The relation between iconicity and subjectification in Portuguese complementation-Complements

Journal of Visual Languages and Computing(2002)13,601^622doi:10.1006/S1045-926X(02)00026-5available online at onThe Role of Visual Perception in DataVisualizationMehdi DastaniInstitute of Information and Computing Sciences,Universiteit Utrecht,P adualaan14,De Uithof3584CH Utrecht,The Netherlands,e-mail:mehdi@cs.uu.nlReceived12July2001;accepted27March2002This paper presents a perceptually motivated formal framework for e¡ective visualiza-tion of relational data.In this framework,the intended structure of data and the per-ceptual structure of visualizations are formally and uniformly de¢ned in terms of relations that are induced on data and visual elements by data and visual attributes, respectively.Visual attributes are analyzed and classi¢ed from the perceptual point of view and in terms of perceptual relations that they induce on visual elements.The presented framework satis¢es a necessary condition for e¡ective data visualizations.This condition is formulated in terms of a structure preserving map between the intended structure of data and the perceptual structure of visualization.r2002 Published by Elsevier Science Ltd.1.IntroductionT heories of e FF ective data visualization aim to analyze and explain the relation between data and their e¡ective visualizations[1^6].This relation is essentially a denota-tion relation in the sense that visual elements and relations denote data elements and relations,respectively.For example,the number of cars produced by di¡erent factories can be visualized by a bar-chart.Each bar represents a car factory and its length represents the number of produced cars by that factory.The perceivable length relations between bars denote relations between the numbers of cars produced by di¡erent factories. The e¡ectiveness of a visualization depends on many conditions such as conventions, subjective user preferences,cultural settings,and the structural correspondence with its denoting data[4,5,7^9].For example,the e¡ectiveness of a temperature map in which temperature values are visualized by color hue values increases when conventional color hue values are used for high and low temperatures,i.e.red for high temperatures and blue for low temperatures.Although the satisfaction of some conditions enhance the e¡ec-tiveness of visualizations,the satisfaction of other conditions are necessary for e¡ective data visualization.In general,we distinguish conditions for e¡ective data visualization into necessary and su⁄cient conditions.This distinction is closely related to the well-known distinction between expressiveness and e¡ectiveness[2,3].There are at least two reasons for not using the expressiveness/e¡ectiveness distinction.First,the expressiveness criterion is de¢ned 1045-926X/02/$-see front matter r2002Published by Elsevier Science Ltd.with respect to a graphical language which we want to abstract from.The aim of this paper is to propose a general framework for e¡ective data visualization,which is inde-pendent of graphical languages and thus not limited to their expressions.Second,the expressiveness criterion is claimed to be satis¢ed if and only if data are represented in the visual structure and the e¡ectiveness criterion is claimed to be satis¢ed if the mechan-ism of the humanvisual system is taken into account[2].W e believe that the expressiveness criterion makes only sense if visual structures are perceivable by human viewers.This implies that both expressiveness and e¡ectiveness criteria depend on the human visual system and that the expressiveness/e¡ectiveness distinction is not a distinction between perceptually depending and perceptually not depending conditions.In the context of e¡ective data visualization,the structural correspondence condition is a necessary condition without which the e¡ectiveness of visualizations cannot be guar-anteed.This condition should be de¢ned between the structure of data and the percep-tual structure of its visualization.The emphasize on perceptual structure in this formulation is based on the assumption that the e¡ectiveness of visualizations depend on the e¡ective use of the capabilities of the human visual system to perceive visual struc-tures.W e argue that a theory of e¡ective data visualization should therefore be based on theories of human visual system that describe human capabilities of perceiving visual relations.For example,the human capability to perceive relation between color hue va-lues is limited to the identity relation.For this reason,a visualization in which the age of individuals is visualized by color hue values is not e¡ective if the intended data relations are quantitative relations between ages.In this paper,we focus on the structural correspondence condition for e¡ective data visualization and propose a formal framework for e¡ective data visualization,which sa-tis¢es the structural correspondence condition.This paper is organized as follows.In Section2,we propose a process model for e¡ective data visualization and identify the steps through which the structural correspondence between data and visualizations should be established.For this model,we discuss the class of input data structures,the projection step from data structures to perceptual structures,the layout of perceptual structures,and¢nally the necessary condition for e¡ective data visualization.In Section 3,some general types of data attributes that are important for data visualization are intro-duced and a classi¢cation of attribute-based data structures is proposed.Similarly,various visual attributes are analyzed from the perceptual point of view and a classi¢cation of perceptual structures is proposed.In Section4,we build on formal de¢nitions of at-tributes and specify data and perceptual structures in a formal way.The structural correspondence condition for e¡ective data visualization is then de¢ned as a structure-preserving relation between data and perceptual structures.Finally,in Section5,we conclude the paper and discuss future research directions.2.A Process Model for E¡ective DataVisualizationThe process of data visualization is considered here as the inverse of the interpretation process,i.e.the visualization process generates for an input data a visualization the interpretation of which results the original data.Since the e¡ectiveness of a visualization can be measured in terms of the easiness and directness of acquiring its intended interpretation and because the interpretation of a visualization depends on human visualperception,an e¡ective visualization should strongly bene¢t from the capabilities of the humanvisual system.W e formulate,therefore,the e¡ectiveness of visualization as follows:a visualizationpresentstheinputdata e¡ectively iftheintended structureofthedata and theperceptualstructure ofthevisualization coincide .Based on this view,a process model for e¡ective data visualization [10]is schematically illustrated in Figure 1.This process starts with an input data which is assumed to consist of data elements.The ¢rst step in data visualization is to determine the structure of data,i.e.how data elements are related to each other.This is not a trivial decision process since for example a set of integers may be related to each other nominally (i.e.integers are used as identi¢ers),ord-inally (i.e.integers are used as ordinals),or quantitatively (integers are used as quantities).This step in the process of data visualization transforms the input data to what is called structured data.The second step in data visualization is to determine visual elements that represent data elements in such a way that the perceptual structure of the decided visual elements represents the structure of the data.W e assume that the result of this step is not necessarily a drawable visualization,but an abstract perceptual structure.Therefore,the third and the last step in data visualization is the layout process which transforms the abstract perceptual structure to a drawable visualization.On the other hand,the interpretation process starts with a visualization,possibly to-gether with a legend.Since the input of the interpretation process is a visualization and because visualizations must be perceived before they can be understood as denoting data,the ¢rst interpretation step is perception.The process of perception determines percei-vable relations among visual elements on the basis of their visual attribute values.As visual attribute values are assigned to visual elements by the layout process,perception is considered as the inverse of the layout process.Subsequently ,visual elements are mapped to data elements.This mapping is possibly supported by a legend that deter-mines the correspondence between visual and data attributes and their values.In the case of e¡ective data visualization,the relations between data elements are structurally iden-tical to the perceivable relations between their representing visual elements.Finally ,given the structured data,the table of attribute values can be trivially generated.In the rest of this section,we elaborate on di¡erent steps of this process model.2.1.Structuringthe Input DataIn this paper,the input data for the visualization process is assumed to be nested rela-tional data.This class of data can be represented as m Ân data tables consisting ofFigure 1.A process model for e¡ective data visualizationm columns and n rows.A column A i ði ¼1;y ;m Þconsists of values of one data attri-bute and a row B j ðj ¼1;y ;n Þconsists of m values each belongs to one of attributes A 1;y ;A m :A row of a table is called a data entry a .The values of attributes A 1;y ;A m may themselves be rows of another k Âl table that consists of A 01;y ;A 0k columns and B 01;y ;B 0l rows.In such a case,attributes A 1;y ;A m are called aggregated attributes.An attribute value,which is a single value,is called an atomic value.For example,consider T able 1,which describes the cooperation degree between pairs of car companies.In this table,company1and company2are aggregated attributes since their values are rows fromT able 2.This table describes the number of sold cars having a certain type and being produced in a certain country .In this data table,car-type ,sold-numbers ,and country are attributes that have atomicvalues.These two data tables c an be c ombined into one single data table by means of aggregated attributes.T able 3is obtained by combining T ables 1and 2.According to the proposed data visualization process model,the ¢rst step is to deter-mine the intended structure of the input data.In general,we assume that an input data can be structured in many di¡erent ways and that the intended structure of the input data is determined by human users.W e also assume that human users have access to di¡erent transformations that provide di¡erent structures of the input data [2].T wo types of data structures are distinguished.The structures of the ¢rst type are con-stituted by relations that are intended to exist among values of individual data attributes.These relations characterize the type of the corresponding attributes.For instance,values of the sold-number attribute are from the domain of integers and they may be intended to be related to each other by some arithmetical relations.These arithmetical relations charac-terize the type of the sold-number attribute.A data attribute together with its type is called a typed data attribute and a table that is de¢ned on typed data attributes is called a typed data table.In a typed data table,relations that are de¢ned on values of individual attri-butes are induced on data entries in the sense that two data entries are related in the same way as their attribute values are related.Relations on data entries may be induced by dif-ferent data attributes.For instance,data entries inT able 2may be considered as equivalentT able 1.A data table describing the cooperationdegrees between car companiescompany1cooperation-degree company 2d 1d 01good d 02d 2d 01best d 03d 3d 02good d 01d 4d 03best d 01d 5d 03normal d 04d 6d 04normald 03aMany researchers in this ¢eld,such as Bertin [1]and Card etal .[2],have used tables as the format of the input data for the visualization process.However,they have used di¡erent terminology than used in this paper.For example,Bertin uses the term data characteristics for data attributes and data objects for data entries ,while Card etal .use data variables for data attributes and data cases for data entries .The terminology used in this paper is motivated by the relational database terminology .or non-equivalent according to the car-type attribute,while they are quantitatively related to each other according to the sold-number attribute.The structures of the second type are constituted by binary relations that are de¢ned on values of two di¡erent aggregated attributes.The values of these aggregated attributes should be the entries of one and the same table.In fact,the characterizing relation for each single aggregated attribute is the identity relation,but since the values of two aggre-gated attributes are from one and the same domain (i.e.the entries from one and the same table),two aggregated attributes together de¢ne a binary relation between domain ele-ments.These binary relations are implicit in the data table and do not characterize any single attribute.For instance,the binary relation de¢ned on the values of thecompany1and company2attributes in T able 1can be represented as:fðd 01;d 02Þ;ðd 01;d 03Þ;ðd 02;d 01Þ;ðd 03;d 01Þ;ðd 03;d 04Þ;ðd 04;d 03Þg :This binary relation relates data entries fromT able 2.Binary relations can be characterized by their structural properties such as functional (injective,surjective,bijective),transitive,symmetric,and re£exive properties.InT able 1the values of the company1and company2attributes de¢ne such a binary relation which is non-functional,symmetric,non-re£exive,and non-transitive.Notice that this type of structures can be extended by allowing n -ary relations between aggregated attribute values.In such a case,data tables consist of n aggregated attributes.2.2.Proje ctingStructure d Data to Pe rce ptual StructureThe second step in visualizing data is the projection of the structured data into a percep-tual structure.A perceptual structure is de¢ned as a typed visual table which is basicallyT able 2.A data table describing car companies