Fuzzy subsethood for fuzzy sets of type-2 and generalized type-n

Authors:
John T. Rickard;Janet Aisbett;Greg Gibbon
Affiliations:
Distributed Infinity, Inc., Larkspur, CO;University of Newcastle, Newcastle, NSW, Australia;University of Newcastle, Newcastle, NSW, Australia and Faculty of Computing and Information Technology, Makerere University, Kampala, Uganda
Venue:
IEEE Transactions on Fuzzy Systems
Year:
2009

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Abstract

In this paper,we use Zadeh's extension principle to extend Kosko's definition of the fuzzy subsethood measure S(G, H) to type-2 fuzzy sets defined on any set X equipped with a measure. Subsethood is itself a fuzzy set that is a crisp interval when G and H are interval type-2 sets. We show how to compute this interval and then use the result to compute subsethood for general type-2 fuzzy sets. A definition of subsethood for arbitrary fuzzy sets of type-n 2 is then developed. This subsethood is a type-(n - 1) fuzzy set, and we provide a procedure to compute subsethood of interval type-3 fuzzy sets.