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Mechanizing set theory cardinal arithmetic and the axiom of choice

Mechanizing Set TheoryCardinal Arithmetic and the Axiom of ChoiceLawrence C.PaulsonComputer Laboratory,University of Cambridgeemail:lcp@Krzysztof GrabczewskiNicholas Copernicus University,Toru´n,Polandemail:kgrabcze@mat.uni.torun.plJanuary1996Minor revisions,November2000Abstract.Fairly deep results of Zermelo-Frænkel(ZF)set theory have been mechanized using the proof assistant Isabelle.The results concern cardinal arithmetic and the Axiom of Choice (AC).A key result about cardinal multiplication isκ⊗κ=κ,whereκis any inﬁnite cardinal. Proving this result required developing theories of orders,order-isomorphisms,order types, ordinal arithmetic,cardinals,etc.;this covers most of Kunen,Set Theory,Chapter I.Further-more,we have proved the equivalence of7formulations of the Well-ordering Theorem and20 formulations of AC;this covers theﬁrst two chapters of Rubin and Rubin,Equivalents of the Axiom of Choice,and involves highly technical material.The deﬁnitions used in the proofs are largely faithful in style to the original mathematics.Key words:Isabelle,cardinal arithmetic,Axiom of Choice,set theory,QED projectContents1Introduction1 2Isabelle and ZF Set Theory2 3The Cardinal Proofs:Motivation and Discussion33.1Inﬁnite Branching Trees33.2Overview of Kunen,Chapter I5 4Foundations of Cardinal Arithmetic74.1Well-orderings74.2Order Types84.3Combining Well-orderings84.4Cardinal Numbers94.5Cardinal Arithmetic10 5Provingκ⊗κ=κ105.1Kunen’s Proof105.2Mechanizing the Proof12 6The Axiom of Choice and the Well-Ordering Theorem15 7Rubin and Rubin’s AC Proofs177.1The Well-Ordering Theorem177.2The Axiom of Choice187.3Difﬁculties with the Deﬁnitions207.4General Comments on the Proofs217.5Consolidating Some Proofs227.6The Axiom of Dependent Choice24 8Proving WO6=⇒WO1248.1The idea of the proof258.2Preliminaries to the Mechanization268.3Mechanizing the Proof27 9Conclusions302Lawrence C.Paulson and Krzysztof GrabczewskiMechanizing Set Theory11.IntroductionA growing corpus of mathematics has been checked by machine.Researchers have constructed computer proofs of results in logic[26],number theory[25], group theory[28],λ-calculus[10],etc.An especially wide variety of results have been mechanized using the Mizar Proof Checker,including the theorem κ⊗κ=κdiscussed below[2].However,the problem of mechanizing mathe-matics is far from solved.The Boyer/Moore Theorem Prover[3,4]has yielded the most impressive results[25,26].It has been successful because of its exceptionally strong sup-port for recursive deﬁnitions and inductive reasoning.But its lack of quantiﬁers forces mathematical statements to undergo serious contortions when they are formalized.Most automated reasoning systems areﬁrst-order at best,while mathematics makes heavy use of higher-order notations.We have conducted our work in Isabelle[20],which provides for higher-order syntax.Other recent systems that have been used for mechanizing mathematics include IMPS[6], HOL[8]and Coq[5].We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice(AC).Paulson has mechanized most of theﬁrst chapter of Kunen[12]and a paper by Abrial and Lafﬁtte[1].Gra¸bczewski has mech-anized theﬁrst two chapters of Rubin and Rubin’s famous monograph[24], proving equivalent eight forms of the Well-ordering Theorem and twenty forms of AC.We have conducted these proofs using an implementation of Zermelo-Frænkel(ZF)set theory in pared with other Isabelle/ZF proofs [15,18,21]and other automated set theory proofs[23],these are deep,abstract and highly technical results.We have tried to reproduce the mathematics faithfully.This does not mean slavishly adhering to every detail of the text,but attempting to preserve its spirit.Mathematicians write in a mixture of natural language and symbols; they devise all manner of conventions to express their ideas succinctly.Their proofs make great intuitive leaps,whose detailed justiﬁcation requires much additional work.We have been careful to note passages that seem unusually hard to mechanize,and discuss some of them below.In conducting these proofs,particularly from Rubin and Rubin,we have tried to follow the footsteps of Jutting[11].During the1970s,Jutting mech-anized a mathematics textbook using the AUTOMATH system[14].He paid close attention to the text—which described the construction of the real and complex numbers starting from the Peano axioms—and listed any deviations from pared with Jutting,we have worked in a more abstractﬁeld,and with source material containing larger gaps.But we have had the advantage of much more powerful hardware and software.We have relied upon Isabelle’s reasoning tools(see§2below)toﬁll some of the gaps for us.2Lawrence C.Paulson and Krzysztof GrabczewskiWe have done this work in the spirit of the QED Project[22],which aims “to build a computer system that effectively represents all important mathe-matical knowledge and techniques.”Our results provide evidence,both posi-tive and negative,regarding the feasibility of QED.On the positive side,we are able to mechanize difﬁcult mathematics.On the negative side,the cost of doing so is hard to predict:a short passage can cause immense difﬁculties. Overview.Section2is a brief introduction to Isabelle/ZF.The remaining sections reportﬁrst Paulson’s work and then Gra¸bczewski’s.Sections3–5dis-cuss the foundations of cardinal arithmetic in increasing detail,culminating in the machine proof of a key result about cardinal multiplication,κ⊗κ=κwhereκis inﬁnite.Section6introduces the Axiom of Choice and describes the mechanization of Abrial and Lafﬁtte.Sections7and8are devoted to the mechanization of parts of Rubin and Rubin.Section9presents some conclu-sions.2.Isabelle and ZF Set TheoryIsabelle[20]is a generic proof assistant.It supports proofs in higher-order log-ic,various modal logics,linear logic,etc.Our work is based upon Isabelle’s implementation of Zermelo-Frænkel(ZF)set theory,itself based upon an imple-mentation ofﬁrst-order logic.Isabelle/ZF arose from early work by Paul-son[17]and No¨e l[15];it is described in detail elsewhere[18,21].There are two key ideas behind Isabelle:−Expressions are typedλ-terms.Thus the syntax is higher-order,giving a uniform treatment of quantiﬁers,descriptions and other binding operators.In Isabelle/ZF,all sets have the same type.But other important objects, such as classes,class relations and class functions,can be expressed using higher types.−Theorems are schematic inference rules.Isabelle’s basic inference mech-anism is to join two schematic rules,in a sort of Horn clause resolution.A typical step in a backward proof consists of joining one rule(typicallya lemma)to another rule(representing the proof state).Thus,theoremsare proved by referring to previous theorems.Proof states may contain unknowns:placeholders for terms that have been left unspeciﬁed.Uniﬁca-tion can incrementally instantiate unknowns,which may be shared among several subgoals.Built around these key ideas are various facilities intended to ease the user’s task.Notations can be deﬁned using a general mixﬁx format,with precedences; variable-binding operators are easily speciﬁed.Isabelle manages a database ofMechanizing Set Theory 3theories and theorems;when asked to load a theory,it automatically loads any other theories that it depends upon.Although Isabelle supports proof checking,users will be more productive if they are provided with automatic tools.−The classical reasoner solves subgoals using methods borrowed from tab-leau provers.It employs user-supplied rules,typically about logical con-nectives or set operators,to break down assertions.