AN ALGEBRAIC APPROACH TO CONSISTENCY CHECKING BETWEEN CLASS DIAGRAMS

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1 AN ALGEBRAIC AROACH TO CONSISTENC CHECKING BETWEEN CLASS DIAGRAMS HIDEKAZU ENJO, MOTONARI TANABU, JUNICHI IIJIMA NTT DATA Corporaton, okohama Natonal Unversty, Tokyo Insttute of Technology --9 Koto-ku, Tokyo 5-867, Japan, 79-4 Tokwada, Hodogaya, okohama , Japan, -- Ookayama, Meguro-ku, Tokyo , Japan , , ABSTRACT There are several dagram methods for data modelng lke a class dagram. It s very hard to descrbe a bg data model of a large enterprse system nto one dagram. A large set of partal data models are used durng desgnng an nformaton system for a large enterprse. The skll of modelers makes fluctuaton and dscrepancy among data models. It s necessary how to keep consstency among data models. There are two knds of nconsstency among data models descrbed n class dagrams. One s nconsstency between two data models lke dfferences between attrbutes, data types, or multplctes of same name classes. The other s nconsstency dependng on order of dvdes and mergers for data models. We present synta and semantcs of a class dagram descrbng a data model for foundaton for class dagram algebra. Then we ntroduce algebrac structure for syntactcal and semantcal merger operatons on class dagrams and the consoldaton condton. Satsfyng the consoldaton condton keeps consstency of class dagrams syntactcally and semantcally. We also show assocatve law and commutatve law of the syntactcal and semantcal merger operatons for the class dagram algebra that keeps avodng nconsstency dependng on order of dvde and mergng. Keywords: class dagram algebra, data modelng, consstency checkng

2 INTRODUCTION In order to desgn a large enterprse system to handle huge comple nformaton, t s mportant to analyze a lot of requrements and make a model of bg complcated nformaton. There are several dagram methods to support data modelng lke class dagrams. However, t s very hard to descrbe a bg data model of a large enterprse system nto one dagram. A large set of partal data models are used durng desgnng an nformaton system for a large enterprse. Ths stuaton rases a rsk of embeddng nconsstency because the skll of modelers makes fluctuaton and dscrepancy among data models. And nconsstency wthn the set of partal data models decreases the qualty of the data model. It s necessary how to keep consstency among data models. The Unfed Modelng Language (UML) (UML, 005) s a standard modelng language, especally for obect orented desgn. Class dagrams, whch are a type of UML dagrams, are descrbng statc vews of data models. So t s necessary to clear how to keep consstency among class dagrams. There are two knds of nconsstency among data models descrbed n class dagrams. One s nconsstency between two data models lke dfferences between attrbutes, data types, or multplctes of same name classes. The other s nconsstency dependng on order of dvdes and mergers for data models. Inconsstency between two parts of a model It s easy to merge two parts of model lke fg. and fg. and the result s fg. because same classes are merged nto one class n a result dagram and same assocatons are merged nto one assocaton of that. Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure Eample () of a artal Data Model Date:Date[..] Quote Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure Eample () of a artal Data Model

3 Quote Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure Result of Mergng Eample () and () However, t s not easy to merge two parts of a model f there are two classes of same name but dfferent set of attrbutes. For eample, fg. and fg.4 have a same named class Detals but a set of attrbutes are dfferent. A class named Detals n fg. has three attrbutes named Goods, Num and Remarks but that of fg.4 has only Goods and Num. Date:Date[..] Quote Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] rce: Int[..] Fgure 4 Eample of an Inconsstent Class There are several ways for mergng. For eample, one s a merge of two classes when same name and same set of attrbutes. The result s fg.5, when mergng fg. and fg.4 n the strct way. However, the class dagram of fg.5 s awkward because two classes wth same name but attrbutes are dfferent n one class dagram.

