## Abstract

This chapter explores many aspects of the type-1 fuzzy system that was introduced in Chap. 1. It provides a very comprehensive and unified description of the two major kinds of type-1 fuzzy systems that are widely used in real-world applications—Mamdani and TSK fuzzy systems. The coverage of this chapter includes rules, singleton, and non-singleton fuzzifiers, input–output formulas for the fuzzy inference engine, type-1 first- and second-order rule partitions, the effects of the two kinds of fuzzifiers on the input–output formulas, combining or not combining fired-rule output sets on the way to defuzzification, defuzzifiers (centroid, height, and center-of-sets), fuzzy basis functions which provide a mathematical description of a fuzzy system from its input to its output, remarks and insights about a type-1 fuzzy system (including layered architecture interpretations for it, universal approximation by it, continuity of it, rule explosion and some ways to control it, and rule interpretability for it). Eighteen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sect. 3.7 that is continued in later chapters. Chap. 9 builds upon the material that is in this chapter.

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