−The simpliﬁer employs user-supplied conditional equalities to rewrite a subgoal.It can make use of contextual information and handles commuta-tive operators using a simple method borrowed from Boyer and Moore [3,page 104].We have found these tools indispensable.But there is much room for improve-ment;mechanizing a page of text can take a week or more.We discuss some reasons for this below.A lengthier introduction to Isabelle and Isabelle/ZF appears elsewhere [18].The Isabelle documentation has been published as a book [20].Figure 1sum-marizes the Isabelle/ZF notation for set theory.Note.Application of the function f to the argument x is formally written f ‘x .In informal mathematics we use the more familiar f (x )for clarity.But a set-theoretic function is just another set,and Isabelle allows the notation f (x )only if f is a meta-level function.This usually corresponds to subscripting in informal mathematics,for example f x .For the Isabelle/ZF development of functions,see Paulson [18,§7.5].3.The Cardinal Proofs:Motivation and DiscussionThe original reason for mechanizing the theory of cardinals was to general-ize Paulson’s treatment of recursive data structures in ZF.The original treat-ment [21]permitted only ﬁnite branching,as in n -ary trees.Countable branch-ing required deﬁning an uncountable ordinal.We are thus led to consider branching of any cardinality.3.1.I NFINITE B RANCHING T REESLet κstand for an inﬁnite cardinal and κ+for its successor cardinal.Branching by an arbitrarily large index set I requires proving the theorem|I |≤κ∀i ∈I αi <κ+( i ∈I αi )<κ+(1)4Lawrence C.Paulson and Krzysztof Grabczewskisyntaxdescription {a 1,...,a n }ﬁnite set <a ,b >ordered pair {x :A.P [x ]}Separation {y .x :A ,Q [x,y ]}Replacement {b [x ].x :A }functional Replacement INT x :A.B [x ]x ∈A .B [x ],general intersection UN x :A.B [x ]x ∈A .B [x ],general union A Int BA ∩B ,intersection A Un BA ∪B ,union A ->BA →B ,function space A *BA ×B ,Cartesian product PROD x :A.B [x ]Πx ∈A .B [x ],general product SUM x :A.B [x ]Σx ∈A .B [x ],general sum THE x.P [x ]ιx.P [x ],deﬁnite description lam x :A.b [x ]λx ∈A .b [x ],abstraction f ‘xf ‘x or f (x ),function application a :Aa ∈A ,membership A <=BA ⊆B ,subset relation ALL x :A.P [x ]∀x ∈A .P [x ],bounded quantiﬁerEX x :A.P [x ]∃x ∈A .P [x ],bounded quantiﬁerFigure 1.ASCII notation for ZFYou need not understand the details of how this is used in order to follow the paper.1Not many set theory texts cover such material well.Elementary texts [9,27]never get far enough,while advanced texts such as Kunen [12]race through it.But Kunen’s rapid treatment is nonetheless clear,and mentions all the essen-tial elements.The desired result (1)follows fairly easily from Kunen’s Lem-ma 10.21[12,page 30]:∀α<κ|X α|≤κ| α<κX α|≤κThis,in turn,relies on the Axiom of Choice and its consequence the Well-ordering Theorem,which are discussed at length below.It also relies on a fundamental result about multiplication of inﬁnite cardinals:κ⊗κ=κ.This is Theorem 10.12of Kunen.(In this paper,we refer only to his Chapter I.)The proof presents a challenging example of formalization,as we shall see.Mechanizing Set Theory5 We could prove A×A≈A,for all inﬁnite sets A,by appealing to AC in the form of Zorn’s Lemma;see Halmos[9,pages97–8].Thenκ⊗κ=κwould follow immediately.But we need to proveκ⊗κ=κwithout AC in order to use it in later proofs about equivalences of AC.In fact,the law A×A≈A is known to be equivalent to the Axiom of Choice.Paulson hoped to proveκ⊗κ=κdirectly,but could notﬁnd a suitable proof.He therefore decided to mechanize the whole of Kunen’s Chapter I, up to that theorem.We suggest this as a principle:theorems do not exist in isolation,but are part of a framework of supporting theorems.It is easier in the long run to build the entire framework,not just the parts thought to be relevant. The latter approach requires frequent,ad-hoc extensions to the framework.3.2.O VERVIEW OF K UNEN,C HAPTER IKunen’sﬁrst chapter is entitled,“Foundations of Set Theory.”Kunen remarks on page1that the chapter is merely a review for a reader who has already stud-ied basic set theory.This explains why the chapter is so succinct,as compared say with Halmos[9].Theﬁrst four sections are largely expository.Section5introduces a few axioms while§6describes the operations of Cartesian product,relations,func-tions,domain and range.Already,§6goes beyond the large Isabelle/ZF the-ory described in earlier papers[18,21].That theory emphasized computa-tional notions,such as recursive data structures,at the expense of traditional set theory.Now it was time to develop some of the missing material.Paul-son introduced some deﬁnitions about relations,orderings,well-orderings and order-isomorphisms,and proved theﬁrst two lemmas by well-founded induc-tion.The main theorem required a surprising amount of further work;see§4.3 below.Kunen’s§7covers ordinals.Much of this material had already been formal-ized in Isabelle/ZF[21,§3.2],but using a different deﬁnition of ordinal.A set A is transitive if A⊆P(A):every element of A is a subset of A.Kunen deﬁnes an ordinal to be a transitive set that is well-ordered by∈,while Isabelle/ZF deﬁnes an ordinal to be a transitive set of transitive sets.The two deﬁnitions are equivalent provided we assume,as we do,the Axiom of Foundation.Our work required formalizing some material from§7concerning order types and ordinal addition.We have also formalized ordinal multiplication.But we have ignored what Kunen calls A<ωbecause Isabelle/ZF provides list(A), the set ofﬁnite lists over A[21,§4.3]for the same purpose.Kunen’s§8and§13address the legitimacy of introducing new notations in axiomatic set theory.His discussion is more precise and comprehensive than Paulson’s defence of the notation of Isabelle/ZF[18,page361].Kunen’s§9concerns classes and recursion.The main theorems of this sec-tion,justifying transﬁnite induction and recursion over the class of ordinals,6Lawrence C.Paulson and Krzysztof Grabczewskiwere already in the Isabelle/ZF library[21,§3.2,§3.4].Kunen discusses here (and with some irony in§12)the difﬁculties of formalizing properties of class-es.Variables in ZF range over only sets;classes are essentially predicates,so a theorem about classes must be formalized as a theorem scheme.Many statements about classes are easily expressed in Isabelle/ZF.An ordi-nary class is a unary predicate,in Isabelle/ZF an object of type i⇒o,where i is the type of sets and o is the type of truth values.A class relation is a binary predicate and has the Isabelle type i⇒(i⇒o).A class function is tradition-ally represented by its graph,a single-valued class predicate[12,page25];it is more easily formalized in Isabelle as a meta-level function,an object of type i⇒i.See Paulson[18,§6]for an example involving the Replacement Axiom.Because Isabelle/ZF is built uponﬁrst-order logic,quantiﬁcation over vari-ables of types i⇒o,i⇒i,etc.,is forbidden.(And it should be;allowing such quantiﬁcation in uses of the Replacement Axiom would be illegitimate.)How-ever,schematic deﬁnitions and theorems may contain free variables of such types.Isabelle/ZF’s transﬁnite recursion operator[21,§3.4]satisﬁes an equa-tion similar to Kunen’s Theorem9.3,expressed in terms of class functions.Isabelle/ZF does not overload set operators such as∩,∪,domain and list to apply to classes.Overloading is possible in Isabelle,but is probably not worth the trouble in this case.And the class-oriented deﬁnitions might be cumber-some.Serious reasoning about classes might be easier in some other axiomatic framework,where classes formally exist.Kunen’s§10concerns cardinals.Some of these results presented great dif-ﬁculties and form one of the main subjects of this paper.But the Schr¨o der-Bernstein Theorem was already formalized in Isabelle/ZF[21,§2.6],and the ﬁrst few lemmas were straightforward.An embarrassment was proving that the natural numbers are cardinals.This boils down to showing that if there is a bijection between an m-element set and an n-element set then m=n.Proving this obvious fact is most tiresome.Rea-soning about bijections is complicated;a helpful simpliﬁcation(due to M.P. Fourman)is to reason about injections instead.Prove that if there is an injec-tion from an m-element set to an n-element set then m≤n.Applying this implication twice yields the desired result.Many intuitively obvious facts are hard to justify formally.This came up repeatedly in our proofs,and slowed our progress considerably.It is a funda-mental obstacle that will probably not yield to improved reasoning tools.Kunen proves(Theorem10.16)that for every ordinalαthere is a larger car-dinal,κ.Under AC this is an easy consequence of Cantor’s Theorem;without AC more work is required.Paulson slightly modiﬁed Kunen’s construction, lettingκbe the union of the order types of all well-orderings of subsets ofα, and found a pleasingly short machine proof.Our main concern,as mentioned above,is Kunen’s proof ofκ⊗κ=κ.We shall examine the machine proof in great detail.The other theorems of Kunen’sMechanizing Set Theory7§10concern such matters as cardinal exponentiation and coﬁnality.We havenot mechanized these,but the only obstacle to doing so is time.The rest of Kunen’s Chapter I is mainly discussion.4.Foundations of Cardinal ArithmeticLet us examine the cardinal proofs in detail.We begin by reviewing the neces-sary deﬁnitions and theorems.Then we look at the corresponding Isabelle/ZF theories leading up to the main result,κ⊗κ=κ.Throughout we shall con-centrate on unusual aspects of the formalization,since much of it is routine.4.1.W ELL-ORDERINGSA relation<is well-founded over a set A provided every non-empty subset of A has a<-minimal element.