Meta-theorems on inequalities for scalar fuzzy set cardinalities

Authors:
B. De Baets;S. Janssens;H. De Meyer
Affiliations:
Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium
Venue:
Fuzzy Sets and Systems
Year:
2006

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Abstract

We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Lukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the Jaccard coefficient, or equivalently, the triangle inequality of the corresponding dissimilarity measures.