Basicintroduction
Thefuzzyrelationequationoccupiesaveryimportantpositioninfuzzymathematicstheoryanditsapplication.SanchezofFrancehasdonepioneeringresearchinthisarea.Hefirstproposedthefuzzyrelationequationandgavethemaximumsolutionoftheequation.FollowingtheJapanesepoliticalelementtogiveaspecificsolutiontothefuzzyrelationequationonafiniteset,mycountry'sLuoChengzhong,XuWenli,CaoZhiqiang,LiBixiang,etc.gaveasimplifiedsolutiontotheequation.
Typesoffuzzyrelationequations
Therearetwotypesoffuzzyrelationequations.
TypeI
GiventhedomainU,V,W,thefuzzyrelationshipisknown
Findthefuzzyrelationship
Satisfytheequation:TypeII
Knownfuzzyrelationship
Findthefuzzyrelationshipandsatisfytheequation:InTypeII,let
,itsmembershipfunction:iscalledasthetranspositionrelationshipof,anditalsostipulates
,thenTsatisfiestheequation:Therefore,thetwotypesofequationscanbetransformedintoeachother.
Fuzzyrelationequationsolution
thatsatisfiesthefuzzyrelationequationiscalledthesolutionoftheequation,iftheequationIfthesection>hasasolution,theequationissaidtobecompatible,otherwisetheequationissaidtobeincompatible.Ifacertainsolutionoftheequation,foranyothersolution,thereisalways,theniscalledthemaximumoftheequationuntie.
Sanchezprovedmostgenerallythatforanyfuzzyrelationequation,ifthereisasolution,theremustbeamaximumsolution.Forthefuzzyrelationequationsinthefiniteuniverse,Zhengyuangivesaspecificsolution,whichiscalledtheZhengyuanmethod.LuoChengzhongandotherssimplifiedtheZhengyuanequation.Seefuzzyrelationship.