Generalized Relative Cardinalities Of Fuzzy Sets

  • Daniel Pilarski
  • Published 2005 in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems


The relative cardinality or relative count of a fuzzy set represents the proportion of elements of that fuzzy set which are in another fuzzy set. This type of a cardinality plays an important role e.g. in questions of interpretation of linguistic quantifiers. Terms like "most", "about a half" are called proportional linguistic quantifiers. Examples of sentences involving such quantifiers are "Many cars are very expensive.", "About a half of the students passed their exams very well", etc. "Q A are B" is the general form of considered sentences, where Q is a fuzzy quantifier and A and B are labels of fuzzy sets (details can be found e.g. in 2 , 5 , 9 ) . This kind of quantifiers can be interpreted as a fuzzy characterization of the relative cardinality of B and A. The procedure of the truth value evaluation of such quantifiers is based on the assumption that the proposition "Q A are I?" is semantically equivalent to "Relative cardinality of B in A is Q" (see 1 ) . Another example of applications of relative scalar cardinalities is the question of group decision making with hesitation factor. A classical definition and properties of a relative count can be found in ,. This paper is an attempt at investigating basic properties of relative cardinality involving cardinality patterns and triangular operations. We consider fuzzy sets with triangular operations, i.e. the sum A Us B of A, B G FFS induced by a t-conorm s with (A Us B)(x) = A(x) s B(x), the intersection Af]t B of A,B G FFS induced by a t-norm t with (Afit B){x) — A(x) t B(x), where FFS is the family of all finite fuzzy sets in a universe M.


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