in terms of car types,the number of sold cars,and thecountries they produce carscar-typesold-number country d 01vw 40000Germany d 02toyota 50000Japan d 03opel 30000Germany d 04mazda20000JapanT able 3.T ables 1and 2are combinedcompany1cooperation-degreecompany 2car-type sold-number country car-type sold-number country vw 40000Germany good toyota 50000Japan vw 40000Germany best opel 30000Germany T oyota 50000Japan good vw 40000Germany Opel 30000Germany best vw 40000Germany Opel 30000Germany normal mazda 20000Japan Mazda20000Japannormal opel30000Germanya table constituted by typed visual attributes.The type of a visual attribute is determined by the relations that can be perceived among values of that attribute.As we will explain in the next section,the perceivable relation for each visual attribute is determined before-hand based on theoretical and empirical studies of human visual perception.The rows of a typed visual table will then be called visual entries.The projection of the structured data into a perceptual structure consists of a bijective function from typed data attributes to typed visual attributes and another bijective func-tion from rows of the data table (i.e.data entries)to rows of the visual table (i.e.visual entries).There are di¡erent functions that map typed data attributes to typed visual attri-butes.These di¡erent functions specify alternative visualizations of data.W e assume that the second function does not map data attribute values to visual attribute values,but to variables that stand for visual attribute values.These variables will be instantiated with actual values by the layout process.As we will explain in Section 4,for e¡ective data visualization the projection should be a structure-preserving projection.This can be guaranteed by ensuring that the ¢rst function maps each typed data attribute to a visual attribute with identical type,and that the second function maps data attribute values to visual variables in such a way that whenever two data attribute values are in a certain relation (characterizing the type of that data attribute),their corresponding visual vari-ables are in the corresponding perceptual relation as well.This projection speci¢es abstract visual entries that are related to each other by percep-tual relations induced by the typed visual attributes.For example,considerT able 1once again.The ¢rst function maps the company1data attribute to the connection1visual attribute,company2to connection2,cooperation-degree to size ,car-type to xpos ,sold-number to ypos ,and country to color-hue .The second function maps data entries to abstract visual entries such that the structure preserving condition is satis¢ed.For instance,consider the data attribute country which is characterized by the identity relation.Whenever twovalues of the country attribute are identical,their corresponding color-hue variables are identical as well.This projection results typed visual tables illustrated inT ables 4and 5.2.3.The Layout of VisualizationThe variables in the typed visual table should be instantiated with visual attribute values to generate a drawable visualization of data.However,the typed visual table consists of only those visual attributes to which a data attribute is projected.Therefore,it may be the case that some visual attributes,the values of which are necessary to generate a drawableT able 4.Avisual table for the data described inT able 1connection1Size connection 2v 1v 01size 2v 02v 2v 01size 3v 03v 3v 02size 2v 01v 4v 03size 3v 01v 5v 03size 1v 04v 6v 04size 1v 03visualization,are not used by the projection.For instance,if the position attribute is not used by a projection,thenvisual entries should receive positionvalues in order to become drawable.These unused visual attributes will be called undecided visual attributes in con-trast to the decided attributes to which data attributes are projected.Consider the visualizations that are illustrated in Figure 2.These are three possible visualizations for the data represented inT ables 1and 2.In visualization 2-A,shape and size are undecided visual attributes.Moreover,in visualization 2-B position and shape,and invisualization 2-C position,are undecided visual attributes.Invisualization 2-C,the label attribute is decided for the car-type ,sold-number ,and country attributes.This example shows that more than one data attribute can be projected to a visual attribute.Of course,this does not hold for all visual attributes.For example,only one data attribute can be projected to the color-hue attribute since otherwise the e¡ectiveness of visualization cannot be guaranteed.W e assume that the projection function is de¢ned based on this general visualization knowledge.Visualization 2-C shows also that a drawable visualization does not need to have values for all visual attributes.For example,in this visualization visual entries do not need shape values to become drawable.W e assume that the combination of decided visual attributes determines which undecided visual attribute values should be speci¢ed for visual entries to become drawable.The generation of values for decided and undecided visual attributes is a constraint satisfaction process.The constraints on the values of the decided visual attributes are imposed by the perceptual structure determined at the previous visualization step.Although the perceptual structure does not impose any constraints on the valuesof1000020000500003000040000(a)(c)Figure 2.Visualizations of the data described byT able 1T able 5.Avisual table for the data described inT able 2xposypos color-hue v 01seg 1pos 4hue 1v 02seg 2pos 5hue 2v 03seg 3pos 3hue 1v 04seg 4pos 2hue 2undecided visual attributes,these values cannot be generated arbitrarily since otherwise unwanted visual implicature [11]may occur.In the next section,we will discuss this issue in more details.The determination of values for decided and undecided visual attributes will be called the layout process.It should be noted that the textual information inserted in these visualizations may either be the values of the label attribute or it may be a part of the interpretation function.In order to distinguish these two kinds of textual information,we use capital letters only to indicate that a textual information is a value of the visual label attribute.2.4.E¡ective DataVisualizationA data table can be visualized in many di¡erent ways by deciding di¡erent visual attributes for each data attribute and by specifying di¡erent visual entries for each data entry .In Figure 3,two di¡erent visualizations for T ables 1and 2are illustrated.In visualization 3-A,the xpos ,ypos ,and color-hue attributes visualize the car-type ,sold-number ,and country attri-butes,respectively .Moreover,the links between visual elements,de¢ned by the connection1and connection2attributes,visualize the company1and company2attributes,respectively .Final-ly ,the thickness attribute of the links visualizes the cooperation-degree attribute.Invisualization 3-B,the shape ,color-hue ,and label attributes visualize the sold-number ,country,and car-type attri-butes,respectively .Like visualization 3-A,the links and their thickness visualize the companies and their degrees of cooperations,respectively .In visualizations 3-A and 3-B the inserted numbers are parts of the interpretation function (i.e.legend).Although these two visualizations represent the same data,visualization 3-B is not an e¡ective visualization if the quantitative relations between the numbers of sold cars are the intended data relations.In this visualization,the perceivable relation induced by the shape attribute is the identity relation which implies that one can only perceive that the numbers of sold cars are equal or not.Of course,one may deduce the intended quanti-tative data relations indirectly by means of the interpretation information that are in-serted into visualization 3-B.For example,the interpretation values 20000and 40000implies that the second is twice the ¢rst.However,this inference is based on the knowl-edge of the perceiver about the structure of real numbers rather than being based on perceivable relations.The e¡ectiveness of visualization is the ability to perceive data rela-tions directly by structurally similar perceivable relations.In order to illustrate the direct perception of relations,consider visualizations 4-A and 4-B (Figure 4).These visualizations are the same as visualizations 3-A and3-B,100002000050000300004000050000vwtoyota opel mazda(a)(b)Figure 3.Alternative visualizations of the data described inT able 1respectively ,except that in these visualizations the interpretation information are not pre-sented.Knowing that the shape attribute visualizes the sold-number attribute in visualization 4-B,it is impossible to perceive any quantitative relation between visual elements and therefore it is impossible to infer any quantitative data relation.In contrast,knowing that the ypos attribute visualizes the sold-number attribute in visualization 4-A,it is easy to per-ceive quantitative relations between visual elements and therefore easy to infer quantita-tive data relations.In the rest of this paper,whenever we refer to the perceptual structure of visualizations we mean the structure of visualizations without using any interpretation information.The above examples show that the choice of visual attributes for data attributes may in£uence the e¡ectiveness of visualizations.However,the speci¢cations of visual entries by the layout process may in£uence the e¡ectiveness of visualizations as well.W e argued that the layout process should specify values for both decided and undecided visual attri-butes.In the context of e¡ective data visualization,the generation of values for the decided visual attributes is constrained by the perceivable relations.For example,in vi-sualization 3-A where the color-hue attribute is decided for the country attribute,the layout process has generated identical color-hue values whenever their corresponding data attri-bute values are identical.The generation of values for the undecided visual attributes may also in£uence the e¡ectiveness of visualizations.For example,consider again visualization 2-A.Although this visualization represents the same information as visualization 3-A,it is not an e¡ec-tive visualization.The reason is that a humanviewer perceives di¡erences between shapes of visual elements and may infer that these di¡erences visualize di¡erences in the pre-sented data.In order to avoid this type of unwanted visual implicatures,the values of undecided visual attributes should be speci¢ed in such a way to induce an identity rela-tion on the involved visual elements.In this way ,visual elements will be perceived as being identical to each other with respect to undecided visual attributes.This issue will be further explained in Section 4.33.An Attribute-based Classi¢cation of StructuresAn important step in data visualization is the projection of data structures (typed data table)to perceptual structures (types visual table).In this section,a unifying approach is proposed to analyze di¡erent classes of data and perceptual structures.This approachisopel toyota vw mazda(a)Figure 4.Visualizations without any interpretation valuesbased on a formalization of attributes in terms of their characterizing relations.The ad-vantage of this approach is that data and perceptual structures are both formalized in a uniform way such that the projection between them can be formulated in a formal and intuitive way .Based on this formalization a classi¢cation of data and perceptual structures is proposed.In order to de¢ne attributes in a formal way ,we use some notions from the measurement theory [12,13].The theory of measurement provides both a classi¢cation of attributes as well as their mathematical de¢nitions by introducing di¡erent types of measurementscales .In fact,a measurement scale is a map from a structured set of elements to a structured set of values like the set of real numbers,the set of integers,or a set of strings.In order to model aggregated attributes,we also allow an attribute to map a structured set of elements to another structured set of elements.In the measurement theory ,relational systems are used to represent structured sets.De¢nition 1.A relational system is a pair /A ;R 1;y ;R n S where A is a set of elements,and R 1;y ;R n are relations de¢ned on A :W e consider an attribute as a measurement scale,which map a structured set into another structured set.De¢nition 2.An attribute is a homomorphism H from a relational system /A ;R 1;y ;R n S into a relational system /B ;S 1;y ;S n S :The set A is the set of ele-ments and the set B is either a set of elements or a set of attribute values such as the set of real numbers,the set of integers,or a set of strings.In the context of data visualization,A can be either a set of visual elements or a set of data elements,R 1;y ;R n can be either perceptual relations or data relations,B is either a set of elements or a set of values,and S 1;y ;S n are characterizing relations de¢ned on B :These characterizing relations are abstract mathematical relations such as ¼;r and þ:The homomorphism guarantees that the relations that an attribute induces on elements have identical structural properties as its characterizing relations.In the next subsections,we introduce data attributes and their corresponding data structures,followed by visual attributes and their corresponding perceptual structures.3.1.Data AttributesBelow are ¢ve di¡erent types of data attributes that are relevant for data visualization.Nominal attributes :A nominal attribute is a homomorphicmap H from a relational system /A ;E S into a relational system /B ;¼S :Note that the homomorphicmap assigns values such as real numbers or strings to elements of A :Ordinal attributes :An ordinal attribute is a homomorphicmap H from a relational sys-tem /A ;$S into the relational system /B ;r S :The homomorphicmap matc hes the ordinal relation among attribute values with the ordinal relation among elements of A :Interval attributes :An interval attribute is a homomorphicmap H from a relational sys-tem /A ;$;3S into the relational system /R k ;r ;~S ;where ~is a quaternary me-trical relation de¢ned on real numbers.In the context of data visualization,we may use the quaternary metrical relation ~de¢ned as:ab ~cd 3j a Àb j r j c Àd j :This attribute。

