Chapter One 1 Basic Concept of Sets Native

Chapter One § 1. Basic Concept of Sets Native set theory: We can define a set A of objects by some property that elements of A may or may not posses, to form the set consisting all elements of A having that property. Commonly, we shall use capital letters A, B, … to denote sets, and lowercase letters a, b, … to denote the objects or elements belonging to these sets. If any object a belongs to set A, we express this fact by the notation If a does not belongs to set A, we express this fact by the writting

Empty set : set having no elements. If the set has only a few elements, one can simply list objects in the set. For example, A = {a, b, c}. In general, we write B = {x A: elements in B should posses. }. is the property N: set of natural numbers; Z: set of integers(整数); Q: set of rational numbers(有理数). Equality means logical identity. A=B iff A B ( ) ( )

Thm 1. Assume A, B, C are sets, then 1. reflexive(反身性) A = A; 2. symmetric(对称性) A = B B = A; 3. transitive(传递性) A = B and B = C A = C. Inclusion: we say A is a subset of B if every element of A is also an element of B. , we express this fact by writing. Thm 2. 1) 2) 3) ; ; . The usual way to prove set equality is using property 2).

Proper inclusion: and . Denoted by A B. Thm 3. 1) A is not proper subset of itself; 2) ; 3) can not be true. Family or collection(集族) of sets: A , B , C , …. set whose elements are sets. Example. A = {{1}, {1, 2}, }; The power set of X is : the family(集族) consists of all subsets of X.

Union, intersection and deference of sets ; ; . Thm 1. 1 2) 3) 4) (commutative) (associative) (distributive) 5) (De Morgan) Thm 2. TFAE 1) 2) 3) Underline set X: specify the objects we interested in.

Definition: X A is called complement of A, often denoted by A. Thm 3. Assume X is underline set , A 1. 2. 3. 4. 5. X, B X, then We can define the union and intersection of finitely many sets. Or even arbitratry unions and intersections.

§ 2. Relation(关系) 1. Definition: Cartesian production(卡式积) of sets Ex. Note. 2. Definition: 3. defined inductively by n times 4. 3. Definition: is called a relation from X to Y. If , we say x is R related to y, written as. Domain of range of image of set A.

Mapping is a relation such that for every one , there exists only Ex. For nonempty sets X and Y, is also a relation from X to Y. Domain of , range of Ex. Empty relation Domain of = range of = 4. Definition: is a relation from Y to X, it is called to be the inverse relation of R. is the -image of set B, it is also called R-pre-image of B.

Thm 1: Assume 1) 2) then 3) Thm 2: 1) 2) Give an example to show does the equality hold? Ex. P 5 5, P 8 4, P 12 6, P 15 3. In general, when

§ 3. Equivalent relation(等价关系) Consider relation R from X to X. 1. Definition: identity relation 1）Reflexive 2）Symmetric 3）Transitive diagonal. i. e. , 2. A relation satisfying 1), 2), 3) is said to be an equivalent relation. If that case Ex. then we say that relation R is anti-symmetric, in and can not be hold simultaneously. Both are equivalent relation.

Ex. Consider relations on equivalence =: { inclusion : { relation, since it is not symmetric. } is an equivalent relation; } is not an equivalent proper inclusion { } is not an equivalent relation , since it is not reflexive. Ex. < = {(x, y): x, y Y, x < y} is not reflexive. Ex. Let be a prime number, is an equivalence relation.

3. Definition: Assume R is an equivalent relation, is said to be R equivalent class of x or equivalent class of x in short, denoted by or , any is called to be a representative of , family is called to be quotient set of X with respect to equivalent relation R, denoted by. Ex. Suppose equivalent, , then , thus 1, 3, and 2, 0 are Thm. Let R be an equivalent relation on X, then 1) and 2) thus equivalent relation R divided X into disjoint nonempty equivalent classes(等价类).

Pf. 2) Assume then there is by symmetry, then since R is transitive. Suppose then i. e. , this shows that similarly therefore A partition of a set X is a collection of disjoint nonempty subsets of X whose union is all of X. Ex. divided Z into many equivalent classes It is called congruent class modulo Ex. In analytic geometry, we often consider free vectors, essentially these are equivalent class.

§ 4. Mapping, 1 -1 mapping 1. Definition: If and for any there exists an 2. unique , then F is called to be a mapping from X into Y. 3. Equivalently, 1) , 2) We write 4. , call it image of point x or value of x. x is called to be a 5. pre-image of y, pre-image is a set. Usually we write: 6. Notation (记号，记法): 1) where 2) where 3) If is a mapping, then is also a mapping.

4) is a relation from Y into X. 5) Domain of F = X, range of F = F(X). 6) If range of F = Y, then we say F is an onto mapping or surjection(满射). Thm. (composition of mappings) Let mappings, then is also a mapping, and be , Thm. Pre-image of mapping preserves the operations of union, intersection and complement. 1) , since is also a relation. 2) We need only to show that Let then i. e. , This shows that

3) We shall use lowercase letters f, g, h, … to denote mappings from now on. Definition: if is called a 1 -1 mapping or injection(单射), Ex. identity mapping(恒等映射) If is surjection as well as injection, we say that f is bijection (双射).

