Norms and Metrics over IFSs or Their Elements

Abstract

First, let us emphasize that here we do not study the usual set-theoretic properties of the IFSs (i.e. properties which follow directly from the fact that IFSs are sets in the sense of the set theory – see Section 2.1). For example, given a metric space E, one can study the metric properties of the IFSs over E. This can be done directly by topological methods (see e.g. [421]) and the essential properties of the IFSs are not used. On the other hand, all IFSs (and hence, all fuzzy sets) over a fixed universe E generate a metric space (in the sense of [421]), but with a special metric (cf., e.g., [301]): one that is not related to the elements of E and to the values of the functions μ _A and ν _A defined for these elements.