新编简明英语语言学教程笔记考试必备Chapter one Introduction一、定义1.语言学LinguisticsLinguistics is generally defined as the scientific study of language.2.普通语言学General LinguisticsThe study of language as a whole is often called General linguistics.3.语言languageLanguage is a system of arbitrary vocal symbols used for human communication. 语言是人类用来交际的任意性的有声符号体系。

4.识别特征Design FeaturesIt refers to the defining poperties of human language that distinguish it from any animal system of communication.语言识别特征是指人类语言区别与其他任何动物的交际体系的限定性特征。

Arbitrariness任意性Productivity多产性Duality双重性Displacement移位性Cultural transmission文化传递⑴arbitrarinessThere is no logical connection between meanings and sounds.P.S the arbitrary nature of language is a sign of sophistication and it makes it possible for language to have an unlimited source of expressions⑵ProductivityAnimals are quite limited in the messages they are able to send.⑶DualityLanguage is a system, which consists of two sets of structures ,or two levels.⑷DisplacementLanguage can be used to refer to contexts removed from the immediate situations of the speaker.⑸Cultural transmissionHuman capacity for language has a genetic basis, but we have to be taught and learned the details of any language system. this showed that language is culturally transmitted. not by instinct. animals are born with the capacity to produce the set of calls peculiar to their species.5.语言能力CompetenceCompetence is the ideal user’s knowledge o f the rules of his language.6.语言运用performancePerformance is the actual realization of this knowledge in linguistic communication.语言运用是所掌握的规则在语言交际中的体现。

