Working with Type-2 Fuzzy Sets

This chapter explains how to work with type-2 fuzzy sets (T2 FSs). Most of its topics are needed in the rest of this book. For a reader who is only interested in interval type-2 fuzzy sets, a shortcut is possible, and is explained in Sect. 7.1. This chapter covers the following topics: set theoretic operations (union, intersection, and complement) for general type-2 fuzzy sets (GT2 FSs) computed by using the Extension Principle; set theoretic operations for interval type-2 fuzzy sets (IT2 FSs); set theoretic operations for GT2 FSs computed by using horizontal slices; type-2 relations and compositions on the same product space and on different product spaces; compositions of a T2 FS with a type-2 relation; type-2 hedges; Extension Principle for IT2 and GT2 FSs; functions of GT2 FSs computed by using α-planes; Cartesian product of T2 FSs; and implications. There are four appendixes: Appendix 1 provides proofs for 10 theorems; Appendix 2 is about the properties of T2 FSs; Appendix 3 derives the union, intersection, and complement of IT2 FSs using type-1 fuzzy set mathematics; and, Appendix 4 is about cardinality and similarity of T2 FSs. 22 examples are used to illustrate the chapter’s important concepts.