4 Quote Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Contact: Strng[..] Tel: Strng[..] Detals Detals Goods: Strng[..*] Num: Int[..] rce: Int[..] Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure 5 Result of Mergng Eample () and Eample of an Inconsstent Class Another way for mergng two class dagrams s to select the class n the frst dagram when there s an nconsstent class. Fg.6 s the result of mergng fg. and fg.4. Quote Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure 6 Another Result of Mergng Eample () and Eample of an Inconsstent Class Also assocatons named n fg. and fg.7 are dfferent because an assocaton named n fg. assocates among classes named, Detals and but that of fg.7 assocates among classes named, Detals and. Date:Date[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure 7 Eample of an Inconsstent Assocaton 4

5 There are also several ways for mergng two assocatons. One s a merger of two assocatons when same name and same set of related classes. The result s fg.8, when mergng fg. and fg.7 n the strct way. However, the class dagram of fg.8 s also awkward because two assocatons wth same name but set of related classes are dfferent n one class dagram. Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure 8 Result of Mergng Eample () and Eample of an Inconsstent Assocaton Modfcaton of assocatons s also other mergng way. The dagram of Fg.9 s the result of assocaton modfcaton durng mergng from a trple assocaton of, Detals and and that of, Detals and to a quadruple assocaton of, Detals,,. There are many ways of mergng but n ths paper we propose a smple operaton of mergng and constrants to class dagrams for clearng the algebrac structure of class dagrams. Date: Date[..] Contact: Strng[..*] Tel: Strng[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] Remarks: Strng[0..*] Fgure 9 Result of Mergng Eample () and Eample of an Inconsstent Assocaton wth Modfcaton 5

6 Inconsstency dependng on a sequence of mergng and dvson There are many sequences of mergng f many class dagrams are merged. For eample, one sequence s mergng a dagram A and a dagram B then mergng a dagram C lke ((A B) C). Another sequence s mergng a dagram B and a dagram C then mergng a dagram A lke ((B C) A). If mergng operaton s assocatve and commutatve, results of all mergng sequences are same. However, a mergng way selectng the class n the frst dagram volates commutatve law. For eample, fg.6 s the result of fg. and fg.4 wth selectng the class from a frst dagram and fg.0 s the result of fg.4 and fg.. Two dagrams are dfferent because mergng operaton wth selectng the class from a frst dagram sn t commutatve. So mergng operaton s preferred commutatve and assocatve. Addtonally, we propose a smple assocatve and commutatve operaton of mergng n ths paper. Quote Contact: Strng[..*] Tel: Strng[..] Date: Date[..] Contact: Strng[..] Tel: Strng[..] Detals Goods: Strng[..*] Num: Int[..] rce: Int[..] Fgure 0 Result of Eample Mergng an Inconsstent Class and Eample () RELATED WORK There are several studes related consstency analyses for class dagrams. Tsolaks and Ehrg (Tsolaks, 000) analyzed the consstency of class and sequence dagrams by usng attrbuted graph grammars. a and Kane (a, 00) analyzed the consstency of class and sequence dagrams. Kösters, S and Wnter (Kösters, 00) analyzed the consstency of use case and class dagrams. eung (eung, 004) analyzed the consstency of class and state dagrams. They focused between dfferent types of UML dagrams. Berard, Calvanese, and Gacomo (Berard, 005) presented the correspondence between class dagrams and Descrpton Logcs, whch enable us to utlze Descrpton Logcs based systems for reasonng on class dagrams. Szlenk (Szlenk, 006) presented the mathematcal defnton of class dagrams and studed the consstency wthn a sngle class dagram. Kanewa and Satoh (Kanewa, 006) propose optmzed algorthms that compute respectve consstences for class dagrams based on frst order predcate logc. Sabetzadeh and Easterbrook (Sabetzadeh, 005) studed the mergng of class dagrams to gan a unfed perspectve. However, they focused how to merge nconsstent and ncomplete class dagrams for requrement engneerng and they haven t been clear detal of algebrac structure for a mergng operaton. 6

