Type-1 Fuzzy Sets and Fuzzy Logic

Abstract

This chapter formally introduces type-1 fuzzy sets and fuzzy logicLogicfuzzy. It is the backbone for Chap. 3and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp setsSetscrisp, type-1 fuzzy setsSetstype-1 fuzzyand associated concepts [including a short biography of Prof. ZadehZadehbiography(the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variablesLinguistic variablesdefined, returning to linguistic variables from a numerical value of a membership functionReturning to linguistic variables from numerical values of MFs, set theoretic operations for crispSet theoretic operations forcrisp setsand type-1 fuzzy sets, crisp and fuzzy relations and compositionsCompositionscrispsame product spaceon theCompositionsfuzzy (type-1)same product spacesameRelationscrispsame product spaceorRelationsfuzzy (type-1)same product spacedifferent product spacesCompositionscrispdifferent product spacesCompositionsfuzzy (type-1)different product spacesRelationscrispdifferent product spacesRelationsfuzzy (type-1)different product spaces, compositions of a type-1 fuzzy set with a type-1 fuzzy relationComposition oftype-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension PrincipleExtension principle(which is about functions of fuzzy sets), α-cutsΑ-cutsrepresenting type-1 fuzzy sets(which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cutsFunctions of type-1 fuzzy sets computed using α-cuts, multivariable membership functionsMultivariable membership functionsand Cartesian productsCartesian products, crisp logicLogiccrisp, going from crisp logic to fuzzy logic, Mamdani (engineering) implicationsMamdani implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy setsType-1 fuzzy setproperties of. 35 examples are used to illustrate this chapter’s important concepts.