A new representational framework for fuzzy sets

One of the main contributions of fuzzy logic is the methodology of computing with words (CW). In that context, a successful representation of a fuzzy set as a bearer of meaning of linguistic information should have close relationship with its linguistic description and reflect various changes resulting from applying some operational primitives on words. Taking these requirements as our main goal, we develop a novel representation of fuzzy sets of interest for practical implementations of the CW framework. The representation includes well-known concepts from fuzzy theory, cardinality, fuzziness and skewness of a fuzzy set, and also a novel concept of medium curve. We show that they have close relationship with linguistic domain and are very useful in describing the meaning of word. Then, we examine their properties and relationships in the case of fuzzy sets with S-shaped gray areas and develop an appropriate geometrical representation. Proposed representation can be easily transformed into the common representations of fuzzy sets and vice-versa by the developed simple transformation formulae. Finally, we show how the representation can be conveniently exploited on several practical examples.