Fuzzy Mathematics Principles and Methods

Author: Chief Editor: Chen Yuhé
Publisher:
Publish Date: 2004-07-01
Features: This book provides an in-depth and systematic introduction to the fundamental theories, methods, and applications of fuzzy mathematics. The book is divided into eight chapters: Chapter 2 introduces preliminary knowledge and the basic theory of fuzzy sets; Chapters 3 to 8 cover fuzzy measures and integrals, fuzzy programming, fuzzy relation equations, fuzzy recognition and clustering analysis, fuzzy decision-making, and commonly used fuzzy mathematical models. The book is well-explained, theoretically rigorous, rich in content, and practical in application. The content of Chapters 4 to 8 is self-contained, with some sections reflecting the latest achievements in this field. Each chapter is followed by exercises for reference. This book is suitable for researchers and practitioners in fuzzy mathematics, as well as graduate students and senior undergraduate students in science and engineering. After selection, the content can also be used as a textbook or reference for similar courses in applied mathematics, computer software, and engineering mechanics. It can also serve as a self-study reference for engineering technicians.
Excerpt: This chapter introduces some concepts and results frequently used in fuzzy mathematics, serving as the foundation for reading this book. Readers who are already familiar with these tools can proceed directly to Chapter 2. Since fuzzy mathematics uses classical mathematical tools to study and handle fuzzy phenomena, commonly used classical mathematical tools are widely applied in fuzzy mathematics. As preliminary knowledge, the content of this chapter primarily involves concepts and results that have direct applications in this book, which may not be familiar to general science and technology workers or undergraduate and graduate students in non-mathematics fields. Due to space constraints, in most cases, we only present the relevant concepts and results, while detailed proof details can be found in related books [18–20].
(III) Associativity ((x ∨ y) ∨ z = x ∨ (y ∨ z)), ((x ∧ y) ∧ z = x ∧ (y ∧ z));
(IV) Absorption x ∧ (x ∨ y) = x, x ∨ (x ∧ y) = x. Conversely, if a binary operation ∨ and ∧ satisfying the above (I)–(IV) is defined on X, and it is stipulated that for any x, y ∈ X, x ≤ y → x ∨ y = y, then (X, ≤) forms a lattice. By Theorem 1, we can derive an equivalent definition of a lattice: If the binary operations ∨ and ∧ on X satisfy the idempotent law, commutative law, associative law, and absorption law, then (X, ∨, ∧) is called a lattice. Below, we introduce several special types of lattices.
Definition 4 Let (X, ∨, ∧) be a lattice:
(Ⅰ) If for any subset A of X, supA and infA exist, then (X, ∨, ∧) is called a complete lattice. In particular, for a complete lattice, supX and infX exist, denoted as supX = 1 and infX = 0, respectively, called the maximum and minimum elements of X.
(Ⅱ) If for any x, y, z ∈ X, the distributive law ((x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)), ((x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z)) holds, then (X, ∨, ∧) is called a distributive lattice.
(Ⅲ) Let the lattice (X, ∨, ∧) have a maximum element 1 and a minimum element 0. If for any x ∈ X, there exists an element xc ∈ X such that x ∨ xc = 1 and x ∧ xc = 0, then (X, ∨, ∧) is called a complemented lattice. The element xc is called the complement or co-element of x.
Theorem 2 Let (X, ∨, ∧) be a distributive complemented lattice. Then, for any element x ∈ X, there exists a unique element xc ∈ X such that x ∨ xc = 1 and x ∧ xc = 0.
Proof If there exists another x' ∈ X such that x ∨ x' = 1 and x ∧ x' = 0, then x' = 1 ∧ x' = (x ∨ xc) ∧ x' = (x ∧ x') ∨ (xc ∧ x') = 0 ∨ (xc ∧ x') = xc ∧ x'.
Thus, x' ≤ xc.
x' = 0 ∨ x' = (x ∧ xc) ∨ x' = (x ∨ x') ∧ (xc ∨ x') = 1 ∧ (xc ∨ x') = xc ∨ x'.
So xc ≤ x', hence x' = xc.
Definition 5 Let (X, ∨, ∧) be a lattice with a maximum element 1 and a minimum element 0. If a mapping C: X → X satisfies the following conditions:
(I) C(0) = 1;
(II) C(x) is strictly decreasing;
(III) C(C(x)) = x for all x ∈ X.
Then, C(x) is called the pseudo-complement of X, and (X, ∨, ∧) is called a co-complemented lattice.