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Chapter 1 Native Set Theory

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

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Chapter 1 Native Set Theory

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  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.

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