7 NOTATION A set Name s a set of name of all elements consst of class dagram ncludng a class name, an assocaton name, an attrbute name and a role name. A set DataType s a set of data type, where type of all data ncludng nteger and character represented by {Int, Bool,Char, Strng, Date,...}. A multplcty (m, n) s a par of lower bound m and upper bound n, where 0 m n,m N 0,n N. A set N 0 s a set of nteger grater than 0. A set N s a set of nteger grater than. A denotes the set of all subsets of A. THE SNTA OF A CLASS DIAGRAM A class dagram s a type of UML and employed to model concepts n statc vews for an nformaton structure of enterprse system consstng of classes and ther nterrelatonshps. A class dagram has been represented by graphcal and abstract synta. In ths secton, we focus abstract synta for smplfyng dscussons. We defne the abstract synta of the class dagram as followed: Syntactcal Defnton of a class A class c s a par (Name(c), Attrs(c)), where Name(c) Name s a name of the class c and Attrs(c) s a lst of attrbutes n the class c. An attrbute lst Attrs(c) s a fnte lst (Attr(c,),...,Attr(c,n)) of fnte number n. An attrbute Attrs(c, ) s a trple of (Name(c, ),Type(c,),Mult(c,)), where Name(c, ) Name s a name of th attrbute and a member of a set Name, Type(c, ) DataType s a type of th attrbute and a member of a set DataType, Mult(c, ) Multplcty s a multplcty of th attrbute and a member of a set Multplc ty. A set Class s a set of all classes. Contact: Strng[..*] Tel: Strng[..] (,(Contact, Strng,(,*)),(Tel, Strng,(,))), Fgure Eample of an Class Syntactcal Defnton of an assocaton An assocaton a s a par (Name(a), Assocs(a)), where Name(a) Name s a name of the assocaton a and Assocs(a) s a lst of assocated classes n the assocaton a. An lst Assocs(a) of assocated classes s a fnte lst (Assoc(a,),...,Assoc(a, n)) of fnte number n. An assocated class Assoc(a, ) s a trple of (Role(a, ),Mult(a,), AC(a, )), where Role(a, ) Name s a name of th assocaton, Mult(a, ) Multplcty s a multplcty of th assocaton, AC(a, ) Class s a related class of th assocaton. A length of the lst Assocs(a) must be at least. A set Assocato n s a set of all assocatons. 7

8 Contact: Strng[..*] Tel: Strng[..] arty.. Detals..* Detals Goods: Strng[..*] Num: Int[..] Request 0.. Date: Date[..] (,((arty,(,),(,(contact,strng,(,*)),(tel, Strng,(,)))), (Request,(0,),(,(Date, Date,(,)))), (Detals,(,*),(Detals,(Goods, Strng,(,*)),(Num, Int,(,)))))) Fgure Eample of an Assocaton Syntactcal Defnton of a class dagram A class dagram s a par (C, A), where C Class s a subset of Class and A Assocaton s a subset of Assocato n. A class dagram (C, A) suffers followng condtons. Condton : f names of two classes are dfferent, two classes are dfferent. c C, c C, Name(c ) = Name(c ) c = c [] Condton : f names of two assocatons are dfferent, two assocatons are dfferent. a A, a A, Name(a ) = Name(a ) a = a [] Condton : all assocated classes of an assocaton are ncluded n the set of classes n the class dagram. a A, (r,m,ac ) Assocs(a) ac C [] In order to reduce the complety, we consder elmnatng some components as operaton and generalzaton. Because we dscuss only behavor of data model rather than obect model. Syntactcal algebra for class dagrams A merger operaton s defned n an analogous to unon of set because same classes are merged nto one class and same assocatons are merged nto one assocaton. We defne a merger operaton as followed: Defnton of a Syntactcal Merger Operaton Gven two class dagrams (C, A ) and (C, A ), a syntactcal merger operaton : (C, A) (C, A) (C, A) s defned as (C C, A A ). (C, A ) (C, A ) (C C, A A ) [4] 8

