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1、OpusculaMathematica•Vol.26•No.3•2006AngelGarridoSPECIALFUNCTIONSINFUZZYANALYSISAbstract.InthetreatmentofFuzzyLogicanusefultoolappears:themembershipfunc-tion,withtheinformationaboutthedegreeofcompletionofaconditionwhichdefinestherespectiveFuzzySetorFuzzyRelation.Withtheirintroduc
2、tion,itispossibletoprovesomeresultsonthefoundationsofFuzzyLogicandopennewwaysinFuzzyAnalysis.Keywords:logicsinA.I.,FuzzySetTheory,FuzzyRealAnalysis,ArtificialIntelligence.MathematicsSubjectClassification:68727,03E72,26E50.1.INTRODUCTIONTOFUZZYTHEORYWhenwesolveproblemsinA.I.,their
3、representationwillbethroughtheFuzzyLogictechniques,averyusefulprocedure.Forinstance,intheproblemsrelatedtothe“realworld”.Asyouknow,itisoneofthe“possibleworlds”only.Wedefinethe“world”as“acompleteandconsistentdescriptionofhowthethingsareorhowtheycouldhavebeen”.Insolvingquestionsof
4、thistypetheMonotonicLogicoftendoesnotwork,whereassuchtypeofLogic,isclassicalinformalworlds,suchasMathematics.OtherNon-MonotonicLogicsmustalsobeintroduced,wherenowtheextensionofthesetofsentencescanmodifytheconclusion.Thishappensfrequentlyintherealworld:forinstance,inthemedicalsc
5、iences,orinthecommonsensereasoning,withpartialinformation,givingtemporal,revisableandprovisionalconclusions.Fortheirpart,weneedFuzzySets,FuzzyRelationsandsoon,todescribethegradationofcertaintyinourworld.Itisshownthroughanewfunction,theafore-mentionedmembershipfunctionµ,whichdes
6、cribesthedegreeoffulfillmentforeachelementofthepropertydefiningtheset.Orthe“degreeofrelation”betweentwodeterminedelements.Such“membershipdegree”valuecanbeassignedbythecorre-spondingµ,the“membershipfunction”,whoserangeistheclosedunitinterval[0,1].Thusµ:C→[0,1].457458AngelGarridoAc
7、cordingtothis,afuzzysetcanbedefinedas:C={x
8、µ(x),∀x∈U}wheretheverticalsymbol
9、doesnotmean“suchthat”,butitadjoinstheinformationonthe“membershipdegree”ofsuchelementtothesetC.nGiven{Ui}i=1,nuniversesofdiscourse,wedefineafuzzyrelationRthroughanmembershipfunctionthattoeachn-tuple,(xi)i=
10、1,wherexi∈Ui,i=1,2,...,n,associatesavalueintheclosedun