第一章1.What is language?Language is a system of arbitrary vocal symbols used for human communication2.Design features of language①Arbitrariness(任意性)refers to the forms of linguistic signs bear no natural relationship to their meaning. (sounds and meanings)②Duality（二层性）:The property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level and each of the two levels has its own principles of organization.③Productivity/creativity(创造性）:Language is productive in that it makes possible the construction and interpretation of new signals by its users.④Displacement(移位性):Human Languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at moment of communication. (p7)3.Functions of language①Informative(信息功能): to give information about facts. (ideational)②Interpersonal(人际功能): to establish and maintain social status in a society.(age, sex, language, background, accent, status)③Performative(施为功能) : language is used to do things, to perform certain actions. (name, promise, apologize, sorry, declare)④. Emotive/Expressive(情感功能): to express feelings and attitudes of the speaker.⑤Phatic communion(寒暄交流) : to use small and meaningless expressions to establish a comfortable relationship or maintain social contact between people without any factual content. (health, weather)⑥Recreational function(娱乐): the use of language for sheer joy. (lyrics, poetry)⑦Metalingual function(元语言功能): to talk about language itself.4. What is linguistics?Linguistics is generally defined as the scientific study of language.5. Important distinctions in linguisticsDescriptive & prescriptiveSynchronic & diachronicLangue & paroleCompetence & performance6.Descriptive(描写/述性)—describe and analyze linguistic facts or the language people actually use (modern linguistic)Prescriptive(规定性)—lay down rules for“correct and standard”linguistic behavior in using language (traditional grammar:“never use a double negative”)7.Synchronic study(共时)—description of a language at some point of time (modern linguistics) Diachronic study (历时)—description of a language as it changes through time (historical development of language over a period of time)第四章1．What is Syntax（句法）?Syntax is the study of the rules governing the ways different constituents are combined to formsentences.句法就是研究语言的不同成分组成句子的规则2．Four Approaches :The traditional approach传统语言观(Parts of speech、Syntactic Function 不考、Category范畴、Concord and government一致关系和支配关系)、The structural approach 结构语言观、The generative approach、The functional approach功能语言观3．The traditional grammar regards sentences as a sequence of words , so it pays great attention to the study of words, such as the classification of words in terms of parts of speech, the identification of function of words in terms of subject, predicate , etc.4. Parts of speechTraditional grammar defines 8 parts of speech: nouns, verbs, pronouns, adjectives, adverbs, prepositions, conjunctions and interjections.5．The term Category范畴in some approaches refers to word classes and functions in its narrow sense,范畴这一术语狭义上是指词类和功能eg. Noun, Verb, Subject, Predicate. More specifically, it refers to the defining properties of these general units:the categories of the noun名词的范畴, include number, gender, case and countability（case）; the categories of the verb动词的范畴: tense, aspect, voice, etc.6．Number is mostly a category of the noun and pronoun名词和不可数名词.Two terms of number in nouns: singular and plural单数和复数Number is also reflected in the inflections of pronouns and verbs7. Gender is also mostly a category of the noun and pronoun.In English, the gender distinctions are on the whole natural, determined by the biological gender of the creature.8. Case is used in the analysis of word classes to identify the syntactic relationship between words in a sentence.在词类分析中，格范畴用来辨别句子中词之间的句法关系In English, pronouns have three cases of nominative主格, accusative受格, and genitive与格. Nouns have two of general and genitive所有格In English, the case of noun is realised in three channels:(a) inflection(b) following a preposition(c) word order9. Tense时态: the absolute location of an event or action on time. It is marked by an inflection of the verb. As a result, there are only two tenses recognized now: past and present.Since the fut ure time does not involve any inflection of the verb, we do not refer to a “future tense”, even though in many different ways we can talk about the future.10. Aspect体: It has nothing with time, and it tells us whether an action is ongoing or completed. Perfective（完成体）and Imperfective（进行体）Perfective and Progressive (in English)11. Voice语态: describe the relationship between verb and subjectPassive被动语态and active主动语态12. Concord and government①Concord (一致关系)refers to agreement between words, especially between a verband the subject of a sentence.②Government (支配关系)is a type of grammatical relationship between two or more elements ina sentence.In traditional grammar, the term government has typically been used to refer to the relationship between verbs and nouns or between prepositions and nouns.13.The Structural Approach，由Ferdinand de Saussure提出14.Syntactic Relations：Positional relations位置关系、Relations of substitutability替代关系、Relations of co-occurrence同现关系15.Immediate constituent(直接成分)is any meaningful constituent at the first step in an analysis.16.An endocentric construction(向心结构)is a construction that contains:1) a head, which is the single obligatory element in the construction;2) one or more optional elements subordinate to the head.17.them e(主位) refers to the known information which is not new to the reader or listener Rheme(述位)refers to the information that is new. The new information is what is to be transmitted to the reader or listenerThe linguists of the Prague school believed that sentence may be analyzed from the functional side as well as the grammatical side.subject, predicate (grammatical side)theme, rheme (functional side)第五章1.What is Semantics?Semantics is the study of the meaning ofwords,phrases andsentences.语义学是研究单词、短语和句子的意义的学科2．Geoffrey Leech利奇Seven types of meaning7种意义类型:①Conceptual meaning概念意义②Connotative meaning内涵意义③Social meaning社会意义④Affective meaning感情意义Associative Meaning联想意义（②——⑥）⑤Reflected meaning反射意义⑥Collocative meaning搭配意义⑦Thematic meaning主位意义3.Conceptual meaning(概念意义)is also called “denotative”（外延义）and it is concerned with the relationship between a word and the thing it refers to.概念意义也叫外延义，它关注词语跟它所指称事物之间的联系Conceptual meaning is meaning given in the dictionary.4.Associative meaning (联想意义)is the total of all the meanings a person thinks of when they hear the wordAssociative meaning is the meaning which a word suggests or implies.5.Thematic meaning (主位意义)is“what is communicated by the way in which the message is organized in terms of orde r and emphasis.”它是由词序和词语重音所决定的6.The Referential Theory（指称理论）：①The Referential Theory②The Semantic Triangle③Sense and Reference7.The referential theory指称理论is the theory of meaning which relates the meaning of a word to the thing it refers to.指称论是把词语意义跟它所指称的事物联系起来的理论8. The semantic triangle语意三角is the indirectrelation between a word and a thing it refers to and it is mediated by concept.语意三角指词和所指事物之间没有直接关系，它们是以概念为中介的9.Sense (涵义) is a set of properties possessed by a name.10.Reference (指称) is the symbolic relationship that a linguistic expression has with the concrete object.11. The sense of an expression is the thought it expresses, while its reference is the object it representsEvery word has a sense, but not every word has a reference.12.Sense Relations涵义关系①Synonymy（同义关系）②Antonymy（反义关系）（Gradable、Complementary、Converse）③Hyponymy（上下义关系）13.But total synonymy is rare. They may differ in style, connotations and dialect.14.Gradable antonymy (等级反义关系)、Complementary antonymy (互补反义关系)、Converse antonymy (反向反义关系)15. Componential analysis is an approach to the study of meaning which analyses a word into a set of meaning components.16. Sentence Meaning17.Sense relations between sentences①Synonymity (同义)a. He was a bachelor all his life.b. He never married all his boy.Sentences a and b are in a synonymous relationship: the truth of one sentence necessarily implies the truth of another sentence②Inconsistency(矛盾)a. Elizabeth II is Queen of England.b. Elizabeth II is a man.Sentences a and b are in a relationship of contradiction: the truth of one sentence necessarily implies the falseness of another sentence.③Entailment (蕴涵)a. He married a blonde heiress.b. He married a blonde.Entailment refers to a kind of meaning inclusion. If x entails y, the meaning of x is included in y.④Presupposition(前提预设)It is what a speaker or writer assumes that the receiver of the message already knows.⑤Contradiction(矛盾)⑥Semantic anomaly(语义反常)18. An integrated theory﹡Compositionality(组合性原则)：the meaning of a sentence depends on the meaning of the constituent words and the way they are combined.﹡This semantic theory is the integration of syntax and semantics﹡Their basic idea is that a semantic theory consists of two parts: a dictionary and a set of projection rules﹡The dictionary provides the grammatical classification and semantic information of words﹡The projection rules are responsible for combining the meanings of words together.19.Logical semantics（逻辑语义学）﹡A proposition(命题) is what is to be expressed by a declarative sentence when that sentence is uttered to make a statement.﹡It is the basic meaning which a sentence express.﹡A very important property of the proposition is that it has a truth value.第七章nguage and Culture：①Sapir-Whorf Hypothesis②Evidence Given by Whorf ③Implication of SWH ④Relation between Language and Culture2．SHW can be broken down into two basic principles:Linguistic determinism (语言决定论):the language we use determines the way we view about the world around us.Language may determine our thinking patterns. (语言决定思维) P1623. Relation between Language and CultureLanguage influences thought and culture，Language varies in categories and concepts, thus reflecting the different world views of different language users, that is, culture and thought are conditioned by languageCulture influences language，Every language is a part of a culture. As such, it can not but serve and reflect cultural needs. When a culture experience radical changes, the vocabulary also undergoes corresponding alterationsnguage and SocietyRelation between Language and SocietyVarieties of language（Dialects、Registers）Bilingualism and DiglossiaPidgin and Creole5.Varieties related to the user are normally known as dialects and varieties related to use as registers.6.Dialectal Varieties：Regional dialect、Social dialect（Sociolect、Language and gender、Language and age、Idiolect、Ethnic dialect）7.Social dialect refers to a variety of language associated with a particular social group, such as a particular social class, or ethnic group, or those based on age, gender and occupation.8.An ethnic dialect is a social dialect of a language that cuts across regional differences; it is mainly spoken by a less privileged population that has experienced some form of social isolation such as racial discrimination or segregation.9.Idiolect refers to the speech variety of an individual. Every speaker has his own way of expressing his or her idea.10.Register refers to the functional variety of language that is defined according to its use in a context of situation.11.Halliday’s Register TheoryLanguage varies as its function varies; it differs in different situations.Halliday distinguishes 3 variables that determine the register:field of discourse (语场)、tenor of discourse (语旨)、mode of discourse (语式)12.Bilingualism (双语制): the use of two languages, esp with equal or nearly equal fluency.13.Diglossia (双语体现象): when two varieties of a language exist side by side; and each is used for different purposes, this is called diglossia.14．A pidgin: it is a special language variety that mixes and blends languages used for communicative purposes by groups of people who do not know each other’s language.15.A creole :when a pidgin has become the primary language of a speech community, and is acquired by the children of that speech community as their native language, it is said to have become a creole.第八章1.What is PragmaticsPragmatics is the study of language in context / use / communication.2 Semantics and PragmaticsSimilarity：Pragmatics and semantics are both linguistic study of meaningDifference：Semantic meaning: the more constant, inherent side of meaning；Pragmatic meaning: the more indeterminate, the more closely related to context；Pragmatic = meaning - semantics 3.Three Contents :Speech Act Theory、The Theory of Conversational Implicature、Post-Gricean Developments4.Speech Act Theory(言语行为理论)：①Performatives and Constatives②A theory of the illocutionary act5.The utterance which performs an act is calleda performative(行事话语)。

包含关系的二元思辨作文英文回答：In the realm of dialectical arguments, the concept of inclusionary relations presents an intriguing topic for philosophical contemplation. Within the sphere of logic, inclusion, or containment, denotes the relationship wherein one set of elements is entirely contained within another. This hierarchical arrangement raises fundamental questions about the nature of logical entities and the implications of their interconnectedness.Firstly, the inclusionary relation challenges the notion of absolute exclusivity in set theory. Thetraditional understanding of sets posits that each element belongs to a single set, fostering the illusion of distinct and independent entities. However, the concept of inclusion introduces the possibility of overlapping membership, blurring the boundaries between sets and introducing a nuanced understanding of set membership.Secondly, inclusionary relations shed light on the complexities of classification systems. In the process of organizing and categorizing objects, we often encounter situations where an entity simultaneously belongs to multiple classes. For instance, a book may fall under the categories of both "literature" and "adventure novels." This overlap highlights the limitations of rigid taxonomies and underscores the fluidity of conceptual boundaries.Thirdly, the inclusionary relation prompts us to question the nature of logical hierarchy. Traditional set theory assumes a clear-cut hierarchy, with smaller sets nested within larger ones. However, the concept ofinclusion suggests that hierarchical structures can be more intricate, with sets exhibiting multiple levels of containment. This complexity challenges the traditional notion of a linear hierarchy and introduces the possibility of more dynamic and interconnected relationships.In conclusion, the exploration of inclusionaryrelations within dialectical arguments unveils profoundinsights into the nature of logical entities, the complexities of classification systems, and the intricate web of logical interconnectedness. It invites us to abandon simplistic notions of exclusivity and embrace a more nuanced understanding of the relationships that shape our logical world.中文回答：包含关系在二元思辨作文中具有重要意义。

12 maximal onset principle states that when there is a choice as to where to place a consonant. it is put into the onset rather than the coda. . The correct syllabification of the word country should be第一章，填空1.The study of the meaning of lingustic words, phrases is callesde mantics.2.Displacement is a design feature of human language that enables speakers to talk about a wild range of things free from barriers caused by4.Morpheme is the smallest meaningful unit of language.5.If a linguistic study describes and analyzes the language people actually use, it is said to be descriptive.6.Chomsky defines “ competencaes ”the ideal user's knowledge of the rules of his language.nguage is a means of verbal communication. It is informative in that communicating by speaking or writing is a purposeful act.8.The link between a linguistic sign and its meaning is a matter ofnguage is distinguished from traffic lights in that the former has the designing feature of duality.10.In linguistics research, bothq uantity and quality approaches are preferred. 判断：11.The writing system of a language is always a later invention used to record speech, thus there are still many languages in today's have no √12.According to Chomsky, the word compentoetn lcime it”ed itso the ability of an ideal native speaker to construct and recognize. ×13.Duality and cultural transmission are two most important design features of human language.×14.Chomsky's competence' and performance are similar in meaning to Saussure 's langue and parole.√15,An important difference between traditional grammarians and modern linguists in their study of language is that the former tended to over-emphasize the written form of language and encourage people to imitate the “bestauthors ”√for languag16.In modern linguistic studies, the written form of language is given more emphasis than the spoken form for a of reasons. √17.Modern linguistics is mainly diachronic.× chochronic共时的ngue and parole is the fundamental distinction discussed by Chomsky in his Aspects of the Theory ofdistinguished the linguistic competence of the speaker and the actual phenomena or data of linguistics as Parole and language√.20. According to Chomsky, the task of a linguist is to determine from the data of performance the underlying system of rules that has been √选择:1.As modern linguistics aims to describe and analyse the language people actually use, and not to lay down rules for correct linguistic behavior, it is said to bed escriptive2.I can refer to Confucius even though he was dead 2000 years ago. This shows thatlanguage has the design feature of displacement.3.“ Don' t end a sentence with a preposition.thisp irse sacnr iepxtiavme rpulele os.f4.Which of the following is most referred to as a branch of the study of meaning in5.The synchronic study of language takes a fixed instant as its point of observation.6.The branch of linguistics that studies how context influences the way speakers interpret sentencesis calledp ragmatics.7.The fact that different languages have different words for the same object is good proof that human language is A 没照下图片arbitrary8.The descriptive of a language as it changes through time is dai achronic study.9.题目没照下来。