9 Contact: Strng[..*] Tel: Strng[..] arty.. Detals..* Detals Goods: Strng[..*] Num: Int[..] Request 0.. Date: Date[..] (,(Contact, Strng,(,*)),(Tel, Strng,(,))), (, (Data, Date,(,))) (Detals,(Goods, Strng,(,*)),(Num, Int,(,))) (,((arty,(,),(,(contact,strng,(,*)),(tel, Strng,(,)))), (Request,(0,),(,(Date, Date,(,)))), (Detals,(,*),(Detals,(Goods, Strng,(,*)),(Num, Int,(,)))))) Fgure Eample of a Class Dagram However, f ths syntactcal merger operaton s appled class dagrams of fg. and fg.4, the result of fg.5 volates condton of class dagram defnton. The result of ths merger operaton on class dagrams of fg. and fg.7 volates condton of class dagram defnton. Although we omt the proof because of space lmtatons, followng condtons are formed. roposton The syntactcal merger operaton ( ) on class dagrams s closed f and only f followng a consoldaton condton s satsfed. Consoldaton condton: Let (C,A ) and (C,A ) be class dagrams. If names of any classes n C and C are same, those classes are same. If name of any assocatons n A and A are same, those classes are same. c C, c a A, a C, Name(c A, Name(a ) = Name(c ) = Name(a ) c ) a = c = a [5] The syntactcal merger operaton ( ) s closed on the class dagrams whch satsfy the consoldaton condton. Ths operaton and condtons are smple enough for mergng. Although we omt the proof because of space lmtatons, the syntactcal merger operaton ( ) s assocatve and commutatve lke set operaton unon. It s a commutatve sem-group that the algebrac structure of class dagrams whch satsfy the consoldaton condton and the syntactcal merger operaton ( ). All results are same where any sequence of mergng wth the syntactcal merger operaton ( ). 9

10 THE SEMANTICS OF A CLASS DIAGRAM An nstance of a class s called an obect. An nstance of an assocaton s called a lnk. A lnk s a connecton between two or more obects of the classes. We ntroduce comple data as a doman and nterpretaton mappng for the semantcs of class dagram as followed: A Doman for the Semantcs of a Class Dagram ** A obect s a par (,(( v,,..., v, ),...,( v,,..., v, ))) ( ID Value ), where ID s a name of dentfer lke O, ( v,,..., v, ) s a value lst lke ( amada, Suzuk) and (( v,..., ),...,(,..., )), v, v, v, s a lst of value lst lke (( amada, Suzuk),( )). A set Obects s a set of all obects. O ( amada Suzuk ) ( O,(( amada, Suzuk),( ))) Fgure 4 Eample of an Obect * A lnk s a par ( y,( o,..., ok )) ( ID Obects ), where y ID s a name of dentfer lke L, ( o,..., o k ) s a lst of obects and k N. A set Lnks s a set of all lnks. L O ( amada Suzuk ) O ( L,( O,(( amada, Suzuk),( ))),( O,(( ))),( O4,(( Lnu),()))) O4 Lnu Fgure 5 Eample of a Lnk A comple data s a par ( O, L) ( Obects, Lnks), where O Obects s a set of obects, L Lnks s a set of lnks, satsfyng ( ( y,(,..., n)) L, k, k n, k O) that means all obects related all lnks L are ncluded n obects O. A set CompleDat a s a set of all comple data. 0

11 L O ( amada Suzuk ) L O O ( C Laptops ) O4 Lnu ({( O,(( amada, Suzuk),( ))),( O,(( ))), ( O,(( C, Laptops),())),( O4,(( Lnu),()))}, {( L,(( O,(( amada, Suzuk),( ))),( O,(( ))),( O,(( C, Laptops),())))), ( L,(( O,(( amada, Suzuk),( ))),( O,(( ))),( O4,(( Lnu),()))))}) Fgure 6 Eample of a Comple Data An Interpretaton Mappng for the Semantcs of a Class Dagram An nterpretaton mappng I( t) a t ( v) of a data type name t s a map to a predcate t ( v) v Dt, where D s all values of data type named t. For eample, data type Date s mapped to Date (v), t where (" " ) s true. Date An nterpretaton mappng I( c) a c ( ) of a class c s a map to a predcate c (), where number of attrbute wthn lower bound and upper bound and a value of a attrbute s a member of the data type related an attrbute. (( w,(( v ( w ID (,..., v,, a (z) ),( v ) ( k, k, )))), Strng ( v, k )) Strng ( v Fgure 7 Eample of a nterpretaton mappng for a Class An nterpretaton mappng I( a) a a ( z) of an assocaton a s a map to a (z), where all obects related lnks are assocated to classes defned as followed: ( a,..., n ))) y ID AC(a, ) ( )... AC(a, n) ( n )) [6], ))

