1 / 30

Chapter 1 Native Set Theory

If a does not belongs to set A , we express this fact by the writting.

millie
Download Presentation

Chapter 1 Native Set Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. If a does not belongs to set A, we express this fact by the writting 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 Chapter 1 Native Set Theory §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.

  2. In general, we write B = {xA: } . is the property elements in B should posses. Equality means logical identity. A = B iff A B  (  )  (  ) 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}. N: set of natural numbers; Z: set of integers(整数); Q: set of rational numbers(有理数).

  3. 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 . Thm2. 1) ; 2) ; 3) . 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. The usual way to prove set equality is using property 2).

  4. Example. A = {{1},{1,2},}; The power set of X is : the family(集族) consists of all subsets of X. Proper inclusion: and . Denoted by AB. 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.

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

  6. Thm3. Assume X is underline set , AX , BX, then • 1. 3. 4. 5. Definition: X \ A is called complement of A, often denoted by \ A. We can define the union and intersection of finitely many sets. Or even arbitratry unions and intersections.

  7. 1. Definition:Cartesian production(卡式积) of sets Ex. Note. • Definition: • defined inductively by n times 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. §2. Relation(关系)

  8. 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. Mapping is a relation such that for every , there exists only one

  9. Give an example to show In general, when does the equality hold? Thm 1:Assume then 1) 2) 3) Thm 2: 1) 2) Ex. P5 5, P8 4, P12 6, P15 3.

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

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

  12. representative of , family is called to be quotient set of X with respect to equivalent relation R, denoted by . 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 Ex. Suppose , then , thus 1, 3, and 2, 0 are equivalent, 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).

  13. Ex. divided Z into many equivalent classes It is called congruent class modulo 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. In analytic geometry, we often consider free vectors, essentially these are equivalent class.

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

  15. Thm. (composition of mappings) Let be mappings, then is also a mapping, and , 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 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).

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

  17. 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

  18. 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 }. §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.

  19. 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). The union and intersection of collection of sets are irrelevant with the order of indexed set.

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

  21. 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) §6. Countable set, uncountable set and cardinality(基数)

  22. 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.

  23. Proof. Let be a surjection. Define by the equation 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. 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.

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

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

  26. Pf. Consider we see that therefore is an uncountable set. Pf. 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. Thm. R is uncountable. P23 4, P29 2, P36 5, P37 1.

  27. 1. Definition: Choice function(选择函数) for nonempty set X, such that Pf. Let X= {A: A  A } , then X is a nonempty set, let be a choice function. Let =A , then  is the required mapping. §7. Axiom of Choice and its equivalent forms 2. 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 . Thm.(AC) Existence of choice set. For A , there exists a set C such that C A   for anyA  A .

  28. 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. Pf. All 3 Thms implies AC, therefore they are equivalent forms of AC. 3.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 .

  29. 4. 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. Recall relation on A is called to be a strict partial order on A provided 1) (non-reflexivity) never holds; 2) (transitivity) 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 xRy or yRx. (2) There is no x such that xRx. (3) If xRy and yRz, then xRz.

  30. Definition: Suppose A is a set ordered by the 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. 5. 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. Definition: A set A with an order relation is said to be well-ordered provided that every nonempty subset A has a smallest element. 6.Well-ordering Thm. Every set A can be well-ordered.

More Related