摘要本文用认知语言学主观性与主观化理论来阐释汉语羡馀并列式词语生成的认知机制，主要体现在音律性、整合性和象似性三方面，三者有机相联。

在认知语用过程中，为了取得良好的交际效果，人们通常用隐喻和转喻等认知手段对羡馀并列式词语构式进行整合，并赋予其主观义。

同时，数量和对称象似性展现了解码语言结构形式和人类认知间的重要联系。

关键词羡馀主观性主观化音律整合象似On Chinese Superfluous Coordinate Words from the Cognitive Perspective//Liu ZijianAbstract In the essay,the cognitive theory of subjectivity andsubjectivisation is used to illustrate forming mechanism of Chi-nese superfluous coordinate words,which is mainly embodied in-to three aspects,temperament,integration and iconicity,which are related.In the process of cognitive pragmatics,in order to get better communicative result,people usually compress construc-tion of superfluous coordinate words by cognitive metaphor and metonym,and give subjective meanings to them.Meanwhile,the iconicity of quantity and symmetry show the important relation-ship between decoding language structural form and human cog-nition.Key words superfluous;subjectivity;subjectivisation;tempera-ment;integration;iconicity1汉语羡馀并列式词语的界定与分类1.1羡馀的界定：羡馀≠赘余“羡馀”在《现代汉语词典》（第6版）中的解释是“多余的”；“赘余”①中构词语素“赘”在词典中的释义为“多余的；无用的”。

Chapter 1 Invitations to Linguisticsnguage is a system of arbitrary vocal symbols used for human communication. To give the barest definition language is a means of verbal communication. It is instrumental social and conventional. Linguistics is usually defined as the science of language or alternatively as the scientific study of language. It concerns with the systematic study of language or a discipline that describes all aspects of language and formulates theories as to how language works.2.Design features refers to the defining properties of human language that distinguish it from any animal system of communication. They are arbitrariness, duality, creativity, displacement etc.Arbitrariness refers to forms of linguistic signs bear no natural relationship to their meaning Language is arbitrary. There is no logical connection between meanings and sounds even with onomatopoeic words.Duality refers to the property of having two levels of structure. The units of the primary level are composed of elements of the secondary level and each of the two levels has its own principles of organization.☺the lower or the basic level---- the sound units or phonemes which are meaningless, but can be grouped and regrouped into words.☺the higher level ----morphemes and words which are meaningfulCreativity refers to Words can be used in new ways to mean new things and can be instantly understood by people who have never come across that usage before.Displacement refers to the fact that language can be used to refer to things which are present or not present real or imagined matters in the past present or future or in far away places. It means that human languages enable their users to symbolize objects events and concepts which are not present in time and space at the moment of communication3. Jakobson’s classification of functions of language.Jakobson : In his article Linguistics and Poetics (1960) defined six primary factors of any speech event: speaker, addressee, context,message, code, contact.1).Referential function 所指功能2).Poetic function诗学功能3).Emotive function感情功能4).Conative function意动功能5).Phatic function交感功能6).Metalingual元语言功能Hu Zhuanglin’ classification of functions of language and use some examples to illustrate them.1).Informative function 信息功能2).Interpersonal function 人际功能3).Performative function 施为功能4).Emotive function 感情功能5).Phatic communion 交感性谈话6).Recreational function 娱乐性功能7).Metalingual function 元语言功能4. What are the major differences between Saussure’s distinction betwe en langue and parole and Chomsky’s distinction between competence and performance?According to Saussure,(1) Langue is abstract, parole is specific to the speaking situation;(2) Langue is not actually spoken by an individual , parole is always a naturally occurring event;(3) Langue is relatively stable and systematic, parole is a mass of confused facts, thus not suitable for systematic investigation.According to N. Chomsky,Competence enables a speaker to produce and understand an indefinite number of sentences and to recognize grammatical mistakes and ambiguities; A speaker’s competence is stable while his performance is often influenced by psychological and social factors. So a speaker's performance does not always match or equal his supposed competence; Chomsky believes that linguists ought to study competence, rather than performance. In other words, they should discover what an ideal speaker knows of his native language.Chapter 2 Speech Sounds1.Phonetics studies how speech sounds are produced, transmitted and received. It is concerned with the actual physical articulation, transmission and perception of speech sounds.Phonology is essentially the description of the systems and patterns of speech sounds. It isconcerned with the abstract and mental aspect of the sounds in language.Phonology aimsto discover how speech sounds in a language form patterns and how these sounds are used to convey meaning in linguistic communication1. Lips2. Teeth3. Teeth ridge (alveolar)齿龈4. Hard palate 硬腭5. Soft palate (velum) 软腭6. Uvula 小舌7. Tip of tongue8. Blade of tongue 舌面9. Back of tongue10. V ocal cords 声带11. Pharyngeal cavity 咽腔 12. Nasal cavity 鼻腔2.Phone （音素): the smallest perceptible discreet segment of sound in a stream of speech. (in the mouth)Phoneme （音位）：A sound which is capable of distinguishing one word or one shape of a word from another in a given language is a phoneme. (in the mind)allophone (音位变体) : phonic variants of a phoneme are called allophone of the same phoneme. / / = phoneme [ ] = phone { } = set of allophonesIPA:the abbreviation of International Phonetic Alphabet .Minimal pairs 最小对立体Three requirements for identifying minimal pairs:1) different in meaning; 2) only one phoneme different;3) the different phonemes occur in the same phonetic environment.e.g. a minimal pair: pat -fat; lit-lip; phone-toneminimal set: pat, mat, bat, fat, cat, hat, etcplementary distribution 互补分布Phonetically similar sounds might be related in two ways.If they are two distinctive phonemes, they might form a contrast; e.g. /p/and /b/ in [pit] and [bit];If they are allophones of the same phoneme, then they don’t distinguish meaning, but complement each other in distribution, i.e. they occur in different phonetic contextSuprasegmental features 超音段特征—features that involve more than single sound segment, such as stress （重音），length（音程）, rhythm （节奏），tone（音调），intonation（语调）及juncture（音渡）．Chapter 3 Lexicon/Morphology1. Word1.1 Three senses of “word”(1) A physically definable unit: a cluster of sound segments or letters between two pause orblank.(2) Word both as a general term and as a specific term.(3) A grammatical unit1.2 Identification of wordsSome factors can help us identify words:(1) Stability(2) Relative uninterruptibility(3) A minimum free form1.3 The classification of wordWords can be classified in terms of:(1) Variable vs. invariable words (可变词/不可变词)(2) Grammatical words vs. lexical words (语法词/词汇词)(3) Closed-class words vs. open-class words (封闭词/开放词)(4) word class(词类)(1) Variable vs. invariable words (可变词/不可变词)the former refers to words having inflective changes（屈折变化）while the latter refers to words having no such endings.Variable words: follow; follows; following; followedInvariable words: since; when; seldom; through; hello(2) Grammatical words vs. lexical words (function words and content words.语法词/词汇词)The former refers to those words expressing grammatical meanings, such as conjunctions(连词), prepositions（介词）, articles(冠词), and pronouns（代词）;the latter refers to words having lexical meanings, those which refer to substance, action etc. such as n., v., adj., and adv..(3) Closed-class words vs. open-class words (封闭词/开放词)the former refers to words whose membership is fixed or limited; e.g. pron., prep., conj., article. the latter of which the membership is infinite or unlimited. e.g.: n., v., adj., adv.(4) word class (词类)The traditionally recognized word classes are: noun, pronoun, adjective, verb, adverb, preposition, conjunction, interjection, article, etc. More word classes have been introduced into grammar: particles 小品词/语助词(go by, look for, come up);auxiliaries 助词(can, be, will);pro-form 替代词(do, so);determiners 前置词/ 限定词(all, every, few, plenty of, this).2. The formation of word2.1 Morphology 形态学Definition：Morphology is a branch of linguistics, which studies the internal structure of words and the rules by which words are formed.The two fields (p64)Inflectional morphology: the study of inflectionsDerivational morphology: the study of word-formation3. Lexical change3.1 Lexical change proper（词本身的变化）Invention 新造词Blending混合词Abbreviation 缩合词Acronym首字母缩略词back-formation 逆构词analogical creation 类比造词Borrowing 借词、外来词definition:1) Morphology:Morphology is a branch of linguistics, which studies the internal structure of words and the rules by which words are formed.2) Terminology 术语解释Morpheme: The smallest unit of meaning, which can not be divided into further smaller units without destroying or drastically altering the meaning, whether it is lexical or grammatical.Free morphemes: morphemes which may constitute words by themselves．Bound morphemes:morphemes which can not be used by themselves, but must be combined with other morphemes to form wordsInflectional morpheme: a kind of bound morphemes which manifest various grammatical relations or grammatical categories such as number, tense, degree and case.Derivational morpheme: a kind of bound morphemes, added to existing forms to create new words. There are three kinds according to position: prefix, suffix and infix.Chapter 4 Syntax From Word to TextSyntax is the study of the rules governing the ways different constituents are combined to form sentences in a language, or the study of the interrelationships between elements in sentence structures.Endocentric Constructions:is one whose distribution is functionally equivalent to that of one or more of its constituents, i.e., a word or a group of words, which serves as a definable centre or head.Exocentric Constructions:refers to a group of syntactically related words where none of the words is functionally equivalent to the group as a whole, that is, there is no definable “Centre” or “Head” inside the group Category: refers to the defining properties of these general units:Categories of the noun: number, gender, case and countabilityCategories of the verb: tense, aspect, voicethree kinds of syntactic relations:relations of position位置关系Positional relation, or WORD ORDER, refers to the sequential arrangement of words in a language.relations of substitutability 可替代性关系The Relation of Substitutability refers to classes or sets of words substitutable for each other grammatically in sentences with the same structure.relations of co-occurrence 同现关系It means that words of different sets of clauses may permit, or require, the occurrence of a word of another set or class to form a sentence or a particular part of a sentence.Immediate Constituent Analysis (IC Analysis)Immediate constituent analysis is a form of linguistic review that breaks down longer phrases or sentences into their constituent parts, usually into single words. This kind of analysis is sometimes abbreviated as IC analysis, and gets used extensively by a wide range of language experts.Endocentric constructions fall into two main types, depending on the relation between constituents: Coordination and subordinationCoordination is a common syntactic pattern in English and other languages formed by grouping together two or more categories of the same type with the help of a conjunction such as and, but and or .Subordination refers to the process or result of linking linguistic units so that they have different syntactic status, one being dependent upon the other, and usually a constituent of the other.Characteristics of subjectsA) Word order: Subject ordinarily precedes the verb in the statementB) Pro-forms(代词形式) : The first and third person pronouns in English appear in a special form when the pronoun is a subjectC) Agreement with the verb: In the simple present tense, an -s is added to the verb when a third person subject is singular, but the number and person of the object or any other element in the sentence have no effect at all on the form of the verbD) Content questions (实意问句): If the subject is replaced by a question word (who or what), the rest of the sentence remains unchangedE) Tag question (反意问句): A tag question is used to seek confirmation of a statement. It always contains a pronoun which refers back to the subject, and never to any other element in the sentence.Explain the difference between sense and reference from the following four aspects:1) A word having reference must have sense;2) A word having sense might not have reference;3) A certain sense can be realized by more than one reference; 4) A certain reference can beexpressed by more than one senseThe distinction between “sense” and “reference” is comparable to that between “connotation” and “denotation”. The former refers to some abstract properties, while the latter refers to some concrete entities.Firstly, to some extent, we can say that every word has a sense, i.e., some conceptual content; otherwise we would not be able to use it or understand it. Secondly, but not every word has a reference. There are linguistic expressions which can never be used to refer to anything, for example, the words so, very, maybe, if, not, and all. These words do of course contribute meaning to the sentences in which they occur and thus help sentences denote, but they themselves do not identify entities in the world. They are intrinsically non-referring terms. And words like ghost and dragon refer to imaginary things, which do not exist in reality. Thirdly, some expressions will have the same reference across a range of utterances, e.g., the Eiffel Tower or the Pacific Ocean. Such expressions are sometimes described as having constant reference. Others have their references totally dependent on context. Expressions like I, you, she, etc. are said to have variable references. Lastly, sometimes a reference may be expressed by more than one sense. For instance, both ‘evening star’ and ‘morning star’(晚星，启明星), though they differ in sense, refer to Venus.Chapter 6 Language and cognition1.What is Cognition?In psychology it is used to refer to the mental processes of an individual with particular relation to a view that argues that the ming has internal mental states and can be understood in terms of information processing.Another denefition is mental process or faculty of knowing, including awareness, perception, reasoning, and judgment.2.Cognitive LinguisticsCognitive linguistics is the scientific study of the relation between the way we communicate and the way we think.It is an approach to language that is based on our experience of the world and the way we perceive and conceptualize it.3.What are the differences between metaphor & metonymy? Give some examples. Metaphor is a conceptual mapping（概念映射）, not a linguistic one, from one domain to another （从一个语域到另一个语域）, not from a word to another.Metonymy is a cognitive process in which one conceptual entity, the vehicle（源域）, provides mental access to another conceptual entity, the target（目标域）, within the same domain. The reference point activates the target.1.Metaphor is used for substitution, while metonymy is used for association.2. Metaphor can mean condensation and metonymy can mean displacement.3. A metonymy acts by combining ideas while metaphor acts by suppressing ideas.4. In a metaphor, the comparison is based on the similarities, while in metonymy thecomparison is based on contiguity.--For example, the sentence ‘he is a tiger in class’ is a metaphor. Here the word tiger is used in substitution for displaying an attribute of character of the person. The sentence ‘the tiger called his students to the meeting room’ is a metonymy. Here there is no substitution; instead the person is associated with a tiger for his nature..Metaphor is actually a cognitive tool that helps us structure our thoughts and experiences in the world around us..Metaphor is a conceptual mapping（概念映射）, not a linguistic one, from one domain to another（从一个语域到另一个语域）, not from a word to another.Metonymy（换喻，转喻）.It is a cognitive process in which one conceptual entity, the vehicle（源域）, provides mental access to another conceptual entity, the target（目标域）, within the same domain.2.Psycholinguistics is the study of psychological aspects of language; it usually studies the psychological states and mental activity with the use of language.Language acquisition (1) Holophrastic stage（单词句阶段）–Language’s sound patterns–Phonetic distinctions in parents’ language.–One-word stage: objects, actions, motions, routines.2) Two-word stage: around 18m3) Three-word-utterance stage4) Fluent grammatical conversation stageChapter 7 Language, culture and society1.the relationship between language and thought?Generally, the relation of L to C is that of part to whole, for L is part of C.The knowledge and beliefs that constitute a people’s culture are habitually encoded and transmitted in L.There exists a close relationship between language and culture. This is evidenced by the findings of anthropologists such as Malinowski, Firth, Baos, Sapir and Whorf. The study of the relation between language and the context in which it is used is the cultural study of language.2.What’s Sapir-Whorf Hypothesis? Give your comment on it.Edward Sapir (1884 - 1939) and Benjamin Lee Whorf (1897-1941)Our language helps mould our way of thinking and, consequently, different languages may probably express speakers’unique ways of understanding the world.Linguistic determinism: L may determine our thinking patterns.Linguistic relativity: a. Similarity between language is relative; b. the greater their structural differentiation is, the diverse their conceptualization of the world will be.Chapter 8 Language in usePragmatics: The study of language in use and the study of meaning in context, as well as the study of speakers’ meaning, utterance meaning& contextual meaning..What’s your understanding of conversational implicature? Use one or two examples to discuss the violation of its maxims.People do not usually say things directly but tend to imply them. CP is meant to describe whatactually happens in conversation. People tend to be cooperative and obey CP in communication. Since CP is regulative, CP can be violated. Violation of CP and its maxims leads to conversational implicature.1.What are the main differences between pragmatics and semantics?Semantics and pragmatics are both linguistic studies of meaning. The essential difference lies in whether in the study of meaning the context of use is considered. If it is not, the study is restricted to the area of traditional semantics; if it is, the study is carried out in the area of pragmatics.Semantics studies sentences as units of the abstract linguistic system while pragmatics studies utterances as instances of the system.The former stops at the sentence level; the latter looks at bigger chunks of conversation. The former regards sentences as stable products; the latter treats utterances as dynamic processes. The former analyses sentences in isolation; The latter analyses utterances in close connection with their contexts of situation.2. What does pragmatics study? How does it differ from traditional semantics?答： Generally speaking, pragmatics is the study of meaning in the context. It studies meaning in a dynamic way and as a process. In order to have a successful communication, the speaker and he arer must take the context into their consideration so as to affect the right meaning and intention. T he development and establishment pragmatics in 1960s and 1970s resulted mainly from the expan sion of the study semantics. However, it is different from the traditional semantics. The major diff erence between them lies in that pragmatics studies meaning in a dynamic way, while semantics st udies meaning in a static way. Pragmatics takes context into consideration while semantics does n ot. Pragmatics takes care of the aspect of meaning that is not accounted for by semantics.3. What are the four maxims of the CP? Try to give your own examples to show how flouting these maxims give rise to conversational implicature?答： Cooperative Principle, abbreviated as CP. It goes as follows:Make your conversational contribution such as required at the stage at which it occurs by the ac cepted purpose or direction of the talk exchange in which you are engaged.To be more specific, there are four maxims under this general principle:(1) The maxim of quantity① Make your contribution as informative as required (for the current purpose of the exchange) . ② Do not make your contribution more informative than is required.(2) The maxim of quality① Do not say what you believe to be false.② Do not say that for which you lack adequate evidence.(3) The maxim of relation Be relevant.(4) The maxim of manner① Avoid obscurity of expression. ② Avoid ambiguity.③ Be brief (avoid unnecessary prolixity).④ Be orderly.Chapter 9 Language and literature1.What is ‘foregrounding’?In a purely linguistic sense, the term ‘foregrounding’ is used to refer to new information, in contrast to elements in the sentence which form the background against which the new elementsare to be understood by the listener / reader.In the wider sense of stylistics, text linguistics, and literary studies, it is a translation of the Czech aktualisace (actualization), a term common with the Prague Structuralists.The English term ‘foregrounding’has come to mean several things at once:-the (psycholinguistic) processes by which - during the reading act - something may be given special prominence;-specific devices (as produced by the author) located in the text itself. It is also employed to indicate the specific poetic effect on the reader;-an analytic category in order to evaluate literary texts, or to situate them historically, or to explain their importance and cultural significance, or to differentiate literature from other varieties of language use, such as everyday conversations or scientific reports.Literal language and figurative language-A language is called literal when what is meant to be conveyed is same as what the word to word meaning of what is said. In contrast the figurative language, the words are used to imply meaning which is other than their strict dictionary meaning.-Literal language refers to words that do not deviate from their defined meaning. Figurative language refers to words, and groups of words, that exaggerate or alter the usual meanings of the component words. Figurative language may involve analogy to similar concepts or other contexts, and may involve exaggerations. These alterations result in figures of speech.Chapter 11 LinguisticsApplied linguistics: is the study of the relation of linguistics to foreign language teaching, of the ways of applying linguistic theories to the practice of foreign language teaching. Universal Grammar:is a theory in linguistics that suggests that there are properties that all Possible natural human languages have. Usually credited to Noam Chomsky, the theory suggests that some rules of grammar are hard-wired into the brain, and manifest themselves without being taught. There is still much argument whether there is such a thing and what it would be. Syllabus: a syllabus is a specification of what take place in the classroom,which usually contains the aims and contents of teaching and sometimes contains suggestions of methodology. Interlanguage: the type of language constructed by second or foreign language learners who are still in the process of learning a language is often referred to as interlanguage.contrastive analysis: A way of comparing L1 and L2 to determine potential errors for the purpose of isolating what needs to be learned and what not. Its goal is to predict what areas will be easyto learn and what will be difficult. Associated in its early days with behaviorism and structuralism. the Input Hypothesis: according to krashen's input hypothesis, learners acquire language as a result of comprehending input addressed to them.Chapter 12 Theories & schools of modern linguisticsTransformational-Generative GrammarThe five stages of development of TG Grammar:1) The classical theory (1957)2) The standard theory (1965)3) Extended standard theory4) GB/PP theory (1981)5) The Minimalist ProgramInnateness hypothesis: Chomsky believes that language is somewhat innate, and that children are born with what he calls a Language Acquisition Device(LAD), which is a unique kind of knowledge that fits them for language learning.CHOMSKY’S TG GRAMMAR DIFFERS FROM THE STRUCTURAL GRAMMARIN A NUMBER OF WAYS1. Rationalism2. innateness 3 deductive methodology4 emphasis on interpretation 5formalization 6.emphasis on linguistic competence 7. strong generative powers 8.emphasis on linguistic universals。