12 a (z),, ))) y ID ( ) rder ( ) rderdeta ls ( ) Fgure 8 Eample of a nterpretaton mappng for an Assocaton An nterpretaton mappng I( Mult(a) ) a (, z) of a multplcty Mult(a) s a map to a Mult(a) predcate (, z), where a number of lnks related each obect s wthn lower bound and upper Mult(a) bound as followed: k, k n ((,..., ( Mult(a) AC(a, k ) ( o, L) AC(a, k ),..., a, k k r, o, s, o,... k+,...,... k+ rk n sk,..., n k... )) L y ID,..., k rk + k+ sk + k+... rn sn, y, )) L y ID,..., n, y, n a a k k, o,, o, k+ k+,..., n,..., n ))))), )))))) [7] (( ( ( ( ( ( Mult() ( o, L) Detals Detals 0,,,, * *,, Mult() a (, z), y,(( y,( o, Mult(), )) L y ID, y,(( y,( o,, )) L y ID, y,(( y,(, o, )) L y ID, y,(( y,(, o, )) L y ID 0, y,(( y,(,, o)) L y ID, y,(( y,(,, o)) L y ID o,, )))), o,, )))),, o, )))),, o, )))),,, o)))),,, o))))) Fgure 9 Eample of a nterpretaton mappng for a Multplcty of an Assocaton An nterpretaton mappng ((C, A) ) a (( O, of a class dagram (C, A) s a map to a predcate I (C, A) (C, A) (( O,, where condton for class s that all predcates c (z) mapped from classes are satsfed, condton for assocaton s that all predcates a (z) mapped from assocatons are satsfed and condton for multplcty s that all predcates Mult(a) mapped from multplcty are satsfed. Condtons are followng. (C, A) (( O, ((( O, L) CompleData),( o O, c C, ), ( l L, a A, ( l)),( o O, a A, a c Mult(a) ( o, ) [8]

13 (C,A) (( O, ( o O,( ( l L, (C, A) a (C, A) (( O, L) CompleData) ( l)) ( o O, (( O, Mult() Detals ( o, )) Fgure 0 Eample of a nterpretaton mappng for a Class Dagram Semantcal algebra for class dagrams A structure I of a class dagram s a par ( D, I M M ), where D M s a doman whch all elements consst of comple data and I M s an nterpretaton mappng for data types, classes, assocatons, multplctes of assocatons and class dagrams defned above. On ths structure I, we can evaluate whether a comple data s a nstance of a class dagram through the comple data s assgned nto the class dagram. L: O: arty Request O: Contact[]=amada Contact[]=Suzuk Tel= Detals arty O: Detals Goods[]=C Goods[]=Laptops Num= L: Request O4: Detals Goods[]=Lnu Num= Date= Detals { (C,A) [({ Detals [( O, (( amada, Suzuk),( )))], [( O,(( C, Laptops),()))], Detals [( O4,(( Lnu),()))]}, [( O, (( )))], [( L,(( O, (( amada, Suzuk),( ))),( O, (( ))),( O,(( C, Laptops),()))))], [( L, (( O,(( amada, Suzuk),( ))),( O,(( ))),( O4,(( Lnu),()))))]})] Fgure Eample of an Assgnment to a Class Dagram Although we omt the proof because of space lmtatons, t s a model that substructure I (C, A) whch comple data n the doman of structure I are replaced wth a set of all nstances of the class dagram (C, A). Defnton of a Semantcal Merger Operaton Gven two predcates,a ) and,a ) mapped from class dagrams (C, A ) and (C, A ), a (C (C semantcal merger operaton s defned as followed:

