∪ 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
, 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 convenie 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
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)
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}.
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.
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.