语言学考研真题和答案第一章语言学Fill in the blanks1. Human language is arbitrary. This refers to the fact that there is no logical or intrinsic connection between a particular sound and the _______it is associated with. （人大2007研）meaning 语言有任意性，其所指与形式没有逻辑或内在联系2. Human languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at the moment of communication. This quality is labeled as _______. （北二外2003研）displacement 移位性指人类语言可以让使用者在交际时用语言符号代表时间和空间上不可及的物体、事件和观点3. By duality is meant the property of having two levels of structures, such that units of the _______ level are composed of elements of the __________ level and each of the two levels has its own principles of organization. （北二外2006研）primary, secondary 双重性指拥有两层结构的这种属性，底层结构是上层结构的组成成分，每层都有自身的组合规则4. The features that define our human languages can be called _______ features. （北二外2006）design人类语言区别于其他动物交流系统的特点是语言的区别特征，是人类语言特有的特征。

Chapter 1 Invitations to Linguistics1.3 Design features of languageThe features that define our human languages can be called design features which can distinguish human language from any animal system of communication.1.3.1 ArbitrarinessArbitrariness refers to the fact that the forms of linguistic signs bear no natural relationship to their meanings.1.3.2 DualityDuality refers to the property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level and each of the twolevels has its own principles of organization.1.3.3 CreativityCreativity means that language is resourceful because of its duality and its recursiveness. Recursiveness refers to the rule which can be applied repeatedly withoutany definite limit. The recursive nature of language provides a theoretical basis for thepossibility of creating endless sentences.1.3.4 DisplacementDisplacement means that human languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at the moment ofconversation.加1 Each sound in the language is treated as discrete.加2 the direct/non-arbitrary/non-symbolic relation between meaning and form. There are resemblances between the language form and whatthey refer to. That relationship is called icon. Iconicity exists in sounds,lexicons and syntax. It is the motivation between language forms andmeanings. It is a relation of resemblance between language form and whatthey refer to.1.5 Functions of languageAs is proposed by Jacobson, language has six functions:1. Referential: to convey message and information;2. Poetic: to indulge in language for its own sake;3. Emotive: to express attitudes, feelings and emotions;4. Conative: to persuade and influence others through commands and entreaties;5. Phatic: to establish communion with others;6. Metalingual: to clear up intentions, words and meanings.three metafunctions:1. function: to convey new information, to communicate a content that isunknown to the hearer;embodying all use of language to express social and personalrelationships;3.of spoken and written discourse into a coherent and unified text and make a livingpassage different from a random list of sentences.According to Hu Zhuanglin, language has at least seven functions:1.5.1 InformativeThe informative function means language is the instrument of thought and people often use it to communicate new information.1.5.2 Interpersonal functionThe interpersonal function means people can use language to establish and maintain their status in a society.1.5.3 PerformativeThe performative function of language is primarily to change the social status of persons, as in marriage ceremonies, the sentencing of criminals, the blessing of children,the naming of a ship at a launching ceremony, and the cursing of enemies.1.5.4 Emotive functionThe emotive function is one of the most powerful uses of language because it is so crucial in changing the emotional status of an audience for or against someone orsomething.1.5.5 Phatic communionThe phatic communion means people always use some small, seemingly meaningless expressions such as Good morning, God bless you, Nice day,etc., tomaintain a comfortable relationship between people without any factual content.1.5.6 Recreational functionThe recreational function means people use language for the sheer joy of using it, such as a baby’s babbling or a chanter’s chanting.1.5.7 Metalingual functionThe metalingual function means people can use language to talk about itself. E.g. I can use the word “book” to talk about a book, and I can also use the expression “theword book” to talk about the sign “b-o-o-k” itself.1.6 What is linguistics?Linguistics is the scientific study of language. It studies not just one language of any one community, but the language of all human beings.1.7 Main branches of linguistics1.7.1 PhoneticsPhonetics is the study of speech sounds, it includes three main areas: articulatory phonetics, acoustic phonetics, and auditory phonetics.1.7.2 PhonologyPhonology studies the rules governing the structure, distribution, and sequencing of speech sounds and the shape of syllables.1.7.3 MorphologyMorphology studies the minimal units of meaning –morphemes and word-formation processes.1.7.4 SyntaxSyntax refers to the rules governing the way words are combined to form sentences in a language, or simply, the study of the formation of sentences.1.7.5 SemanticsSemantics examines how meaning is encoded in a language.1.7.6 PragmaticsPragmatics is the study of meaning in context.1.8 Macrolinguistics1.8.1 Psycholinguistics1.8.2 Sociolinguistics1.8.3 Anthropological linguistics1.8.4 Computational linguisticsTo say that linguistics is a descriptive science is to say that the linguist tries to discover and record the rules to which the members of a language-community actuallyconform and does not seek to impose upon them other rules, or norms, of correctness.Prescriptive linguistics aims to lay down rules for the correct use of language and settle the disputes over usage once and for all.For example, “Don’t say X.” is a prescriptive command; “People don’t say X.” is a descriptive statement. The distinction lies in prescribing how things ought to be anddescribing how things are. In the 18th century, all the main European languages werestudied prescriptively. However, modern linguistics is mostly descriptive because thenature of linguistics as a science determines its preoccupation with description insteadof prescription.1.9.2A synchronic study takes a fixed instant (usually at present) as its point ofobservation. Saussure’s diachronic description is the study of a language through thecourse of its history. E.g. a study of the features of the English used in Shakespeare’stime would be synchronic, and a study of the changes English has undergone since thenwould be a diachronic study. In modern linguistics, synchronic study seems to enjoypriority over diachronic study. The reason is that unless the various state of a languageare successfully studied it would be difficult to describe the changes that have takenplace in its historical development.1.9.3Saussure distinguished the linguistic competence of the speaker and the actual phenomena or data of linguistics as langue and parole. Langue is relative stable andsystematic, parole is subject to personal and situational constraints; langue is not spokenby an individual, parole is always a naturally occurring event. What a linguist should do,according to Saussure, is to draw rules from a mass of confused facts, i.e. to discoverthe regularities governing all instances of parole and make them the subject oflinguistics.1.9.4According to Chomsky, a language user’s underlying knowledge about the system of rules is called the linguistic competence, and the actual use of language in concretesituations is called performance. Competence enables a speaker to produce andunderstand and indefinite number of sentences and to recognize grammatical mistakesand ambiguities. A speaker’s competence is stable while his performance is ofteninfluenced by psychological and social factors. So a speaker’s performance does notalways match his supposed competence. Chomsky believes that linguists ought to studycompetence, rather than performance. Chomsky’s competence-performance distinctionis not exactly the same as, though similar to, Saussure’s langue-parole distinction.Langue is a social product and a set of conventions of a community, while competenceis deemed as a property of mind of each individual. Saussure looks at language morefrom a sociological or sociolinguistic point of view than Chomsky since the latter dealswith his issues psychologically or psycholinguistically.1.9.5 Etic vs. emicBeing etic means researcher s’making far too many, as well as behaviorally and inconsequential, differentiations, just as often the case with phonetics vs. phonemicsanalysis in linguistics proper.An emic set of speech acts and events must be one that is validated as meaningful via final resource to the native members of a speech community rather than via appealto the investigator’s ingenuity or intuition alone.Chapter 2 Speech Sounds2.1 Speech production and perceptionPhonetics is the study of speech sounds. It includes three main areas:1. – the study of the production of speech sounds2. –the study of the physical properties of the sounds produced inspeech3. – the study of perception of speech soundsMost phoneticians are interested in articulatory phonetics.2.3 Segments, divergences, and phonetic transcription2.3.2 Phonetic transcriptionInternational Phonetic Alphabet (IPA): the system of symbols for representing the pronunciation of words in any language according to the principles of the InternationalPhonetic Association. The symbols consist of letters and diacritics. Some letters aretaken from the Roman alphabet, some are special symbols.2.4 Consonants2.4.3 Manners of articulation1. Stop/plosive:2. Fricative:3. (Median) approximant:4. Lateral (approximant):2.4.4 Places of articulation1. Bilabial: A speech sound which is made with the two lips.2. Labiodental: A speech sound which is made with the lower lip and the upperfront teeth.3. Dental: A speech sound which is made by the tongue tip or blade and theupper front teeth.4. Alveolar: A speech sound which is made with the tongue tip or blade and thealveolar ridge.5. Postalveolar: A speech sound which is made with the tongue tip and the backof the alveolar ridge.6. Retroflex: A speech sound which is made with the tongue tip or blade curledback so that the underside of the tongue tip or blade forms a stricture with theback of the alveolar ridge or the hard palate.7. Palatal: A speech sound which is made with the front of the tongue and thehard palate.8. Velar: A speech sound which is made with the back of the tongue and the softpalate.9. Uvular: A speech sound which is made with the back of the tongue and theuvula, the short projection of the soft tissue and muscle at the posterior end ofthe velum.10. Pharyngeal: A speech sound which is made with the root of the tongue and thewalls of the pharynx.11. Glottal: A speech sound which is made with the two pieces of vocal foldspushed towards each other.2.4.5 The consonants of EnglishReceived Pronunciation (RP): The type of British Standard English pronunciation which has been regarded as the prestige variety and which shows no regional variation.It has often been popularly referred to as “BBC English” or “Oxford English” because itis widely used in the private sector of the education system and spoken by mostnewsreaders of the BBC network.[b] voiced bilabial stop[s] voiceless alveolar fricative[z] voiced alveolar fricative[m] bilabial nasal[n] alveolar nasal[l] alveolar lateral[j] palatal approximant[h] glottal fricative[r] alveolar approximantChapter 3 Lexicon3.1 What is word?1. What is a lexeme?A lexeme is the smallest unit in the meaning system of a language that can bedistinguished from other similar units. It is an abstract unit. It can occur in many different forms in actual spoken or written sentences, and is regarded as the same lexeme even when inflected. E.g. the word “write” is the lexeme of “write, writes, wrote, writing and written.”2. What is a morpheme?A morpheme is the smallest unit of language in terms of relationship betweenexpression and content, a unit that cannot be divided into further smaller units without destroying or drastically altering the meaning, whether it is lexical or grammatical. E.g. the word “boxes” has two morphemes: “box” and “es,” neither of which permits further division or analysis shapes if we don’t want to sacrifice its meaning.3. What is an allomorph?An allomorph is the alternate shapes of the same morpheme. E.g. the variants of theplurality “-s” makes the allomorphs thereof in the following examples: map – maps, mouse –mice, ox – oxen, tooth – teeth, etc.4. What is a word?A word is the smallest of the linguistic units that can constitute, by itself, a completeutterance in speech or writing.3.1.1 Three senses of “word”1. A physically definable unit2. The common factor underlying a set of forms3. A grammatical unit3.1.2 Identification of words1. Stability2. Relative uninterruptibility3. A minimum free form3.1.3 Classification of words1. Variable and invariable words2. Grammatical words and lexical words3. Closed-class words and open-class words4. Word class3.2 The formation of word3.2.1 Morpheme and morphologyMorphology studies the internal structure of words, and the rules by which words are formed.3.2.2 Types of morphemes1.Free morphemes: Those which may occur alone, that is, those which may constitute words by themselves, are free morphemes.Bound morphemes: Those which must appear with at least another morpheme are called bound morphemes.2. Root, affix and stemA root is the base form of a word that cannot further be analyzed. An affix isthe collective term for the type of formative that can be used only when added toanother morpheme. A stem is any morpheme or combination of morphemes towhich an inflectional affix can be added.3. Inflectional affix and derivational affixInflection is the manifestation of grammatical relationships through the addition of inflectional affixes, such as number, person, finiteness, aspect and case,which do not change the grammatical class of the stems to which they are attached.The distinction between inflectional affixes and derivational affixes is sometimes known as a distinction between inflectional morphemes andderivational morphemes. We can tell the difference between them with thefollowing ways:(1) Inflectional affixes very often add a minute or delicate grammaticalmeaning to the stem. E.g. toys, walks, John’s, etc. Therefore, they serveto produce different forms of a single word. In contrast, derivationalaffixes often change the lexical meaning. E.g. cite, citation, etc.(2) Inflectional affixes don’t change the word class of the word they attachto, such as flower, flowers, whereas derivational affixes might or mightnot, such as the relation between small and smallness for the former, andthat between brother and brotherhood for the latter.(3) Inflectional affixes are often conditioned by nonsemantic linguisticfactors outside the word they attach to but within the phrase or sentence.E.g. the choice of likes in “The boy likes to navigate on the internet.” isdetermined by the subject the boy in the sentence, whereas derivationalaffixes are more often based on simple meaning distinctions. E.g. Thechoice of clever and cleverness depends on whether we want to talkabout the property “clever” or we want to talk about “the state of beingclever.”(4) In English, inflectional affixes are mostly suffixes, which are alwaysword final. E.g. drums, walks,etc. But derivational affixes can beprefixes or suffixes. E.g. depart, teacher, etc.3.2.3 Inflection and word formation1. InflectionInflection is the manifestation of grammatical relationships through the addition of inflectional affixes, such as number, person, finiteness, aspect and case,which do not change the grammatical class of the stems to which they are attached.2. Word formationWord formation refers to the process of word variations signaling lexical relationships. It can be further subclassified into the compositional type (compound)and derivational type (derivation).(1) CompoundCompounds refer to those words that consist of more than one lexical morpheme, or the way to join two separate words to produce a single form,such as ice-cream, sunrise, paper bag, railway, rest-room, simple-minded,wedding-ring, etc.The head of a nominal or an adjectival endocentric compound is deverbal, that is, it is derived from a verb. Consequently, it is also called a verbalcompound or a synthetic compound. Usually, the first member is a participantof the process verb. E.g. Nouns: self-control, pain-killer, etc. Adjectives:virus-sensitive, machine washable, etc. The exocentric compounds are formedby V + N, V + A, and V + P, whereas the exocentric come from V + N and V+ A. E.g. Nouns: playboy, cutthroat, etc. Adjectives: breakneck, walk-in, etc.(2) Derivation。

