Mathematics of Fuzzy Sets and Fuzzy Logic

The concept of a set is fundamental in Mathematics and intuitively can be described as a collection of objects possibly linked through some properties. A classical set has clear boundaries, i.e. x ∈ A or x ∉ A exclude any other possibility.

As we have seen in the previous section, we can identify the operations between fuzzy sets by the corresponding operations on the unit interval, fuzzy set operations being defined point-wise. This implies that we can study operations between fuzzy sets by the corresponding operations over the real unit interval.

(Classical relation). A subset R ⊆ X×Y where X and Y are classical sets, is a classical relation.

Fuzzy numbers generalize classical real numbers and roughly speaking a fuzzy number is a fuzzy subset of the real line that has some additional properties. Fuzzy numbers are capable of modeling epistemic uncertainty and its propagation through calculations. The fuzzy number concept is basic for fuzzy analysis and fuzzy differential equations, and a very useful tool in several applications of fuzzy sets and fuzzy logic.

Often, we have to perform operations with uncertain parameters. In this case we will have to define the fuzzy counterparts of the classical operations between real numbers.

We begin our discussion on fuzzy arithmetic with Zadeh’s extension principle. It serves for extending a real-valued function into a corresponding fuzzy function.

Reasoning with imprecise information is one of the central topics of fuzzy logic. A fuzzy inference system consists of linguistic variables, fuzzy rules and a fuzzy inference mechanism. Linguistic variables allow us to interpret linguistic expressions in terms of fuzzy mathematical quantities. Fuzzy Rules are a set of rules that make association between typical input and output data sometimes in an intuitive way, or, on other occasions, in a data driven way. A fuzzy inference mechanism is able to model the process of approximate reasoning, through interpolation between the fuzzy rules. Of course good interpolations are also approximations, and in this way approximate reasoning is performed.

In the previous chapter we have described in detail Fuzzy Inference Systems. These have fuzzy sets as inputs, and their output is a fuzzy set as well, so these systems work exclusively in a fuzzy setting. Often, in practical applications we need to be able to accept crisp inputs and also, the system needs to produce a crisp number for the output. Surely this is often a well defined classical functional relationship between inputs and outputs. Naturally raises the question why do we need fuzzy systems when we have a crisp relationship, crisp input and also a crisp output for a classical system. The reason for this fact lies in epistemic uncertainty. However we have a crisp relationship, this is often unknown or only partially known to us.

The topological structure of Fuzzy Numbers was investigated in detail by several authors (e.g., Diamond-Kloeden[44], Puri-Ralescu [123], Ma [104], Goetschel-Voxman [74]). There are some properties that in a classical Mathematical structure (e.g. that of a Banach space) are easily fulfilled, while in the fuzzy setting they do not hold. In this sense, in this chapter we present several negative results through several counterexamples. Some of these results are known but the counterexamples presented in Sections 8.2, 8.3, 8.4 are new, being published for the first time here. Mathematical Analysis on Fuzzy Number’s space is an interesting topic (see Anastassiou [4], Bede-Gal [18], [20], [19], Chalco-Cano-Román-Flores-Jiménez-Gamero [35] Gal [66], Lakshimikantham-Mohapatra [98] Wu-Gong [151]). We study in this chapter mainly integration and differentiability of fuzzy-number-valued functions.

Fuzzy differential equations (FDEs) appear as a natural way to model the propagation of epistemic uncertainty in a dynamical environment. There are several interpretations of a fuzzy differential equation. The first one historically was based on the Hukuhara derivative introduced in Puri-Ralescu [123] and studied in several papers (Wu-Song-Lee [150], Kaleva [83], Ding-Ma-Kandel [46], Rodriguez-Lopez [125]). This interpretation has the disadvantage that solutions of a fuzzy differential equation have always an increasing length of the support. This fact implies that the future behavior of a fuzzy dynamical system is more and more uncertain in time. This phenomenon does not allow the existence of periodic solutions or asymptotic phenomena. That is why different ideas and methods to solve fuzzy differential equations have been developed. One of them solves differential equations using Zadeh’s extension principle (Buckley-Feuring [30]), while another approach interprets fuzzy differential equations through differential inclusions. Differential inclusions and Fuzzy Differential Inclusions are two topics that are very interesting but they do not constitute the subject of the present work (see Diamond [45], Lakshmikantham-Mohapatra [98]). Recently new approaches have been developed based on generalized fuzzy derivatives discussed in the previous chapter. In the present work we will work with the interpretations based on Hukuhara differentiability, Zadeh’s extension principle and the strongly generalized differentiability concepts.

In fuzzy set theory the membership function of a fuzzy set is a classical function A : X → [0, 1]. In some applications the shape of the membership function is itself uncertain. This problem appears mainly because of the subjectivity of expert knowledge and imprecision of our models. In these situations we can use a higher order extension of fuzzy set theory.

Intuitively, clustering means partitioning a data set into clusters (subsets) whose objects share similar properties, i.e., they are near to each other in some well defined sense for “near” (see e.g., Jain-Dubes [82]). In this section we discuss the classical k-means clustering algorithm which is one of the basic classical clustering methods. It also stands at the basis of the subsequently described corresponding fuzzy techniques. Also, Faber [57] has proposed a continuous k-means clustering method.We discuss here a fuzzy version of Faber’s algorithm the continuous fuzzy c-means method in Section 12.3. Those ideas are published here for the first time up to the author’s best knowledge.

Fuzzy Transform was proposed in Perfilieva in [121] and Perfilieva [120] and it is an approximation method based on fuzzy sets. Also, it can be seen as a fuzzy set-based analogue of the Fourier Transform. In the present chapter we follow in great lines the presentation and discussion in Bede-Rudas [23].

Computational Intelligence is a discipline within Artificial Intelligence and it studies the topics of Fuzzy Sets and Systems, Neural Networks, Genetic Algorithms, Swarm Intelligence and combination of these topics (see Engelbrecht [55]). The present chapter will present an introduction to the Theory of Neural Networks, and also the combination of Neural and Fuzzy systems, i.e., the Adaptive Network-based Fuzzy Inference System (Jang [81]).