Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets

The main aim of this paper is to propose new subsethood measures for continuous, general type-2 fuzzy sets. For this purpose, we introduce inclusions and subsethood measures for interval-valued fuzzy sets first. Then, using an α-plane representation for type-2 fuzzy sets, we extend these inclusions and subsethood measures to general type-2 fuzzy sets. Subsethood measures for interval-valued fuzzy sets (hence, also for type-2 fuzzy sets) rely on already known subsethood measures for ordinary fuzzy sets. We focus on a special subsethood measure for ordinary fuzzy sets, based on α-cut representation, and show, how to compute subsethood measures for continuous type-2 fuzzy sets with no need for discretizing of universe. This is a very interesting and useful property of proposed subsethood measures, which is one of the reason, why our approach has less computational demand than the others.