CDMTCSResearchReportSeriesHigman’s EmbeddingTheorem.An Elementary ProofLuminit¸a DediuUniversity“Dun˘a rea de Jos”,RomaniaCDMTCS-010December1995Centre for Discrete Mathematics andTheoretical Computer ScienceHigman’s Embedding Theorem.An Elementary Proof∗Luminit¸a Dediu†AbstractIn1961G.Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions aboutfinitely presented groups.The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman’s Theorem.All the necessary preliminary material,including elements of group theoryand recursive functions theory,is systematically presented and with complete proofs.The aquaintedreader may skip thefirst sections and proceed immediately to the last.1SubgroupsWe assume familiarity with the concept of subgroup.We shall use the standard notation,i.e.H≤G means that H is a subgroup of G.Fact1.1.If H is a subgroup of a group G and K is a subset of H,then K is a subgroup of H iffK is a subgroup of G.Fact1.2.For any family{H i}i∈I of subgroups of a group G the intersectioni∈I H i is also a subgroupof G.Proof.First,we havei∈I H i=∅.Indeed,if H i≤G,for all i∈I,then1∈H i,i∈I,so1∈i∈IH i.Letx,y∈i∈I H i.Then x,y∈H i for all i∈I.As H i≤G,for all i∈I,xy−1∈H i,hence xy−1∈i∈IH i.PFact1.3.Let f:G−→G be a group morphism,H⊆G and K⊆G .a)If H is a subgroup of G,then f(H)is a subgroup of G .b)If K is a subgroup of G ,then f−1(K)is a subgroup of G.∗Paper written to be admitted for PhD studies at Bucharest University under the guidance of Professor Cristian Calude.†Department of Mathematics,University“Dun˘a rea de Jos”of Galatzi,RomaniaProof.Since H≤G we have1G∈H and f(1G)=1G ∈f(H)(f is a morphism).Hence f(H)=∅. Let x,y∈f(H).There exist two elements h1,h2∈H such that x=f(h1),y=f(h2).Then we can write(using the properties of f):xy−1=f(h1)[f(h2)]−1;since H≤G it follows that h1h−12∈H,so xy−1=f(h1h−12)∈f(H).The proof of the second statement is similar:if K≤G ,then1G∈K,so1G=f−1(1G )∈f−1(K) and f−1(K)=∅.Accordingly,if x,y∈f−1(K),then for some k1,k2∈K we have x=f−1(k1),y= f−1(k2)and xy−1=f−1(k1)[f−1(k2)]−1=f−1(k1k−12)∈f−1(K)(K≤G and k1k−12∈K).PFrom Fact1.3the preimage under the morphism f of the trivial subgroup{1G }of G is a subgroup of G.This is a special subgroup and it is called the kernel of the morphism f.We denote it by Kerf:Kerf={x∈G:f(x)=1G }.In the same way,the image by f of the group G,denoted by Imf or f(G),is a subgroup of G .2Generated Subgroups.GeneratorsLet E be a subset of a group G.Definition2.1The subgroup generated by the subset E(in G)is the intersection of all the subgroups of G containing E.Such subgroups do exist:for instance,the trivial subgroup G.Proposition2.2The subgroup G generated by a subset E consists of allfinite products of elements of E and inverses of these elements.Proof.Following the definition we have:H.G =H≤GH⊇ELet G be the set of allfinite products of elements of E and their inverses.We shall prove that G =G .By definition it follows that E⊆G .On the other hand,from Fact1.2,G is a subgroup of G and it is immediate now that G ⊆G .Conversely,it is sufficient to show that G ≤G and G ⊇E.First,note that G contains the“simple products”,i.e.all products formed by a single element x of E or x−1,so E⊆G .Let a=x1x2...x m and b=y1y2...y n be two elements of G ,where x i,y j∈E,1≤i≤m,1≤j≤n.Therefore we have:ab−1=(x1...x m)(y1...y n)−1=x1...x m y−1n...y−11,and this is afinite product of elements of E and inverses of elements of E,hence ab−1∈G .PFrom2.2it follows that if H1,H2are subgroups of a group G,then the subgroup generated by H1,H2 is formed by allfinite products of elements in H1,H2and their inverses.We shall denote this subgroup by H1,H2 or Gp{H1,H2}.As a particular case,if G is a commutative group,then the subgroup H1,H2 consists of all elements of the form h1h2with h1∈H1,h2∈H2.This subgroup is denoted by H1H2and we call it the product of subgroups H1,H2.Definition2.3A group G is said to befinitely generated if there exists afinite set E of elements of G that generates it.If E generates G we say that E is a set of generators for G.Example1.The aditive group of integers(Z Z,+)can be generated by the element1or by its oposite -1,since every integer n=0can be written as a sum of n terms equal to1if n>0or as a sum of−n terms equal to-1if n<0.Groups that can be generated by a single element are called cyclical groups.3Equivalence Relations.Quotient SetLet A be a set.Definition3.1A binary relation,usually denoted by“∼”,is said to be an equivalence relation on A provided the following three conditions hold:i)a∼a(reflexivity),ii)a∼b=⇒b∼a(symmetry),iii)a∼b,b∼c=⇒a∼c(transitivity).Examples2.1.Let n be a positive integer and define on Z Z the following binary relation denoted by“≡mod n”:for every a,b∈Z Za≡b mod n⇐⇒n/(a−b).2.The divisibility relation on I N is not an equivalence relation since it is not symmetric.If“∼”is an equivalence relation on A,then for each a∈A we define the set:ˆa={b∈A:b∼a}called the equivalence class of the elementˆa.Theorem3.2The equivalence classes determined by“∼”on A have the following properties:1)a∈ˆa,for all a∈A(henceˆa=∅),2)ˆa=ˆb⇐⇒a∼b,3)for all a,b∈A:ˆa=ˆb orˆa ˆb=∅,4)A=a∈Aˆa.Proof.1).Since a∼a it follows that a∈ˆa and soˆa=∅.2).From1)we have a∈ˆa.Ifˆa=ˆb,then a∈ˆb i.e.a∼b.Assume now that a∼b.Let x∈ˆa,then x∼a and,by transitivity,x∼b,i.e.x∈ˆb.This way we haveˆa⊆ˆb.Changing roles between a and b weobtainˆb⊆ˆa.Henceˆa=ˆb.3).Let a,b∈A such thatˆa ˆb=∅.Then there exists x∈ˆaˆb,i.e.x∼a and x∼ingtransitivity we have a∼b and from2)it followsˆa=ˆb.4).For every set A we can write A=a∈A a.From1)a∈ˆa,then A=a∈Aˆa.POne can note that using3)we can write A as a union of sets pairwise disjoint.In this case,we say that the equivalence classes of A realise a partition of A.The set of equivalence classes determined by the relation“∼”on A is denoted by A/∼and is said to be the quotient set of A relative to“∼”.The map p:A−→A/∼which carries every element a of A into his equivalence classˆa is a surjection that we call the canonical surjection.Definition3.3Let H be a subgroup of a group G.We define on G the relation“≡l mod H”given by:x≡l y mod H⇐⇒x−1y∈H for all x,y∈G,and we call it the left congruence modulo H.We shall prove that the above relation is an equivalence on G.For all x,y,z∈G we have(using the properties of a subgroup):i)x−1x=1∈H=⇒x≡l x mod H.ii)By definitionx≡l y mod H⇐⇒x−1y∈Hso,(x−1y)−1∈H⇐⇒y−1x∈H⇐⇒y≡l x mod H.iii)By definitionx≡l y mod H,y≡l z mod H⇐⇒x−1y∈H,y−1z∈Hso,(x−1y)(yz−1)∈H⇐⇒x−1z∈H⇐⇒x≡l z mod H.P Thus we can construct the quotient set G/≡l mod H,which is usually denoted by(G/H)l;its elements are called left equivalence classes modulo H.Following the definition,the class of an element x∈G is the set defined by{y∈G:x−1y∈H}={y∈H:y∈xH}=xH.If X∈(G/H)l,i.e.X is an equivalence class,an element x∈X is called a representative of X. Obviously,x∈G is a representative of X iffX=xH.Note that if x,y∈G and xH=yH,then x≡l y mod H and x−1y∈H.Similarly we can define the right congruence modulo H byx≡r y mod H⇐⇒xy−1∈H.The right equivalence class modulo H of an element x∈G is Hx and the quotient set G/≡r mod H is denoted by(G/H)r.Proposition3.4The quotient sets(G/H)l and(G/H)r have the same cardinal.Proof.Let X∈(G/H)l,X=xH with x∈G.Then we have:X−1=(xH)−1=H−1x−1=Hx−1∈(G/H)r.(We have used the relation H−1=H.Indeed,if x∈H we can put x=(x−1)−1and since x−1∈H(H is a subgroup),then x=(x−1)−1∈H−1.If x−1∈H−1,then x∈H,so x−1∈H.Hence H−1=H.) In the same manner as above we can prove that for every Y∈(G/H)r one has Y∈(G/H)l.In fact we can define the mapsΦ:(G/H)l−→(G/H)r carrying X onto X−1andΨ:(G/H)r−→(G/H)l carrying Y onto Y−1.These maps are inverse one to the other:Φ(Ψ(Y))=Φ(Y−1)=(Y−1)−1=Y,Ψ(Φ(X))=Ψ(X−1)=(X−1)−1=X,so that they establish a bijection between(G/H)l and(G/H)r.PExample 3.Consider S 3the group of all permutations of a 3element set andH = 123123 , 123213a subgroup of S 3.We shall construct the left and right equivalence classes modulo H.For all σ,τ∈S 3we have:σ≡l τmod H ⇐⇒σ−1τ∈H .On the other hand one can note that σ∈H iffσ(3)=3,hence σ−1τ∈H ⇐⇒(σ−1τ)(3)=3⇐⇒τ(3)=σ(3).In this way we get the following relation:σ≡l τmod H ⇐⇒σ(3)=τ(3).The left equivalence classes modulo H are:C l 1=H,C l 2= 123132 , 123312 ,C l 3= 123231 , 123321.Similarly,we deduce that σ≡r τmod H ⇐⇒σ−1(3)=τ−1(3).Since for every transposition τ=(i,j )∈S 3we have τ−1=τ,the inverses of the elements of S 3are:e −1=e,(1,2)−1=(1,3), 123312 −1= 123231,and we obtain the following right classes modulo H :C r 1= 123312 ,(1,3) ,C r 2= 123231 ,(2,3) ,C r 3= 123123,(1,2) .We can see now that the quotient sets (S 3/H )l and (S 3/H )r are not equal.4Normal Subgroups.Quotient GroupsProposition 4.1Let H be a subgroup of a group G .The following statements are equivalent:a)xHx −1⊆H ,for all x ∈G ,b)xHx −1=H ,for all x ∈G ,c)xH =Hx ,for all x ∈G ,d)(G/H )l =(G/H )r .Proof.•a)=⇒b):For all x ∈G we have xHx −1⊆H ;but x −1∈G and thus x −1Hx ⊆H .Putting H =x (x −1Hx )x −1we obtain H ⊆xHx −1(from the last inclusion),hence H =xHx −1.•b)=⇒a)is immediate.•b)=⇒c):xHx −1=H and by multiplication to right by x we obtain xH =Hx .•c)=⇒b):xH =Hx =⇒xHx −1=(Hx )x −1=H .•c)=⇒d):For all x ∈G we denote by ˆx l (and respectively ˆxr )the left (right)equivalence class modulo H.We saw that ˆx l =xH (ˆx r =Hx ).From c)we obtain ˆx l =ˆx r ,for all x ∈G ,hence (G/H )l =(G/H )r .•d)=⇒c):Let x∈G.Thenˆx l=xH∈(G/H)l=(G/H)r,so we canfind y∈G such thatˆx l=y r,i.e.xH=Hy.Putting x=x.1∈xH=Hy it follows that x∈Hy,i.e.x≡r y mod H.Hencex is a representative of Hy and so we can write Hx=Hy.Now we have Hx=Hy=xH,i.e.xH=Hx.P Definition4.2A subgroup H of a group G is said to be normal if it satisfyes one of the equivalent conditions in Proposition4.1.We write this by H¢G.Examples4.1.G and e are(trivial)normal subgroups of G.2.If G is an abelian group,then for all x∈G and for any subgroup H of G we have xH=Hx.Therefore,any subgroup of an abelian group is a normal subgroup.3.Let H be the subgroup considered in Example3.As we have seen,the sets of left and rightequivalence classes modulo H are not equal,hence H is not a normal subgroup.4.Any intersection of normal subgroups is a normal subgroup.Definition4.3Let R be a subset of a group G.The normal subgroupR=N.N¢GN⊇Ris called the normal closure of R in G.If N is a normal subgroup of G,then by definition the left and right congruence relations modulo N on G coincide,hence we can simply talk about the congruence relation on G modulo N.We shall denote it by“≡mod N”and the quotient set by G/N.Proposition4.4If N is a normal subgroup of a group G,then G/N can be organized with a group structure and the canonical surjection p:G−→G/N,p(x)=ˆx,for all x∈G becomes a group morphism.Proof.For all x,y∈G we define the product of two elementsˆx,ˆy∈G/N byˆx.ˆy= x y.First we have to check that the operation given above makes sense:ifˆx=ˆx andˆy=ˆy ,then x−1.x ∈N and y−1.y ∈N,hence there exist h1,h2∈N such that h1=x−1x and h2=y−1y .It follows that x =xh1,y =yh2,so we deduce that x y =(xh1)(yh2)=x(h1y)h2.Since N is a normal subgroup we get Ny=yN,i.e.there exists h3∈N such that h1y=yh3:x y =x(yh3)h2=(xy)(h3h2)∈(xy)N.Accordingly x y ≡(xy)mod N,and so the definition of the internal operation is correctly defined.We show now that G/N is a group:•Associativity:for allˆx,ˆy,ˆz∈G/N we have(ˆxˆy)ˆz=( x y)ˆz=x(yz)=ˆx( y z)=ˆx(ˆyˆz).(xy)z=•If e denotes the identity element of G,then for each x∈G/N we haveˆxˆe= x e=ˆx= e x=ˆeˆx, henceˆe is the identity element of G/N.•The inverse of an elementˆx isx−1.Indeed:x−1.xx−1=ˆxx−1x=ˆe=x−1ˆx=Tofinish the proof we have to verify that the canonical surjection p that carries an element x∈G into his classˆx∈G/N is a group morphism:p(xy)=p(x)p(y).Indeed:p(xy)= x y=ˆxˆy=p(x)p(y) and this concludes the proof.P Definition4.5The group G/N costructed above is called the quotient group of G relative to the normal subgroup N.Example5.Let us construct the quotient groups of the aditive group(Z Z,+).We assume familiarity with the fact that all the subgroups of Z Z have the form nZ Z with n≥0.In the same time,(Z Z,+)is an abelian group,so it is clear that any subgroup of it is normal.Hence we want to construct a quotient group of the form Z Z/nZ Z,n≥0.Then,we consider two cases:•n=0:Z Z/(0)=Z Z.