(This implies that<admits no inﬁnite decreas-ing chains···<x n<···<x2<x1within A.)If furthermore A,< is a linear ordering then we say that<well-orders A.A function f is an order-isomorphism(or just an isomorphism)between two ordered sets A,< and A ,< if f is a bijection between A and A that preserves the orderings in both directions:x<y if and only if f(x)< f(y) for all x,y∈A.Write A,< ∼= A ,< if there exists an order-isomorphism between A,< and A ,< .If A,< is an ordered set and x∈A then pred(A,x,<)def={y∈A|y< x}is called the(proper)initial segment determined by x.We also speak of A itself as an initial segment of A,< .Kunen develops the theory of relations in his§6and proves three funda-mental properties of well-orderings:−There can be no isomorphism between a well-ordered set and a proper initial segment of itself.A useful corollary is that if two initial segments are isomorphic to each other,then they are equal.−There can be at most one isomorphism between two well-ordered sets.This result sounds important,but we have never used it.2−Any two well-orderings are either isomorphic to each other,or else one of them is isomorphic to a proper initial segment of the other.Kunen’s proof of the last property consists of a single sentence:Let f={ v,w |v∈A∧w∈B∧ pred(A,v,<A) ∼= pred(B,w,<B) };note that f is an isomorphism from some initial segment of A onto some initial segment of B,and that these initial segments cannot both be proper. This gives the central idea concisely;Suppes[27,pages233–4]gives a much longer proof that is arguably less clear.However,the assertions Kunen makes are not trivial and Paulson needed two days and a half to mechanize the proof.4.2.O RDER T YPESThe ordinals may be viewed as representatives of the well-ordered sets.Every ordinal is well-ordered by the membership relation∈.What is more important, every well-ordered set is isomorphic to a unique ordinal,called its order type and written type(A,<).Kunen[12,page17]proves this by a direct construc-tion.But to mechanize the result in Isabelle/ZF,it is easier to use well-founded recursion[21,§3.4].If A,< is a well-ordering,deﬁne a function f on A by the recursionf(x)={f(y)|y<x}for all x∈A.Thentype(A,<)def={f(x)|x∈A}.It is straightforward to show that f is an isomorphism between A,< and type(A,<),which is indeed an ordinal.Our work has required proving many properties of order types,such as methods for calculating them in particular cases.Our source material contains few such proofs;we have spent much time re-discovering them.4.3.C OMBINING W ELL-ORDERINGSLet A+B def=({0}×A)∪({1}×B)stand for the disjoint sum of A and B, which is formalized in Isabelle/ZF[21,§4.1].Let A,<A and B,<B be well-ordered sets.The order types of certain well-orderings of A+B and A×B are of key importance.The sum A+B is well-ordered by a relation<that combines<A and<B, putting the elements of A before those of B.It satisﬁes the following rules:a <A aInl(a )<Inl(a)b <B bInr(b )<Inr(b)a∈A b∈BInl(a)<Inr(b)The product A×B is well-ordered by a relation<that combines<A and<B,lexicographically:a <A ab ,b∈B,b a∈A b <B bCardinal=OrderType+Fixedpt+Nat+Sum+constsLeast::(i=>o)=>i(binder"LEAST"10)eqpoll,lepoll,lesspoll::[i,i]=>o(infixl50)cardinal::i=>i("|_|")Finite,Card::i=>odefsLeast_def"Least(P)==THE i.Ord(i)&P(i)&(ALL j.j<i-->˜P(j))"eqpoll_def"A eqpoll B==EX f.f:bij(A,B)"lepoll_def"A lepoll B==EX f.f:inj(A,B)"lesspoll_def"A lesspoll B==A lepoll B&˜(A eqpoll B)"Finite_def"Finite(A)==EX n:nat.A eqpoll n"cardinal_def"|A|==LEAST i.i eqpoll A"Card_def"Card(i)==(i=|i|)"endFigure2.Isabelle/ZF Theory Deﬁning the Cardinal Numbers The well-orderings of A+B and A×B are traditionally used to deﬁne the ordinal sum and product.We do not require ordinal arithmetic until we come to the proofs from Rubin and Rubin.But we require the well-orderings themselves in order to proveκ⊗κ=κ.That proof requires yet another well-ordering construction:inverse image.If B,<B is an ordered set and f is a function from A to B then deﬁne <A byx<A y⇐⇒f(x)<B f(y).Clearly<A is well-founded if<B is.If f is injective and<B is a well-ordering then<A is also a well-ordering.If f is bijective then obviously f is an isomor-phism between the orders A,<A and B,<B ;it follows that their order types are equal.Sum,product and inverse image are useful for expressing well-orderings; this follows Paulson’s earlier work[16]within Constructive Type Theory.4.4.C ARDINAL N UMBERSFigure2presents the Isabelle/ZF deﬁnitions of cardinal numbers,following Kunen’s§10.The Isabelle theoryﬁle extends some Isabelle theories(Order-Type and others)with constants,which stand for operators or predicates.The constants are deﬁned essentially as follows:−The least ordinalαsuch that P(α)is deﬁned by a unique description[18, pages366–7]and may be written LEASTα.P(α).−Two sets A and B are equipollent if there exists a bijection between them.Write A≈B or,in Isabelle,A eqpoll B.−B dominates A if there exists an injection from A into B.Write A B or A lepoll B.−B strictly dominates A if A B and A≈B.Write A≺B orA lesspoll B.−A set isﬁnite if it is equipollent to a natural number.−The cardinality of A,written|A|,is the least ordinal equipollent to A.Without AC,no such ordinal has to exist;we might then regard|A|as undeﬁned.But everything is deﬁned in Isabelle/ZF.The operator THE returns0unless the description identiﬁes an object uniquely.Thus,an “undeﬁned”cardinality equals0;this conveniently ensures that|A|is always an ordinal.−A set i is a cardinal if i=|i|;write Card(i).Reasoning from these deﬁnitions is entirely straightforward except for the “obvious”facts aboutﬁnite cardinals mentioned above.4.5.C ARDINAL A RITHMETICLetκ,λ,µrange overﬁnite or inﬁnite cardinals.Cardinal sum and product are deﬁned in terms of disjoint sum and Cartesian product:κ⊕λdef=|κ+λ|κ⊗λdef=|κ×λ|These satisfy the familiar commutative,associative and distributive laws.The proofs are uninteresting but non-trivial,especially as we work without AC.We do so in order to use the results in proving various forms of AC to be equivalent (see below);but frequently this forces us to construct well-orderings explicitly.5.Provingκ⊗κ=κWe begin with an extended discussion of Kunen’s proof and then examine its formalization.5.1.K UNEN’S P ROOFKunen calls this result Theorem10.12.His proof is admirably concise.0,κκ,0α,00,α0,0Figure 3.Predecessors of α,β ,with β≤αTheorem.If κis an inﬁnite cardinal then κ⊗κ=κ.Proof.By transﬁnite induction on κ.Assume this holds for smaller cardi-nals.Then for α<κ,|α×α|=|α|⊗|α|<κ(applying Lemma 10.10when αis ﬁnite).3Deﬁne a well-ordering on κ×κby α,β γ,δ iff max(α,β)<max(γ,δ)∨[max(α,β)=max(γ,δ)∧α,β precedes γ,δ lexicographically ].Each α,β ∈κ×κhas no more than|succ(max(α,β))×succ(max(α,β))|<κpredecessors in ,so type(κ×κ, )≤κ,whence |κ×κ|≤κ.Since clearly |κ×κ|≥κ,|κ×κ|=κ.The key to the proof is the ordering ,whose structure may be likened to that of a square onion.Let αand βbe ordinals such that β≤α<κ.The predecessors of α,β include all pairs of the form α,β for β <β,and all pairs of the form α ,α for α <α;these pairs constitute the αth layer of the onion.The other predecessors of α,β are pairs of the form γ,δ such that γ,δ<α;these pairs constitute the inner layers of the onion.(See Figure 3.)The set of all -predecessors of α,β is a subset of succ(α)×succ(α), which gives an upper bound on its cardinality.Kunen expresses this upper bound in terms of max(α,β)to avoid having to assumeβ≤α.To simplify the formal proofs,Paulson used the more generous upper bound |succ(succ(max(α,β)))×succ(succ(max(α,β)))|.This is still a cardinal belowκ.As Kunen notes,there are two cases.Ifαorβis inﬁnite then succ(succ(max(α,β)))<κbecause max(α,β)<κand because inﬁnite cardinals are closed under successor;therefore,the inductive hypothesis realizes our claim.On the other hand,ifαandβare bothﬁnite, then so is succ(succ(max(α,β))),whileκis inﬁnite by assumption.To complete the proof,we must examine the second half of Kunen’s sen-tence:“so type(κ×κ, )≤κ,whence|κ×κ|≤κ.”Recall from§4.2that there is an isomorphismf:κ×κ→type(κ×κ, )such thatf(α,β)={f(γ,δ)| γ,δ α,β }.Thus,f(α,β)is an ordinal with the same cardinality as the set of predecessors of α,β .This implies f(α,β)<κfor allα,β<κ,and therefore type(κ×κ, )≤κ.Because f is a bijection betweenκ×κand type(κ×κ, ),we obtain|κ×κ|≤κ.The opposite inequality is trivial.5.2.M ECHANIZING THE P ROOFProvingκ⊗κ=κrequires formalizing the relation .Kunen’s deﬁnition looks complicated,but we can get the same effect using our well-ordering constructors(recall§4.3).Note that is an inverse image of the lexicographic well-ordering ofκ×κ×κ,under the function g:κ×κ→κ×κ×κdeﬁned byg(α,β)= max(α,β),α,β ;this function is trivially injective.Figure4presents part of the Isabelle theoryﬁle for cardinal arithmetic.It deﬁnes as the constant csquare rel.Here is a summary of the operators appearing in its deﬁnition:−rvimage(A,f,<)is the inverse image ordering on A derived from< by f.−lam<x,y>:K*K.<x Un y,x,y>is the function called g above.The pattern-matching in the abstraction expands internally to the con-stant split,which takes apart ordered pairs[18,page367].Finally Un denotes union;note that max(α,β)=α∪βfor ordinalsαandβ.。

