高级微观经济学(上海财经大学 陶佶)note01

Figure 1. Venn Diagrams The Venn diagrams above show four standard binary operations on sets. 1. The union (并集) of A and B is the set A ∪ B = {x: x ∈ A or x ∈ B}.
∪ i∈I Si = {x: x ∈ Si for some i ∈ I};
b) ∩ i∈I Si = {x: x ∈ Si for all i ∈ I}. DeMorgan’s Law can be generalized to indexed collections. Theorem 3. Let A be a set and {Si}i∈I be an indexed collection of sets, then a) A \ ∪ i∈I Si = ∩ i∈I (A \ Si); b) A \ ∩ i∈I Si = ∪ i∈I (A \ Si). Problem 2. Prove Theorem 3. Definition 4. Given any set A, the power set (幂集) of A, written P(A) is the set consisting of all subsets of A; i.e., P(A) = {B: B ⊂ A}. Problem 3. If a set S has n elements, how many elements are there in P(S)? Definition 5. The Cartesian product (笛卡尔乘集) of two sets A and B (also called the product set or cross product) is defined to be the set of all points (a, b) where a ∈ A and b ∈ B. It is denoted
n
2 3
and two-
. An n-dimensional space is defined as the set product
≡
× ×
.xn ) of
×
n
≡ {( x1 , x2 ,
, xn ) | xi ∈ , i = 1, 2,
, n} .
The element ( x1 , x2 ,
is an n-dimensional ordered tuple, or vector, usually denoted
上海财大经济学院
2
作者：陶佶
2005 年秋季
高等微观经济学 I
实分析简介
A × B, and is called the Cartesian product since it originated in Descartes’ formulation of analyLeabharlann Baiduic geometry. The main examples of cross products are Euclidean three-dimensional space dimensional space
, S,
, Z . A set can consist of any type of element. Even sets can be
elements of some set. A consumption set is a collection of consumption plans. The typical sets we deal with have real numbers as their elements. If a is an element of A, we write a ∈ A. If a is not an element of A, we write a ∉ A. If all the elements of A are also elements of B, then A is a subset of B. We write either A ⊂ B or
with boldface or underscored type. This note uses x ≡ ( x1 , Vector Relation: for any two vectors x and y in and x y if xi > yi , i = 1,
n
, xn ) for convenience.
2005 年秋季
高等微观经济学 I
实分析简介
Lecture Note I
1. Logic Consider two statements, A and B. Suppose B ⇒ A is true. 1. A is necessary (必要条件) for B. 2. B is sufficient (充分条件) for A. Contra-positive (逆否) form of B ⇒ A: ~A ⇒ ~B. If both A ⇒ B and B ⇒ A are true, then A and B are equivalent: A ⇔ B. 2. Set Theory We begin with a few definitions. A set (集合) is a collection of objects called elements (元素). Usually, sets are denoted by the capital letters A, B,
B ⊃ A . If A ⊂ B and B ⊂ A , we say that A and B are equal: A = B.
A set S is empty (空集) if it contains no elements at all. An empty set denoted as ∅ is a subset of any set.
n
, we say that x ≥ y if xi ≥ yi , i = 1,
, n;
, n.
Definition 6. S ⊂
is a convex set (凸集) if for all x and y ∈ S , we have tx + (1 − t ) y ∈ S
for all t in the interval [0, 1]. Intuitively, a set is convex iff we can connect any two points in the set by a straight line that lies entirely within the set. Note that convex sets play a fundamental role in microeconomic theory. In theoretical analysis, convexity is assumed by economists to get well-behaved analytical results. Remark 7. The intersection of convex sets is convex, but the union of them is not. 3. A Little Topology We begin with a rigorous definition of metric space. A metric space (测度空间) is a set S with a global distance function (the metric d) that, for every two points x and y in S, gives the distance between them as a nonnegative real number d(x, y). A metric space must satisfy: 1. d(x, y) = 0 iff x = y; 2. d(x, y) = d(y, x); 3. The triangle inequality d(x, y) + d(y, z) ≥ d(x, z). A natural example is the Cartesian plane . Define the distance function
上海财大经济学院
1
作者：陶佶
2005 年秋季
高等微观经济学 I
实分析简介
2. The intersection (交集) of A and B is the set A ∩ B = {x: x ∈ A and x ∈ B}. 3. The difference of A and B is the set A \ B = {x: x ∈ A and x ∉ B}. 4. The symmetric difference of A and B is the set A Δ B = (A ∪ B)\(A ∩ B). It can be easily seen that A Δ B = (A \ B) ∪ (B \ A). Another common set operation is complementation (补集). Let U be a well-defined universal set that contains all the elements in the question. Then the complementation of a set A ⊂ U is Ac = U \ A . Theorem 1. Let A, B, and C be sets. a) A \ (B ∪ C) = (A \ B) ∩ (A \ C); b) A \ (B ∩ C) = (A \ B) ∪ (A \ C). Proof: Theorem 1 can be proved as a sequence of equivalences. Problem 1. Prove Theorem 1. The familiar DeMorgan’s Law is an obvious consequence of Theorem 1 when there is a universal set to make the complementation well-defined. Corollary 2. (DeMorgan’s Law) Let A, and B be sets. a) (A ∪ B)c = Ac ∩ Bc; b) (A ∩ B)c = Ac ∪ Bc. To deal with large collections of sets, we use index set (索引集) I = {1, 2, 3,…} and denote the collection of sets as {Si}i∈I. Union and Intersection can be extended to work with indexed collections. In particular, we define a)
d x , y ≡ ( x1 − y1 ) 2 + ( x2 − y2 ) 2 ≡ x − y
for x and y in . It is obvious to see that the space with the metric d above is a metric space. The metric d called as Euclidean metric or Euclidean norm (欧几里德范数) can be generalized to an n-dimensional Euclidean space. Definition 8. Open and Closed ε -Balls (开球和闭球): Let ε be a real positive number.