•n>0:x≡y mod nZ Z⇐⇒x−y∈nZ Z⇐⇒n/(x−y)⇐⇒x≡y mod n, so the congruence modulo nZ Z on Z Z is reduced to the congruence modulo n on Z Z.Finally,Z Z/nZ Z=Z Z/(≡mod n)=Z Z n.5The Fundamental Isomorphism Theorem for GroupsAs we have seen before if f is a group morphism from G into G ,then Kerf is a subgroup of G.We shall prove here that Kerf is a normal subgroup(that is Kerf satisfies one of the equivalent conditions of4.1)and then we shall construct the quotient group G/Kerf.Let x∈G and y∈Kerf.Then xyx−1∈xKerfx−1and f(xyx−1)=f(x)f(y)f(x−1).Since y∈Kerf,f(y)=e (where e is the identity element of G )andf(xyx−1)=f(x)e f(x−1)=f(xx−1)=f(e)=e .Hence xyx−1∈Kerf for all x∈G and y∈Kerf,thus Kerf is normal in G.Now we may consider the quotient group G/Kerf and present one of the theorems which play a major role in what will follow.Theorem5.1(The Fundamental Isomorphism Theorem)If f:G−→G is a group morphism, then there exists a group isomorphism between G/Kerf and Imf,i.e.G/Kerf Imf.Proof.Define f:G/Kerf−→Imf⊆G by f(ˆx)=f(x).First verify that the definition makes sense: letˆx,ˆy∈G/Kerf such thatˆx=ˆy.That means that x≡y mod(Kerf),hence x−1y∈Kerf and so f(x−1y)=e .Since f is a morphism we can write:e =f(x−1y)=[f(x)]−1f(y);it follows that f(x)=f(y),therefore f(ˆx)=f(ˆy).By definition Imf={f(ˆx):ˆx∈G/Kerf}={f(x):x∈G}=Imf,thus f is a surjection.In order to show that f is injective,letˆx,ˆy∈G/Kerf be such that f(ˆx)=f(ˆy).It follows that f(x)=f(y)and hencef(x)[f(y)]−1=e ⇐⇒f(xy1)=e ⇐⇒xy−1∈Kerf⇐⇒ˆx=ˆy,proving that f is bijective.All that remains to do is to prove that f is a group morphism.Indeed,since f is a morphism we get:f(ˆxˆy)=f( x y)=f(xy)=f(x)f(y)=f(ˆx)f(ˆy),for allˆx,ˆy∈G/Kerf.PAs a corrollary we note that in case f is a surjective group morphism,then G =Imf,and thus G is isomorphic to G/Kerf.This result is very often used to establish isomorphisms between groups.Example6.Let(I C∗,.)be the multiplicative group of the non-zero complex numbers and let f:I C∗−→I R∗+be the morphism defined by f(z)=|z|,for all z∈I C∗.It is immediate that f is a surjective group morphism:for all a∈I R∗+there exists z∈I C∗,z=a+0i,such that f(z)=f(a+0i)=a.For all z1,z2∈I C∗we havef(z1z2)=|z1z2|=|z1||z2|=f(z1)f(z2),thus f is a surjective group morphism from(I C∗,.)into(I R∗+,.)and from the fundamental theorem we deduce that I R∗+ I C∗/Kerf whereKerf={z∈I C∗:|z|=1}.6Free Groups.Defining RelationsLet M be afinite(or not)set of symbols(letters)xα,xβ,...and assign to these symbols(by an one-to-one correspondence)another set of symbols denoted by x−1α,x−1β,...called inverses of symbols inM.Definition6.1A word is afinite ordered sequence of symbols(letters)of the form w=xε1α1...xεnαn,εi=+1,1≤i≤n,where no succesive symbols are inverse one to the other,i.e.xεiαi=x−εi−1αi−1,2≤i≤n.The number n is called the length of the word and is usually denoted by|w|=n.The word that contains no symbol is called the empty word and is denoted by1orλ.Its length is obviously0.Example7.Consider M={a,b,c}.Then w1=ab−1,w2=ba−1cb−1a are words and|w1|=2,|w2|= 5.Let X be the set containing the symbols of M and their inverses.The set of all words written with the symbols in X becomes a group together with the following internal operation called juxtaposition:if w1=xε1α1...xεnαn,εi=+1,1≤i≤n,w2=xδ1β1...xδmβm,δj=+1,1≤j≤m are words,their productis defined byw1w2=xε1α1...xεnαnxδ1β1...xδmβm,i.e.by joining the second word to thefirst.If xαn =xβ1andεn+δ1=0(i.e.xεnαnand xδ1β1are inverse)we can cancel the sequence involved andrepeat the operation with the next symbols if necessary.Letter cancelation in a word product is also calledreduction.For instance,if w1=xαx−1βxδand w2=x−1δxβxαxδ,then w1w2=xαx−1βxδx−1δxβxαxδ=xαxαxδ.As we see in this example,some sequences of identical repeated letters of the formxx (x)n times or x−1x−1...x−1n timesmay appear.We agree to denote these sequences by x n and,respectively,x−n.Thus,a word can bedefined as beeing an orderedfinite sequence of letters of the form w=xε1α1...xεnαn,withεi∈Z Z,1≤i≤nand xεiαi =x−εi−1αi−1(no cancelation is possible).In this way the word w=xαxαxβx−1δx−1δx−1δcan berepresented as w=x2αxβx−3δ.The juxtaposition operation is associative.The empty word1is considered to be the identity elementof the set of all words.The inverse of a word w=xε1α1...xεnαnis the word w−1=x−εnαn...x−ε1α1.So wecan speak by now of the group of words written with the symbols of X,called the free group generated by M.The elements of M are called free generators of the free group.The cardinality of M is said to be the rank of the free group.Theorem6.2Any group is isomorphic to some quotient group of a free group.Proof.Let G be a group and M a set of generators of G.Denote those generators by aα,aβ,.... Consider a free group F whose generators system has the same cardinality as M.Between the elements of M and the free generators of F we can establish a bijection and we agree to denote by xαthe element of F associated to aαin M.The map aα xαdefines a group morphism which associates to eachelement g∈G,g=aε1α1...aεnαna word w=xε1α1...xεnαn,εi=+1,1≤i≤n(orεi∈Z Z).According tothe fundamental isomorphism theorem we have G F/N,where N is the kernel of the above morphism. PRemarks.1)The normal subgroup N contains the words whose image in G is equal to the identity.2)The representation of a group as a quotient group of a free group is not unique:it depends on thechoice of M.Keeping the same notations as above,consider a group G.We have seen that there exists a free groupF and a normal subgroup N of F such thatG F/N.Let w∈N,w=xε1α1...xεnαn,εi∈Z Z,1≤i≤n.The image in G of w is the identity element and so we obtain an equality of the formaε1α1...aεnαn=e(1)called relation between the generators aα1,aα2,...of the group G.If we consider in N a subset R such that the normal subgroup generated by R in F coincides to N, then the system of relations of the form(1)which corresponds in G to elements of R is called a set of defining relations of the group G.All remaining relations between the generators of G are considered as consequences of the defining relations,since every element of N can be generated by the elements of R and their conjugates.We state that the group G is completely defined when the defining relations are given.Since R generates N,then the quotient group F/N is completely determined.We conclude that any group can be described by a set of defining relations between some given generators.Thus,a group can be viewed as a pair of sets(X,R),where X is a set of generators and R is the set of defining relations(by a mild abuse of notation we use the letter R also for the set of defining relations).Section8deals with some more details.Examples8.1.Let us define a group that has a single generator a and a single defining relation a n=e.The freegroup F generated by the symbol x,x←→a,is the cyclical infinite group F= x .The set of words corresponding to the defining relation contains a single element:R={x n}.The normal subgroup generated by x n in F is N= x n .Thus G= a x / x n .But since any cyclical infinite group is isomorphic to the aditive group of integers(Z Z,+)and this isomorphism brings x n onto nZ Z,we conclude that:G x / x n Z Z/nZ Z=Z Z n.2.The symmetric group S3can be defined by two generators a and b and the defining relationsa3=e,b2=e,abab=e,where a=123231,b=(1,2).The reader can easily check that ab=(1,3),ba=(2,3).Since any permutation is a product of transpositions it follows that a and b generate S3.If we consider the free group F generated by a and b,then S3 F/H and F will contain the following6elements:a,b,a2,b2=e,ab,ba=a2b.Indeed,since abab=e we get:ab=b−1a−1=⇒a=b−1a−1b−1=⇒ba=a−1b−1;on the other hand,a3=e=⇒a−1=a2and b2=e=⇒b=b−1,so ba=a2b;any other combination of a’s and b’s gives one of the above6elements.Thus we can represent S3by its generators and defining relations as follows:S3= a,b;a3=e,b2=e,abab=e .7Recursive Functions and SetsHistorically,the notion of a computable function has been developed between1931-1947by different, but equivalent,tools:formal equations(J.Herbrand,K.G¨o del,S.C.Kleene),Turing machines(A.M. Turing),Post systems(E.L.Post),and Markov normal algorithms(A.Markov).A remarkable fact is that all these notions are equivalent,in the sense that they“generate”the same class of functions defined on I N,the recursive functions.Definition7.1A function f:I N k−→I N is said to be computable if there exists an algorithm which computes f(n1,...,n k),for every(n1,...,n k)∈I N k.Examples9.1.Let f(x)be the x th prime ing as algorithm the method described by the sieve ofEratostheneses we can compute f.2.Let f(x,y)be the greatest common divisor(g.c.d.)of the positive integers x,y.This function,aswe all kwow,can be computed by Euclid’s algorithm(considering f(x,0)=f(0,y)=0).Definition7.2The following functions are said to be initial(basic)functions:1.The projection functions:P(n)m ,1≤m≤n,defined by P(n)m(x1,...,x n)=x m.2.The constant functions:C(n)m ,m∈I Nfixed,defined by C(n)m(x1,...,x n)=m.3.The succesor function defined by Succ(x)=x+1.Definition7.3A function defined on I N or I N k is said to be recursive if it is an initial function or if it can be generated by an initial function in afinite number of steps,using the following three rules:a)Functional composition(substitution).If f is a k-variables function,and g1,...,g k arefunctions of n variables theng(x1,...,x n)=f(g1(x1,...,x n),...,g k(x1,...,x n))determinates a n-variables function.b)Primitive recursion.If f is a(k+1)-variables function and g is a(k−1)-variables function,then the following system of equations defines an unique k-variables function:f(x1,...,x k−1,0)=g(x1,...,x k−1),f(x1,...,x k−1,y+1)=h(x1,...,x k−1,y,f(x1,...,x k−1,y)).(Dedekind proved the existence and the unicity of these functions.)c)Minimization.If f is a(k+1)-variables function such that for every k-tuple(x1,...,x k)ofnatural numbers there exists a number y with f(x1,...,x k,y)=0,then one can determine a new function g by the condition:g(x1,...,x k)=µy[f(x1,...,x k,y)=0],whereµy[...]means“the least y such that...”.Examples10.1.The sum function f(x,y)=x+y can be obtained by primitive recursion from h(x,y,z)=z+1and P(1)1(x)=x as follows:f(x,0)=x+0=P11(x)=x,f(x,y+1)=x+(y+1)=(x+y)+1=h(x,y,x+y).On the other hand we can write:h(x,y,z)=Succ(P(3)3(x,y,z)), hence we conclude that h is recursive and therefore f will be recursive too.2.The product function f(x,y)=xy is recursive:f(x,0)=x.0=C(1)(x)=0,f(x,y+1)=x(y+1)=x.y+x.1=g(x,y,xy),where g(x,y,z)=P(3)1(x,y,z)+P(3)3(x,y,z)=x+z is a recursive function.3.In the same manner,the exponential function f(x,y)=x y is recursive:f(x,0)=x0=C(1)1(x)=1,f(x,y+1)=x y+1=x y.x=t(x,y,x y),where t(x,y,z)=P(3)1(x,y,z)P(3)3(x,y,z)is recursive(we agree that00=1).Fix a set S⊆I N.Following Leibniz there are two basic algorithmically ways to“define”S:a decision method and a generating ing the notion of recursive function,we can give a precise version of these two techniques:•The generating methodDefinition7.4A set S of natural numbers is called recursively enumerable if S=∅or if there exists a recursive function f:I N−→I N whose range coincides with S.We shall say in this case that the function f generates(enumerates)the elements of S.•The decision methodDefinition7.5A set S of natural numbers is said to be decidable(recursive)if its characteristicfunction f S,f S(n)=1,if n∈S,0,otherwise,is recursive.If f S is recursive,one can decide whether an arbitrary natural number n is or not an element of S.Examples11.The following sets are recursive:1.S={2n:n≥0},2.S={n:n is prime}.Remark.A set S is recursive iffboth S and its complement I N−S are recursively enumerable.So,if S is a recursive set,then S is recursively enumerable.The converse is not true(see Section9).The notions of recursive function,recursive and recursively enumerable set can be extended from I N to I N n and,then,to Z Z n,for every n>0.Theorem7.6Cantor’s function J:I N2−→I N defined byJ(x,y)=(x+y)(x+y+1)/2+xis a recursive bijection.The reader interested on the proof is referred to[3].Thus we have got an encoding scheme which allows us to identify,in a recursive manner,the sets I N and I N2.J is called the Cantor numbering and the number J(x,y)is referred to as the Cantor number associated to the pair(x,y).The decoding associates of J are usually denoted by K and L and so we have:J(K(z),L(z))=z,K(J(x,y))=x,L(J(x,y))=y.Cantor’s function can be extended to a recursive bijection J(n):I N n−→I N,for every n≥3withdecoding associates denoted by I(n)1,...,I(n)n.Definition7.7A functionΦ:I N n−→I N n is recursive if there exists a recursive funtion f:I N−→I N such thatΦ◦I=I◦f,where I:I N−→I N n,I=(I(n)1,...,I(n)n)is the inverse of the generalized Cantor’s bijection.Consider now a bijectionα:I N−→Z Z and its extensionα(n):I N n−→Z Z n,α(n)(x1,...,x n)=(α(x1),...,α(x n)).Definition7.8A functionΨ:Z Z n−→Z Z n is recursive if there exists a recursive functionΦ:I N n−→I N n such thatΨ◦α(n)=α(n)◦Φ.Thus,the definitions for recursively enumerable and recursive sets can be extended for the subsets of I N n and,respectively,of Z Z n.Let F be a non-empty set such that there exists a one-to-one correspondence f between F and the natural numbers.We say that f is a G¨o del numbering for F and the set Im f is called the set of code-numbers(or G¨o del numbers)of F.Definition7.9The set F is said to be recursive(recursively enumerable)if the corresponding set of code-numbers is recursive(recursively enumerable).8Recursively Presented Groups.Free Products with Amalga-mation.HNN-ExtensionsGroups are often described as quotient groups of some free groups:G F/N.If F is a free group with basis X and N is the normal closure in F of a set R,then we say that the pair(X;R)is a presentation for G,and-by convention-we write G=(X;G)(see examples given in Section6).If X is the set of images x in G of all elements x of X by the canonical surjection(x→x∈F/N)then, since a surjective morphism carries systems of generators into systems of generators,X is generating G. If r=r(x1,...,x n)is an element of R(hence a word written with generators x1,...,x n),then we get r(x1,...,x n)=e in G,that is a defining relation.The elements of R will be called relators.A presentation(X;R)isfinitely generated if X isfinite and isfinite if both X and R arefinite sets. In this last case we often say that the group G has afinite presentation or isfinitely presented.We say that a presentation(X;R)is recursive if X isfinite and R is recursively enumerable.。