专题05 阅读理解D篇(2024年新课标I卷) (专家评价+三年真题+满分策略+多维变式) 原卷版

《2024年高考英语新课标卷真题深度解析与考后提升》专题05阅读理解D篇（新课标I卷）原卷版(专家评价+全文翻译+三年真题+词汇变式+满分策略+话题变式)目录一、原题呈现P2二、答案解析P3三、专家评价P3四、全文翻译P3五、词汇变式P4(一)考纲词汇词形转换P4(二)考纲词汇识词知意P4(三)高频短语积少成多P5(四)阅读理解单句填空变式P5(五)长难句分析P6六、三年真题P7(一)2023年新课标I卷阅读理解D篇P7(二)2022年新课标I卷阅读理解D篇P8(三)2021年新课标I卷阅读理解D篇P9七、满分策略（阅读理解说明文）P10八、阅读理解变式P12 变式一：生物多样性研究、发现、进展6篇P12变式二：阅读理解D篇35题变式（科普研究建议类）6篇P20一原题呈现阅读理解D篇关键词: 说明文;人与社会;社会科学研究方法研究;生物多样性; 科学探究精神;科学素养In the race to document the species on Earth before they go extinct, researchers and citizen scientists have collected billions of records. Today, most records of biodiversity are often in the form of photos, videos, and other digital records. Though they are useful for detecting shifts in the number and variety of species in an area, a new Stanford study has found that this type of record is not perfect.“With the rise of technology it is easy for people to make observation s of different species with the aid of a mobile application,” said Barnabas Daru, who is lead author of the study and assistant professor of biology in the Stanford School of Humanities and Sciences. “These observations now outnumber the primary data that comes from physical specimens(标本), and since we are increasingly using observational data to investigate how species are responding to global change, I wanted to know: Are they usable?”Using a global dataset of 1.9 billion records of plants, insects, birds, and animals, Daru and his team tested how well these data represent actual global biodiversity patterns.“We were particularly interested in exploring the aspects of sampling that tend to bias (使有偏差) data, like the greater likelihood of a citizen scientist to take a picture of a flowering plant instead of the grass right next to it,” said Daru.Their study revealed that the large number of observation-only records did not lead to better global coverage. Moreover, these data are biased and favor certain regions, time periods, and species. This makes sense because the people who get observational biodiversity data on mobile devices are often citizen scientists recording their encounters with species in areas nearby. These data are also biased toward certain species with attractive or eye-catching features.What can we do with the imperfect datasets of biodiversity?“Quite a lot,” Daru explained. “Biodiversity apps can use our study results to inform users of oversampled areas and lead them to places – and even species – that are not w ell-sampled. To improve the quality of observational data, biodiversity apps can also encourage users to have an expert confirm the identification of their uploaded image.”32. What do we know about the records of species collected now?A. They are becoming outdated.B. They are mostly in electronic form.C. They are limited in number.D. They are used for public exhibition.33. What does Daru’s study focus on?A. Threatened species.B. Physical specimens.C. Observational data.D. Mobile applications.34. What has led to the biases according to the study?A. Mistakes in data analysis.B. Poor quality of uploaded pictures.C. Improper way of sampling.D. Unreliable data collection devices.35. What is Daru’s suggestion for biodiversity apps?A. Review data from certain areas.B. Hire experts to check the records.C. Confirm the identity of the users.D. Give guidance to citizen scientists.二答案解析三专家评价考查关键能力，促进思维品质发展2024年高考英语全国卷继续加强内容和形式创新，优化试题设问角度和方式，增强试题的开放性和灵活性，引导学生进行独立思考和判断，培养逻辑思维能力、批判思维能力和创新思维能力。

