Equipotencies

Abstract

Creating a nonclassical cardinality theory for VD-objects, we will use their approximative, unifying representation proposed in Chapter 4. More precisely, Chapters 5–14 contain a nonclassical cardinality theory which is based on the representation from Sections A–C of Chapter 4, and can be applied to VD-objects characterized by means of arbitrary functions M→I, precisely determined or not. Therefore, principally, it does not involve subdefinite sets, unless they are modelled by means of flou sets. Instead, Chapter 15 contains another, modified formulation of that nonclassical cardinality theory, where VD-objects are described by free representing pairs from Section 4-D. It can be applied to (proper) VD-objects characterized by imprecisely determined membership functions, and to subdefinite sets represented by arbitrary twofold fuzzy sets or flou sets. So, that modification leads to a partial generalization of the nonclassical cardinality theory presented in Chapters 5–14 and, on the other hand, it is an expansion of the idea of cardinality of twofold fuzzy sets proposed by Dubois and Prade (see also [FCR#17]).