Type-1 Fuzzy Sets and Fuzzy Logic

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets; type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)]; type-1 fuzzy set defined; type-1 fuzzy numbers; linguistic variables; returning to linguistic variables from a numerical value of a membership function; set theoretic operations for crisp and type-1 fuzzy sets; crisp and fuzzy relations and compositions on the same or different product spaces; compositions of a type-1 fuzzy set with a type-1 fuzzy relation; hedges; the Extension Principle (which is about functions of fuzzy sets); α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals); functions of type-1 fuzzy sets computed by using α-cuts; multivariable membership functions and Cartesian products; crisp logic; going from crisp logic to fuzzy logic; Mamdani (engineering) implications; and some final remarks. There are two appendixes: Appendix 1 is about properties of type-1 fuzzy sets, and Appendix 2 is about cardinality and similarity of type-1 fuzzy sets. A total of 36 examples are used to illustrate this chapter’s important concepts.