期权期货与其他衍生产品第九版课后习题与答案Chapter (14)

(2 3 9 3) (6 27)
The change during the next three years has the probability distribution
(3 3 16 3) (9 48)
The change during the six years is the sum of a variable with probability distribution (6 27) and a variable with probability distribution (9 48) . The probability distribution of the change is therefore (6 9 27 48)
CHAPTER 14 Wiener Processes and Itô’s Lemma
Practice Questions
Problem 14.1. What would it mean to assert that the temperature at a certain place follows a Markov process? Do you think that temperatures do, in fact, follow a Markov process? Imagine that you have to forecast the future temperature from a) the current temperature, b) the history of the temperature in the last week, and c) a knowledge of seasonal averages and seasonal trends. If temperature followed a Markov process, the history of the temperature in the last week would be irrelevant. To answer the second part of the question you might like to consider the following scenario for the first week in May: (i) Monday to Thursday are warm days; today, Friday, is a very cold day. (ii) Monday to Friday are all very cold days. What is your forecast for the weekend? If you are more pessimistic in the case of the second scenario, temperatures do not follow a Markov process. Problem 14.2. Can a trading rule based on the past history of a stock’s price ever produce returns that are consistently above average? Discuss. The first point to make is that any trading strategy can, just because of good luck, produce above average returns. The key question is whether a trading strategy consistently outperforms the market when adjustments are made for risk. It is certainly possible that a trading strategy could do this. However, when enough investors know about the strategy and trade on the basis of the strategy, the profit will disappear. As an illustration of this, consider a phenomenon known as the small firm effect. Portfolios of stocks in small firms appear to have outperformed portfolios of stocks in large firms when appropriate adjustments are made for risk. Research was published about this in the early 1980s and mutual funds were set up to take advantage of the phenomenon. There is some evidence that this has resulted in the phenomenon disappearing. Problem 14.3. A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does the company’s initial cash position have to be for the company to have a less than 5% chance of a negative cash position by the end of one year? Suppose that the company’s initial cash position is x . The probability distribution of the cash position at the end of one year is ( x 4 05 4 4) ( x 2016) where (m v) is a normal probability distribution with mean m and variance v . The probability of a negative cash position at the end of one year is

