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# Fuzzy relation equation

## 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.