Basics of Fuzzy Sets
In this chapter, we focus on the introduction of fundamentals in fuzzy set theory, including some set-theoretic operations and their extensions, the decomposition of a fuzzy set, and mathematical representations of fuzzy sets in terms of a nest of sets. T
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Basics of Fuzzy Sets
In this chapter, we focus on the introduction of fundamentals in fuzzy set theory, including some set-theoretic operations and their extensions, the decomposition of a fuzzy set, and mathematical representations of fuzzy sets in terms of a nest of sets. Towards the end of the chapter, fuzzy sets taking values in [0, 1] are extended to those on a lattice and a similar investigation is carried out.
2.1
Fuzzy Sets and Their Set-Theoretic Operations
According to Cantor, a set consists of some elements which are definite. In other words, for a given element, whether it belongs to the set or not should be clear. As a consequence, a set can only be employed to describe a concept which is crisply defined. For example, a collection of cities with the population more than 5 millions forms a set since we can judge that a given city is in this set or not without vagueness. In traditional mathematics, all the involved concepts ranging all the way from the complex numbers and matrices to geometric transformations and algebraic structures are in this category. However, in the real world, mankind often uses concepts which are quite vague. For example, we say that a man is young or middle-aged, an object is expensive or cheap, a tomato is red and mature, a number is large or small, a car is slow or fast and so on. Let us take young as an illustration. Suppose A is a 20-year-old man. Maybe you think A is certainly young. Now comes a man B only one day elder than A. Of course, B is still young. Then how about a man only one day elder than B. Continuing in this way, you will find it difficult to determine an exact age beyond which a man will be middle-aged. As a matter of fact, there is no sharp line between young and middle-aged. The transition from one concept to the other is gradual. This gradualness results in the vagueness of the concept young, which in return makes the boundary of the set of all young men unclear. In 1965, Zadeh introduced the concept of fuzzy sets just in order to represent this class of X. Wang et al.: Mathematics of Fuzziness – Basic Issues, STUDFUZZ 245, pp. 21–64. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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2 Basics of Fuzzy Sets
sets. In his seminal paper Fuzzy Sets [171], Zadeh assigns a number to every element in the universe, which indicates the degree (grade) to which the element belongs to a fuzzy set. In this interpretation, everybody has a degree to which he/she is young (eventually the degree may be 0 or 1). The people with different ages may have different degrees. To formulate this concept of fuzzy set mathematically, we present the following definition. Definition 2.1. Let X be the universe. A mapping A : X → [0, 1] is called a fuzzy set on X. The value A(x) of A at x ∈ X stands for the degree of membership of x in A. The set of all fuzzy sets on X will be denoted by F (X). A(x) = 1 means full membership, A(x) = 0 means non-membership and intermediate values between 0 and 1 mean partial membership. A(x) is referred to as a membership function as x varies i
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