(完整word版)国际经济学第九版英文课后答案第7单元

CHAPTER 7ECONOMIC GROWTH AND INTERNATIONAL TRADEOUTLINE7.1 Introduction7.2 Growth of Factors of Production7.2a Labor Growth and Capital Accumulation Over Time7.2b The Rybczynski Theorem7.3 Technical Progress7.3a Neutral, Labor-Saving, and Capital-Saving Technical Progress7.3b Technical Progress and the Nation's Production FrontierCase Study 7-1: Changes in Relative Resource Endowments of Various Countries and Regions Case Study 7-2: Change in Capital-Labor Rations in Selected Countries7.4 Growth and Trade: The Small Country Case7.4a The Effects of Growth on Trade7.4b Illustration of Factor Growth, Trade, and Welfare7.4c Technical Progress, Trade, and WelfareCase Study 7-3: Growth of Output per Worker from Capital Deepening, TechnologicalChange, and Improvements in Efficiency7.5 Growth and Trade: The Large-Country Case7.5a Growth and the Nation's Terms of Trade and Welfare7.5b Immiserizing Growth7.5c Illustration of Beneficial Growth and TradeCase Study 7-4: Growth, Trade, and the Giants of the Future7.6 Growth, Change in Tastes, and Trade in Both Nations7.6a Growth and Trade in Both Nations7.6b Change in Tastes and Trade in Both NationsCase Study 7-5: Change in the Revealed Comparative Advantage of Various Countries or RegionsCase Study 7-6: Growth, Trade, and Welfare in the Leading Industrial NationsAppendix: A7.1 Formal Proof of Rybczynski TheoremA7.2 Growth with Factor ImmobilityA7.3 Graphical Analysis of Hicksian Technical ProgressKey TermsComparative statics Antitrade production and consumptionDynamic analysis Neutral production and consumptionBalanced growth Normal goodsRybczynski theorem Inferior goodsLabor-saving technical progress Terms-of-trade effectCapital-saving technical progress Wealth effectProtrade production and consumption Immiserizing growthLecture Guide1.This is not a core chapter and it is one of the most challenging chapters in international tradetheory. It is included for more advanced students and for completeness.2.If I were to cover this chapter, I would present two sections in each of three lectures.Time permitting, I would, otherwise cover Sections 1 and 2, paying special attention to the Rybczynski theorem.Answer to Problems1. a) See Figure 1.b) See Figure 2c) See Figure 3.2. See Figure 4.3. a) See Figure 5.b) See Figure 6.c) See Figure 7.4. Compare Figure 5 to Figure 1.Compare Figure 6 to Figure 3. Note that the two production frontiers have the same vertical or Y intercept in Figure 6 but a different vertical or Y intercept in Figure 3.Compare Figure 7 to Figure 2. Note that the two production frontiers have the samehorizontal or X intercept in Figure 7 but a different horizontal or X intercept in Figure 2.5. See Figure 8 on page 66.6. See Figure 9.7. See Figure 10.8. See Figure 11.9. See Figure 12.10. See Figure 13 on page 67.11. See Figure 14.12. See Figure 15.13.The United States has become the most competitive economy in the world since the early1990’s while the data in Table 7.3 refers to the 1965-1990 period.14.The data in Table 7.4 seem to indicate that China had a comparative advantage incapital-intensive commodities and a comparative disadvantage in unskilled-labor intensive commodities in 1973. This was very likely due to the many trade restrictions and subsidies, which distorted the comparative advantage of China.Its true comparative advantage became evident by 1993 after China had started to liberalize its economy.App. 1a. See Figure 16.1b. For production and consumption to actually occur at the newequilibrium point after the doubling of K in Nation 2, we mustassume either than commodity X is inferior or that Nation 2 is toosmall to affect the relative commodity prices at which it trades.1c. Px/Py must rise (i.e., Py/Px must fall) as a result of growth only.Px/Py will fall even more with trade.1. If the supply of capital increases in Nation 1 in the production of commodity Yonly, the VMPLy curve shifts up, and w rises in both industries. Some labor shifts to the production of Y, the output of Y rises and the output of X falls, r falls, and Px/Py is likely to rise.2. Capital investments tend to increase real wages because they raise the K/L ratioand the productivity of labor. Technical progress tends to increase K/L and realwages if it is L-saving and to reduce K/L and real wages if it is K-saving. Multiple-Choice Questions1. Dynamic factors in trade theory refer to changes in:a. factor endowmentsb. technologyc. tastes*d. all of the above2. Doubling the amount of L and K under constant returns to scale:a. doubles the output of the L-intensive commodityb. doubles the output of the K-intensive commodityc. leaves the shape of the production frontier unchanged*d. all of the above.3. Doubling only the amount of L available under constant returns to scale:a. less than doubles the output of the L-intensive commodity*b. more than doubles the output of the L-intensive commodityc. doubles the output of the K-intensive commodityd. leaves the output of the K-intensive commodity unchanged4. The Rybczynski theorem postulates that doubling L at constant relative commodity prices:a. doubles the output of the L-intensive commodity*b. reduces the output of the K-intensive commodityc. increases the output of both commoditiesd. any of the above5. Doubling L is likely to:a. increases the relative price of the L-intensive commodityb. reduces the relative price of the K-intensive commodity*c. reduces the relative price of the L-intensive commodityd. any of the above6.Technical progress that increases the productivity of L proportionately more than the productivity of K is called:*a. capital savingb. labor savingc. neutrald. any of the above7. A 50 percent productivity increase in the production of commodity Y:a. increases the output of commodity Y by 50 percentb. does not affect the output of Xc. shifts the production frontier in the Y direction only*d. any of the above8. Doubling L with trade in a small L-abundant nation:*a. reduces the nation's social welfareb. reduces the nation's terms of tradec. reduces the volume of traded. all of the above9. Doubling L with trade in a large L-abundant nation:a. reduces the nation's social welfareb. reduces the nation's terms of tradec. reduces the volume of trade*d. all of the above10.If, at unchanged terms of trade, a nation wants to trade more after growth, then the nation's terms of trade can be expected to:*a. deteriorateb. improvec. remain unchangedd. any of the above11. A proportionately greater increase in the nation's supply of labor than of capital is likely to result in a deterioration in the nation's terms of trade if the nation exports:a. the K-intensive commodity*b. the L-intensive commodityc. either commodityd. both commodities12. Technical progress in the nation's export commodity:*a. may reduce the nation's welfareb. will reduce the nation's welfarec. will increase the nation's welfared. leaves the nation's welfare unchanged13. Doubling K with trade in a large L-abundant nation:a. increases the nation's welfareb. improves the nation's terms of tradec. reduces the volume of trade*d. all of the above14. An increase in tastes for the import commodity in both nations:a. reduces the volume of trade*b. increases the volume of tradec. leaves the volume of trade unchangedd. any of the above15. An increase in tastes of the import commodity of Nation A and export in B:*a. will reduce the terms of trade of Nation Ab. will increase the terms of trade of Nation Ac. will reduce the terms of trade of Nation Bd. any of the aboveADDITIONAL ESSAYS AND PROBLEMS FOR PART ONE1.Assume that both the United States and Germany produce beef and computer chipswith the following costs:United States Germany(dollars) (marks)Unit cost of beef (B) 2 8Unit cost of computer chips (C) 1 2a) What is the opportunity cost of beef (B) and computer chips (C) in each country?b)In which commodity does the United States have a comparative cost advantage?What about Germany?c)What is the range for mutually beneficial trade between the United States andGermany for each computer chip traded?d)How much would the United States and Germany gain if 1 unit of beef isexchanged for 3 chips?Ans. a) In the United States:the opportunity cost of one unit of beef is 2 chips;the opportunity cost of one chip is 1/2 unit of beef.In Germany:the opportunity cost of one unit of beef is 4 chips;the opportunity cost of one chip is 1/4 unit of beef.b) The United States has a comparative cost advantage in beef with respect toGermany, while Germany has a comparative cost advantage in computer chips.c)The range for mutually beneficial trade between the United States and Germanyfor each unit of beef that the United States exports is2C < 1B < 4Cd) Both the United States and Germany would gain 1 chip for each unit of beeftraded.2.Given: (1) two nations (1 and 2) which have the same technology but differentfactor endowments and tastes, (2) two commodities (X and Y) produced under increasing costs conditions, and (3) no transportation costs, tariffs, or other obstructions to trade. Prove geometrically that mutually advantageous trade between the two nations is possible.Note: Your answer should show the autarky (no-trade) and free-trade points of production and consumption for each nation, the gains from trade of each nation,and express the equilibrium condition that should prevail when trade stops expanding.)Ans.: See Figure 1 on page 74.Nations 1 and 2 have different production possibilities curves and different community indifference maps. With these, they will usually end up with different relative commodity prices in autarky, thus making mutually beneficial trade possible.In the figure, Nation 1 produces and consumes at point A and Px/Py=P A in autarky, while Nation 2 produces and consumes at point A' and Px/Py=P A'. Since P A < P A',Nation 1 has a comparative advantage in X and Nation 2 in Y. Specialization inproduction proceeds until point B in Nation 1 and point B' in Nation 2, at which P B=P B' and the quantity supplied for export of each commodity exactly equals the quantity demanded for import. Thus, Nation 1 starts at point A in production and consumption in autarky, moves to point B in production, and by exchanging BC of X for CE of Y reaches point E in consumption. E > A since it involves more of both X and Y and lies on a higher community indifference curve. Nation 2 starts at A' in production and consumption in autarky, moves to point B' in production, and by exchanging B'C' of Y for C'E' of X reaches point E'in consumption (which exceeds A').At Px/Py=P B=P B', Nation 1 wants to export BC of X for CE of Y, while Nation 2 wants to export B'C' (=CE) of Y for C'E' (=BC) of X. Thus, P B=P B'is the equilibrium relative commodity price because it clears both (the X and Y) markets.3.Draw a figure showing: (1) in Panel A a nation's demand and supply curve for Atraded commodity and the nation's excess supply of the commodity, (2) in Panel C the trade partner's demand and supply curve for the same traded commodity and its excess demand for the commodity, and (3) in Panel B the supply and demand for the quantity traded of the commodity, its equilibrium price, and why a price above or below the equilibrium price will not persist. At any other price, QD QS, and P will change to P2.Ans. See Figure 2 on page 74.The equilibrium relative commodity price for commodity X (the traded commodityexported by Nation 1 and imported by Nation 2) is P2 and the equilibrium quantityof commodity X traded is Q2.4.a) Identify the conditions that may give rise to trade between two nations.b) What are some of the assumptions on which the Heckscher-Ohlin theory isbased?c) What does this theory say about the pattern of trade and effect of trade on factorprices?Ans. a) Trade can be based on a difference in factor endowments, technology, or tastes between two nations. A difference either in factor endowments or technology results in a different production possibilities frontier for each nation, which, unless neutralized by a difference in tastes, leads to a difference in relative commodity price and mutually beneficial trade. If two nations face increasing costs and have identical production possibilities frontiers but different tastes, there will also be a difference in relative commodity prices and the basis for mutually beneficial trade between the two nations. The difference in relative commodity prices is then translated into a difference in absolute commodity prices between the two nations, which is the immediate cause of trade.b) The Heckscher-Ohlin theory (sometimes referred to as the modern theory – asopposed to the classical theory - of international trade) assumes that nations have the same tastes, use the same technology, face constant returns to scale (i.e., a given percentage increase in all inputs increases output by the same percentage) but differ widely in factor endowments. It also says that in the face of identical tastes or demand conditions, this difference in factor endowments will result in a difference in relative factor prices between nations, which in turn leads to a difference in relative commodity prices and trade. Thus, in the Heckscher-Ohlin theory, the international difference in supply conditions alone determines the pattern of trade. To be noted is that the two nations need not be identical in other respects in order for international trade to be based primarily on the difference in their factor endowments.c) The Heckscher-Ohlin theorem postulates that each nation will export thecommodity intensive in its relatively abundant and cheap factor and import the commodity intensive in its relatively scarce and expensive factor. As an important corollary, it adds that under highly restrictive assumptions, trade will completely eliminate the pretrade relative and absolute differences in the price of homogeneous factors among nations. Under less restrictive and more usual conditions, however, trade will reduce, but not eliminate, the pretrade differences in relative and absolute factor prices among nations. In any event, the Heckscher-Ohlin theory does say something very useful on how trade affects factor prices and the distribution of income in each nation. Classical economists were practically silent on this point.5. consumers demand more of commodity X (the L-intensive commodity) and less ofcommodity Y (the K- intensive commodity). Suppose that Nation 1 is India, commodity X is textiles, and commodity Y is food. Starting from the no-trade equilibrium position and using the Heckscher-Ohlin model, trace the effect of this change in tastes on India's(a) relative commodity prices and demand for food and textiles,(b) production of both commodities and factor prices, and(c) comparative advantage and volume of trade.(d) Do you expect international trade to lead to the complete equalization ofrelative commodity and factor prices between India and the United States?Why?Ans. a. The change in tastes can be visualized by a shift toward the textile axis in India's indifference map in such a way that an indifference curve is tangentto the steeper segment of India's production frontier (because of increasingopportunity costs) after the increase in demand for textiles. This will causethe pretrade relative commodity price of textiles to rise in India.b. The increase in the relative price of textiles will lead domesticproducers in India to shift labor and capital from the production of food tothe production of textiles. Since textiles are L-intensive in relation to food,the demand for labor and therefore the wage rate will rise in India. At thesame time, as the demand for food falls, the demand for and thus the priceof capital will fall. With labor becoming relative more expensive,producers in India will substitute capital for labor in the production of bothtextiles and food.Even with the rise in relative wages and in the relative price of textiles,India still remains the L-abundant and low-wage nation with respect to anation such as the United States. However, the pretrade difference in therelative price of textiles between India and the United States is nowsomewhat smaller than before the change in tastes in India. As a result thevolume of trade required to equalize relative commodity prices and hencefactor prices is smaller than before. That is, India need now export asmaller quantity of textiles and import less food than before for therelative price of textiles in India and the United States to be equalized.Similarly, the gap between real wages and between India and the UnitedStates is now smaller and can be more quickly and easily closed (i.e., witha smaller volume of trade).c. Since many of the assumptions required for the complete equalization ofrelative commodity and factor prices do not hold in the real world, greatdifferences can be expected and do in fact remain between real wages inIndia and the United States. Nevertheless, trade would tend to reduce thesedifferences, and the H-O model does identify the forces that must beconsidered to analyze the effect of trade on the differences in the relative andabsolute commodity and factor prices between India and the United States.5.(a) Explain why the Heckscher-Ohlin trade model needs to be extended.(b) Indicate in what important ways the Heckscher-Ohlin trade model can beextended.(c) Explain what is meant by differentiated products and intra-industry trade.Ans. (a) The Heckscher-Ohlin trade model needs to be extended because, while generally correct, it fails to explain a significant portion of international trade, particularly the trade in manufactured products among industrial nations.(b)The international trade left unexplained by the basic Heckscher-Ohlin trade modecan be explained by(1) economies of scale,(2) intra-industry trade, and(3) trade based on imitation gaps and product differentiation.(c)Differentiated products refer to similar, but not identical, products (such as cars,typewriters, cigarettes, soaps, and so on) produced by the same industry or broad product group. Intra-industry trade refers to the international trade in differentiated products.。