14 ( ( O, L) CompleData, ( O ( O (C, A ), L (C, A ) ( c C O, L ( a A ) CompleData, ( O (( O (C,A ), L )(( O, L ) = ( O, L), )), (C,A ) (( O a c, L, L )), C, o O, ( o O A, l L, ( l) ( l L ) CompleData,, o O, l L )) )), [9] The semantcal merger operaton ( ) s closed on a structure I mapped from class dagrams whch satsfy the consoldaton condton. Ths semantcal operaton s smple enough for mergng. Although we omt the proof because of space lmtatons, followng relaton s formed. roposton Let (C,A ) and (C,A ) be class dagrams and I be an nterpretaton mappng. There s a relaton between the syntactcal ( ) and semantcal ( ) merger operatons on class dagrams wth the consoldaton condton as followed: ( I( (C, A )) I( (C, A )))(( O, ( (C, A ) (C,A ))(( O, (C C, A A ) I( (C C, A A ))(( O, I( (C, A ) (C, A ))(( O, (( O, [0] The semantcal merger operaton ( ) s assocatve and commutatve as well as the syntactcal merger operaton. It s also a commutatve sem-group that the algebrac structure of class dagrams whch satsfy the consoldaton condton and the semantcal merger operaton ( ). All results are same where any sequence of mergng wth the semantcal merger operaton ( ). CONCLUSION We present synta and semantcs of a class dagram descrbng a data model for foundaton for class dagram algebra. Then we ntroduce algebrac structure for a merger operaton on class dagrams and the consoldaton condton. Satsfyng the consoldaton condton keeps consstency of class dagrams syntactcally and semantcally. We also show assocatve law and commutatve law of the syntactcal and semantcal merger operaton for class dagram algebra that keeps avodng nconsstency dependng on order of mergng. It means that the merger operaton has good propertes for avodng nconsstency to handle class dagrams f the consoldaton condton s satsfed. The consoldaton condton and the merger operaton are smple but very powerful because t s only necessary for any modelers to check the consoldaton condton for keepng consstency durng desgnng enterprse systems. Moreover, t s possble to mplement functons of checkng the consoldaton condton and the merger operaton nto 4

15 UML modelng tools. However, another operaton lke dfference s needed because the merger operaton sn t able to dvde a bg class dagram nto partal class dagrams. REFERENCES Berard, D., Calvanese, G., Gacomo G. D. (005), Reasonng on UML class dagrams, Artfcal Intellgence, Vol. 68, Issue, pp Kanewa, K. and Satoh, K. (006), Consstency Checkng Algorthms for Restrcted UML Class Dagrams, roceedngs of the Fourth Internatonal Symposum on Foundatons of Informaton and Knowledge Systems (FoIKS006). Kösters, G., S, H.W. and Wnter, M. (00), Couplng Use Cases and Class Models as a Means for Valdaton and Verfcaton of Requrements Specfcatons, Requrements Engneerng, Vol.6, Issue, pp. -7. Sabetzadeh, M., Easterbrook, S. (005), An Algebrac Framework for Mergng Incomplete and Inconsstent Vews, roceedngs of the th IEEE Internatonal Conference on Requrements Engneerng. Szlenk, M. (006), Formal Semantcs and Reasonng about UML Class Dagram, roceedngs of the Internatonal Conference on Dependablty of Computer Systems, pp Tsolaks, A. and Ehrg, H. (000), Consstency Analyss of UML Class and Sequence Dagrams usng Attrbuted Graph Grammars, roceedngs of Jont ALIGRAH/GETGRATS Workshop on Graph Transformaton Systems, Berln. UML.0 Superstructure Specfcaton (formal/ ), (005), Obect Management Group. a, F. and Kane, G. S. (00), Defnng the Semantcs of UML Class and Sequence Dagrams for Ensurng the Consstency and Eecutablty of OO Software Specfcaton, st Int'l Workshop on Automated Technology for Verfcaton and Analyss, Tawan. eung, (004), Checkng Consstency between UML Class and State Models Based on CS and B, Journal of Unversal Computer Scence, Vol.0, Issue, pp

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1) Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A