Thm. 1) If is a bijection, then is also a mapping, furthermore it is a bijection(双射). 2) If both are bijection, then is also a bijection. Definition: (Restriction and Extension) Suppose satisfy then we say g is the restriction of f on A, f is the extension of g, denoted by particularly is called to be an embedding(嵌入). Definition: Natural projection(canonical projection(正则投射)) is defined by

§ 5. The union and intersection of collection of sets Given a collection A of sets, the union of the elements of A is defined by {A: A A } = {x: x A for at least one A A }. The intersection of the elements of A is defined by {A: A A } = {x: x A for all A A }. The union or intersection are denoted by A or A in short. Sometimes, a collection A of sets is given by using an index set , for every there corresponds a element of A , then A is called an indexed(带下标的) collection of sets. Let A = be an indexed collection of sets, then the union of the elements of A is defined by { : } = {x: x for at least one }. { : } = {x: x for all }.

The union and intersection of collection of sets are irrelevant with the order of indexed set. Thm. 1) for any 2) (distributive) 3) De Morgen Note. If then 1) 2) X is the underline set（基本集 或 基础集）.

Thm. Suppose subsets of Y, then is a mapping, is a collection of

§ 6. Countable set, uncountable set and cardinality(基数) Definition: We say iff there is an injection iff there is a bijection iff and Thm: For any X, Y, Z, 1) 2) 3) This is called an isomorphic relation, it is an equivalent relation. Thm: (Cantor Beinstein or Schroder Beinstein) If and then Thm: 1) 2)

A set A is said to be finite if it is empty or if there is a bijection for some positive integer n. In the former case, we say that A has cardinality 0; in the latter case, we say that A has cardinality n. A set A is said to be infinite if it is not finite. It is said to be countably infinite if there is a bijection A set is said to be countable if it is either finite or countably infinite. A set that is not countable is said to be uncountable. Fact: finite set can’t be isomorphic(同构的) to its proper subset.

Thm. The subset of a countable set N is countable. Proof. We only need to prove that every infinite subset of N is countable. Arrange the subset in an increasing order. Thm. The image B of a countable set N is countable. Proof. Let by the equation be a surjection. Define Thm. Let B be a nonemptyset. Then the following are equivalent: (1) B is countable. (2) There is a surjection. (3) There is an injection.

Thm. Cartesian product countable. of countable sets X and Y is Pf. Diagonal process. Or let be injection. Denote by the prime number starting from 2. Define then h is injective, this justifies that is countable. Thm. The union of countable many countable sets is countable. Pf. Suppose is countable, and for any , we shall show is countable,

For each , let surjection, define surjection. Since be a surjection, is countable, be also a then is also countable. Thm. Suppose then where is mapping}. Think f as characteristic function of subsets of X, actually Pf. Claim 1: where of singleton {x}. Claim 2: Define , it’s enough to define by , actually is the characteristic function Suppose not. Let such that for any be a bijection. where

is a function from X to Y, then we see that for any Since and h is bijection, that is not possible, because for any mapping Thm. There exists an uncountable set. Pf. Consider uncountable set. we see that Thm. R is uncountable. Pf. P 23 4, P 29 2, P 36 5, P 37 1. therefore is an

§ 7. Axiom of Choice and its equivalent forms 1. Definition: Choice function(选择函数) for nonempty set X, such that Axiom of Choice(选择公理): There exists choice function for any nonempty set. Thm. (AC) Let A be a nonempty collection of nonempty sets, then there is a mapping : A {A: A A } such that (A) A for any A A. Pf. Let X= {A: A A } , then X is a nonempty set, let be a choice function. Let = A , then is the required mapping. Thm. (AC) Existence of choice set. For A , there exists a set C such that C A for any A A.

Pf. Let C = (A ) = { (A): A A }, then (A) C A. Thm. Let A be a nonempty disjoint collection of nonempty set, then there exists a set C such that C A is a singleton for any A A. Remark: All 3 Thms implies AC, therefore they are equivalent forms of AC. 2. Turkey lemma A collection A of subsets of a set X is said to be of finite type, provided that a subset B of X belongs to A iff every finite subset of B belongs to A. If A is of finite type, then A has an maximal element, an element which is properly contained in no other element of A.

3. A relation R on a set A is called an order relation (or linear order) if it has the following properties: (1) For x y, either x. Ry or y. Rx. (2) There is no x such that x. Rx. (3) If x. Ry and y. Rz, then x. Rz. A relation on A is called to be a strict partial order on A provided 1) (non-reflexivity) never holds; 2) (transitivity) Hausdorff maximum principle: Let A be a set, let be a strict partial order on A, then there exists a maximal linearly ordered subset B of A.

4. Definition: Suppose A is a set ordered by an order relation <. We say the subset B is bound above if there is an element b of A such that x b for each x B. The element b is called an upper bound for B. Zorn’s lemma: Let A be a set that is strict partially ordered. If every linearly ordered subset of A has an upper bound, then A has a maximal element. 5. Definition: A set A with an order relation is said to be wellordered provided that every nonempty subset A has a smallest element. Well-ordering Thm. Every set A can be well-ordered.