Chapter 7 seven

Applications of the principal-agent model
Adverse Selection
• Players: The principal and the agent • The Order of Play: • (1) Nature chooses the agent's ability a, unobserved by the principal, according to distribution F(a). • (2) The principal offers the agent one or more wage contracts w1(q); w2(q); …
• (3) The agent accepts one contract or rejects them all. • (4) Nature chooses a value for the state of the world, according to distribution G(θ). Output is then q q(a; θ ).
• This covers a broad range of models, We will divide games of asymmetric information into five categories.
• 1. Moral hazard with hidden actions • 2. Adverse selection • 3 Mechanism Design in Adverse Selection and in Moral Hazard with Hidden Information • 4. Signalling • 5. Screening

Level 3 Chapter 7

New Era Interactive English Level 3
Chapter 7 Afflenuza
Li Xin Faculty of College English Teaching
What is affluenza?
Affluenza n.
A serious social disease - caused by consumerism, commercialism and rampant materialism - that is having a devastating impact on our families, communities, and the environment. We have more stuff, but less time, and our quality of life seems to be deteriorating.
Since adv. (L10)
• prep.自…以来； since the war • conj.自…以后； What have you been doing since I last saw you? 既然；因为 Since you ask, I will tell you. • adv. 自从以后 adv. 自从…以后 They left town and haven't been here since.
Detailed Study of the Text
Comparing……, we……(L2)
• Distinguish the present participle and past participle of “compare” L2—“Comparing Comparing…,” ; Comparing L82—“ Compared to with…” to/

计量经济学重点知识点考试必备

1.Econometrics（计量经济学）：the social science in which the tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena.the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.2.Econometric analysis proceeds along the following lines计量经济学分析步骤1)Creating a statement of theory or hypothesis.建立一个理论假说2)Collecting data.收集数据3)Specifying the mathematical model of theory.设定数学模型4)Specifying the statistical, or econometric, model of theory.设立统计或经济计量模型5)Estimating the parameters of the chosen econometric model.估计经济计量模型参数6)Checking for model adequacy : Model specification testing.核查模型的适用性：模型设定检验7)Testing the hypothesis derived from the model.检验自模型的假设8)Using the model for prediction or forecasting.利用模型进行预测Step2：收集数据Three types of data三类可用于分析的数据1)Time series(时间序列数据):Collected over a period of time, are collected at regular intervals.按时间跨度收集得到2)Cross-sectional截面数据:Collected over a period of time, are collected at regular intervals.按时间跨度收集得到3)Pooled data合并数据（上两种的结合）Step3：设定数学模型1.plot scatter diagram or scattergram2.write the mathematical modelStep4：设立统计或经济计量模型CLFPR is dependent variable应变量CUNR is independent or explanatory variable独立或解释变量（自变量）We give a catchall variable U to stand for all these neglected factorsIn linear regression analysis our primary objective is to explain the behavior of the dependent variable in relation to the behavior of one or more other variables, allowing for the data that the relationship between them is inexact.线性回归分析的主要目标就是解释一个变量（应变量）与其他一个或多个变量（自变量）只见的行为关系，当然这种关系并非完全正确Step5：估计经济计量模型参数In short, the estimated regression line gives the relationship between average CLFPR and CUNR 简言之，估计的回归直线给出了平均应变量和自变量之间的关系That is, on average, how the dependent variable responds to a unit change in the independent variable.单位因变量的变化引起的自变量平均变化量的多少。

