Fuzzy-Valued Transitive Inclusion Measure, Similarity Measure and Application to Approximate Reasoning

In fuzzy set theory, inclusion measure indicates the degree to which a given fuzzy set is contained in another fuzzy set. Many inclusion measures taking values in [0,1] have been made in the literature. This paper proposes a series of fuzzy-valued inclusion measures which, by a relation view, are reflexive, antisymmetric and T -transitive where T is a left-continuous triangular norm; In addition, they possess most of the axiomatic properties which are postulated by Sinha and Dougherty for an inclusion measure. Fuzzy-valued similarity measures are also defined by the fuzzy-valued inclusion measures; They have T -transitivity and properties introduced by Liu for a similarity measure. Lastly two methods for inference in approximate reasoning based on the fuzzy-valued inclusion measure and the fuzzy-valued similarity measure are studied.

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