 Inner Product, Norm, and Euclidean Metric 1.37. Definition. For ,nx yR , we define: 1. x y re
28、ad x is greater than or equal to y iff iix y , for 1, , .in= null 2. x y read x is semi-greater than y iff x y , but yx . 3. x ynull read x is strictly greater than y iff iix y , for 1, , .in= null 1.42. Definition. For ,nx yR , we define x y , the inner product of x and y, by: 1niiix yxy=. 1.43. Pr
29、oposition. (Basic properties of the inner product). If x, y and z are elements of nR , and aR , then: 1. 0xx and 0xx 0x=. 2. x yyx=. 3. () ( )()x ay a x y ax y= 4. ()x yz xyxz+=+ 1.44. Lemma. If a, b, and c are real numbers satisfying: 2():2 0,Ra b c + + (1.28) then we must have 0, 0,a c and 2ac b .
30、 1.45. Definition. For nx R , we define x , the Euclidean norm of x, by 1/221/21()niix xxx=. 21.46. Theorem. (Cauchy-Schwarz Inequality). For all ,nx yR , we have |x yxy . Proof: ()()0xy xy + +. 1.47. Theorem. The Euclidean norm, i , satisfies, for each x, y, z R , and each R : 1. 0x Page 50, proble
31、m 1, 2. 1Lecture Note #4 Chapter 2 Sequences and Infinite Series 2.1 Sequences of Real Numbers The set of natural numbers N=1, 2, 3, . , 2.1 Definition. An (infinite) sequence of real numbers is a function RNf : . We used to denote )(nf as nx . And we use nx to denote the sequence itself. Ex: 1+= nx
32、n, nyn= , )sin(nzn= , nwn/1= . 2.2. Definition. Let nx be a sequence, and let Rx *. The sequence nx is said to converge to *x iff |:|)()(0(*xxmnNmn. In this case, *x is said to be the limit of the sequence nx (and nx is said to be convergent), and we write *lim xxnn=, or *xxn . 1. The limit of a seq
33、uence, if it exits, is unique. 2. The alteration of a finite number of terms of a sequence has no effect on convergence, divergence, or limit. 3. If all but a finite number of terms of a sequence are equal to some constant, then the sequence converges to that constant. 2.3. Definition. If a sequence
34、 converges to some real number Rx it is said to be convergent; otherwise it is said to be divergent. 2.4. Definition. A sequence nx is said to have the limit + , or to diverge to + , iff ( R )( Nm ) nxmn :)( . In which case we write +=nnxlim , or +nx . Ex: “ ,1,31,21nxn= ; “,114,83,52,21=nx ; nnx )1
35、(= ; nnx )2(= ; nxn= . 2.6. Definition. A sequence nx is said to be bounded below iff )(:)( nxNnR . The sequence nx is said to be bounded iff it is bounded both above and below. 2.7. Proposition. A sequence nx is bounded iff )|(|:)( :),( , and 2. inixyNi = :)( 2.23. Proposition. If nx is a sequence
36、such that *xxn , and inx is any subsequence of nx , then *xxin . 2.24. Definition. We say Rx *is a cluster point (or accumulation point) of the sequence nx iff, given any 0 and any positive integer, m, there exists nm such that , there exists Np such that, for all pnm , , is a sequence, the expressi
37、on 1nna=is called an infinite series; the sequence ns converges or diverges. If ns s , we write 1nns a=and say that the series converges to s. Another expression: ,1npnp m n p nmnSass+=+=2.36. Proposition. The infinite series, 1nna=, converges iff 0, ,nN such that ,| |nppNS is bounded. 32.42.Proposi
38、tion. A nonnegative series, 1nna=, is convergent iff it is bounded. (2.20) 2.43.Theorem. Suppose 1nnb=is a nonnegative series, and let 1nna=be a second series such that there exists an integer m satisfying : | |nnab , for n=m+1, m+2, Then if 1nnb=is convergent, 1nna=is absolutely convergent. 2.44.Ex
39、amples Consider 1/nan= and nbn= where ( 1 ). We have |1/nnannb . Homework: Page 95, problem 7, 8. Lecture Notes 6 Chapter 3 Continuity 3.1 Continuous Vector-Valued Functions 3.1 Definition. Let :mf XR , where X is a non-empty subset of nR . We say that f is continuous at *x X iff, for every R+ , the
40、re exists R+ such that *( ( , ) ) :| ( ) ( ) |xNx X fx fx 0 , R+ such that: *(,):|()|xNx X fx y =. Consider x=0 3. :f XR , XR+= , sin(1/ ) 0()10xxfxx=. Consider x=0 4. :f XR , XR+= , sin(1/ ) 0()10xxxfxx=. Consider x=0 3.6 Proposition. Let :mf XR , where X is a non-empty subset of nR , and let *x X
41、be a limit point of X. Then f is continuous at *x X iff *lim ( ) ( )xxf xfx=. 3.7 Definition. Let if :mf XR , where X is a non-empty subset of nR , and let A be a subset of X. We shall say that f is bounded on A iff there exists aR+ such that ():|()|x Afxa . If A=X, then we say f is bounded. 3.8. Pr
42、oposition. If :mf XR , where X is a non-empty subset of nR , *x is a limit point of X, and myR is such that *lim ( )xxf xy=, then there exists a positive real number, , such that f is bounded on *(,)N xX . One of the reasons for the importance of continuous functions is that arithmetic operations on
43、 continuous functions preserve continuity. 3.9. Theorem. Suppose :mf XR and :mg XR , where X is a non-empty subset of nR , let * nx R be a limit point of X, and suppose ,myz R are such that *lim ( )xxf xy= and *lim ( )xxg xz= . We have: 1. *lim ( ) ( )xxf xgx yz+=+, 2. for any R , *lim ( )xxf xy = ,
44、 3. *lim ( ) ( )xxf xgx yz=. 3.10. Theorem. (Corollary) Suppose :mf XR and :mg XR , where X is a non-empty subset of nR . If f and g are both continuous at *x X , then we have: 1. f g+ is continuous, 2. for any R , f is continuous, 3. f g is continuous. Proof: (Thm 3.9.) 3.11. Theorem. Suppose :mf X
45、R and :mg XR , where X is a non-empty subset of nR . And both f and g are continuous at *x X . If *()0gx , then the function () ()/ ()hx f x gx= is well defined in a neighborhood of *x , and is continuous at *x . 3.12. Theorem. Let :nf XR and :pg XR , where mXR and ()nf xYR ; and suppose f is contin
46、uous at *x X , and that g is continuous at *()yfxY=. Then the composite function, hgf= D , is continuous at *x . 3.13. Examples: 1. :nnf RR , ()f xx= , then f is continuous onnR . 2. :nf RR , ()f xbx=, then f is continuous onnR . 3. If nN , define :f RR , ( )nf xx= , then f is continuous on R . 4. :
47、mif XR is continuous for 1,.,ik= , the function f defined on X by 1() ()kiiif xafx=is also continuous. 5. 1:mf XR is continuous, where 1X is a non-empty subset of nR , let 2X be a subset of pR , and define 12XXX= and ( )12 1,(F xx fx=）. Then F is continuous on X. 6. Suppose :nf XR and :pg XR are continuous, where mXR , then () ( (), ()F xfxgx= is continuous on X. 7. Suppose :miif XR , where iniXR , is continuous, for 1, 2i = and define f on 12XXX= by 11 2() ( ) ( )f xfxfx=. Then f is continuous on X. 3.14. Proposition. Suppose :mf XR , where X is a non-empty subset of nR , and let *x X .