理想国英文原文第七章

理想国英文原文第七章Title: Analysis of Chapter 7 of "The Republic" by PlatoIntroduction:"The Republic" is a philosophical work written by Plato, a renowned ancient Greek philosopher. Chapter 7 of this influential book explores various aspects of the ideal state or "Ideal Republic." This article aims to provide an accurate and comprehensive analysis of Chapter 7, highlighting its main points and their significance.Body:1. The concept of the philosopher-king:1.1 The philosopher-king's role as the ruler: Plato argues that the ideal state should be governed by philosopher-kings who possess knowledge and wisdom.1.2 The philosopher-king's education: Plato emphasizes the importance of a rigorous education for the philosopher-king, including training in mathematics, philosophy, and dialectics.1.3 The philosopher-king's detachment from material possessions: According to Plato, philosopher-kings should be free from material desires and focused solely on pursuing wisdom and the well-being of the state.2. The three classes of society:2.1 The Guardians: Plato proposes a division of society into three classes, with the Guardians being the ruling class responsible for protecting the state and maintaining order.2.2 The Auxiliaries: This class comprises warriors and soldiers who assist the Guardians in defending the state from external threats.2.3 The Producers: The third class consists of farmers, artisans, and craftsmen who provide for the material needs of the state.3. The abolition of private property and family:3.1 Common property: Plato argues that in the ideal state, property should be held collectively, with no individual ownership. This ensures equality and prevents corruption.3.2 The abolition of the family: Plato suggests that the traditional family structure should be replaced by a communal system, where children are raised collectively to eliminate personal attachments and promote unity.4. The role of women in the ideal state:4.1 Equality of women: Plato advocates for the equality of men and women in the ideal state, emphasizing that women should receive the same education and have the same opportunities as men.4.2 Women as Guardians: Plato proposes that women should also serve as Guardians, participating in the defense and governance of the state alongside men.4.3 The selective breeding of Guardians: Plato suggests a system of selective breeding to ensure that only the most capable individuals become Guardians, regardlessof gender.5. The importance of justice in the ideal state:5.1 Justice as harmony: Plato argues that justice is the fundamental virtue of the ideal state, where each individual performs their designated role harmoniously, leading to a balanced and prosperous society.5.2 The role of education in fostering justice: Plato emphasizes that education plays a crucial role in cultivating virtuous individuals who understand and uphold justice in the state.5.3 The philosopher-king as the embodiment of justice: Plato asserts that the philosopher-king, possessing wisdom and knowledge, represents the epitome of justice and is responsible for ensuring its prevalence in the state.Conclusion:Chapter 7 of "The Republic" by Plato delves into the concept of the ideal state, discussing the philosopher-king, the three classes of society, the abolition of private property and family, the role of women, and the significance of justice. Plato's ideas in this chapter have had a profound influence on political philosophy and continue to provoke thoughtful discussions on the nature of an ideal society.。

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The $429.66 represents an average amount paid by each policy-holder to cover insurance company expenses (and produce a profit for the insurance companies).
random variables, then: σ2x+y = σ2x + σ2y σ2x-y = σ2x + σ2y
Rules for Variance
If X and Y have a correlation ρ, then σ2x+y = σ2x + σ2y + 2ρσxσy σ2x-y = σ2x + σ2y - 2ρσxσy
(.99996)
(.00002) (.00002)
So, the average long-run payoff is
($0*.99996)+($100*.00002)+($22,000*.0
0002) = $0.442
Does this make sense?
It does if you consider the long run. Also you could think about it from the fair’s point of view. From their side you can see that they pay out an average of 44 cents per raffle ticket sold.
μx+y = μx + μy
Rules for Variances:
Rule 1. If X is a random variable and a and b are fixed numbers, then:
σ2a+bx = b2σ2x Rule 2. If X and Y are independent
Law of Large numbers:
Draw independent observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean x-bar of the observed values eventually approaches the mean μ of the population.
Write the distribution like this:
Value of X: x1
X2
Probability: p1
p2
……….. xk ……….. pk
To find μ, multiply each possible value, x, by its probability, then sum the products.
P(A) = 14,100,0000.116
121,100,000
Which means P(A’) 0.884. You can calculate the payoffs for A and A’ as follows. If the person had an accident, he or she paid $638 and received $1796, so the person received $1796 - $638 = $1158.
Rules for Means:
Rule 1. If X is a random variable and a and b are fixed numbers, then:
μa+bx = a + bμx
Rule 2. If X and Y are random variables, then:
X0123
P(X) 0.05 0.25 0.40 0.30
Extra Examples: #1
Suppose you buy a raffle ticket at the fair. The ticket costs $1 and the prize is a new truck. The new truck is valued at approximately $22,000. Second prize is a $100 gift certificate to Southpoint Mall. The fair sells 50,000 raffle tickets. What are your average winnings per ticket?
Further, you could figure their cost to be .442 * $1(50000 tickets sold) = $22,100
Further, you could figure their profit to be (1-.442) * $1(50000 tickets sold) = $27,900
Chapter 7.2
Warm Up Do #’s 36,39,42 Homework Due 12/16th and 12/19th # 37,38, 44, 45, 46, 55, 56, 57, 61, 68 POP Mini Quiz Fri/Mon Review Worksheet over Break Chapter 7 Quiz on Jan 4th and Jan 5th
Solution: Let event A represent having an automobile accident. Then event A’ represents not having an automobile accident. In 1993, the probability of having an automobile accident was:
Warm Up 12/16 and 12/17
At a large university students have either a final exam or a final paper at the end of a course. The table below lists the distribution of the number of final exams that students at the university will take, and their associated probabilities. What are the mean and standard deviation of this distribution (remember to show ALL work)?
Chapter 7.2 Means and Variances of Random Variables
Notations
The mean, x-bar, of a set of observations is their ordinary average.
The mean μ is the average of the possible values of the random variable X. Also called the expected value of X.
Chapter 7.2
Warm Up Do #’s 36,39,42 Homework Due 12/16th and 12/17th # 37,38, 44, 45, 46, 55, 56, 57, 61, 68 POP Mini Quiz Fri/Mon Review Worksheet over Break Chapter 7 Quiz on Jan 4th and Jan 5th
Value of X: x1
X2
……….. xk
Probability: p1
p2
……….. pk
Variance is found by:
σ2x =(x1 – μ)2●p1 + (x2 – μ)2●p2 +…+ (xk - μ)2●pk
To find the standard deviation, you must square root the variaa value of x.
X x 1 p 1 x 2 p 2 x k p k x ip i
Finding the Variance of a Discrete Random Variable.
Write the distribution like this:
Since we’ll be finding means for different random variables, we’ll use μx for the mean of the random variable X.
Finding the Mean of a Discrete Random Variable.
Make a probability distribution table to represent, X, the payoff.
The probability distribution of X is :
PayoffX:
$0
$100
$22,000
Probability: 49998/50000 1/50000 1/50000
Another Example #2
Finding an Expected Value.
In 1993, there were 121,100,000 cars in use in the United States. That year, there were 14,100,000 automobile accidents. The average premium paid in 1993 for automobile collision insurance was $638, and the average automobile collision claim paid by insurance companies was $1796. What was the expected value of the insurance coverage for a person who paid $638 for collision insurance in 1993?