Ordering, distance and closeness of fuzzy sets

In dense rule bases where the observation usually overlaps with several antecedents in the rule base, various algorithms are used for approximate reasoning and control. If the antecedents are located sparsely, and the observation does not overlap as a rule with any of the antecedents, function approximation techniques combined with the Resolution Principle lead to applicable conclusions. This kind of approximation is possible only if a new concept of ordering and distance, i.e. a metric in the state space, and a partial ordering among convex and normal fuzzy sets (CNF sets) is introduced. So, the fuzzy distance of two CNF sets can be defined, and by this distance, closeness and similarity of CNF sets, as well.