Libro inteligencia artificial

CHAPTER

2

CLASSICAL SETS AND FUZZY SETS

Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extentthat mathematical structures are used to model physical actualities, it is often an unrealistic requirement.. . . Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox. James Bezdek Professor, Computer Science, 1981

As alluded to in Chapter 1, the universe of discourseis the universe of all available information on a given problem. Once this universe is defined we are able to define certain events on this information space. We will describe sets as mathematical abstractions of these events and of the universe itself. Figure 2.1a shows an abstraction of a universe of discourse, say X, and a crisp (classical) set A somewhere in this universe. A classical set isdefined by crisp boundaries, i.e., there is no uncertainty in the prescription or location of the boundaries of the set, as shown in Fig. 2.1a where the boundary of crisp set A is an unambiguous line. A fuzzy set, on the other hand, is prescribed by vague or ambiguous properties; hence its boundaries are ambiguously specified, as shown by the fuzzy boundary for set A in Fig. 2.1b. ∼ In Chapter 1 weintroduced the notion of set membership, from a one-dimensional viewpoint. Figure 2.1 again helps to explain this idea, but from a two-dimensional perspective. Point a in Fig. 2.1a is clearly a member of crisp set A; point b is unambiguously not a member of set A. Figure 2.1b shows the vague, ambiguous boundary of a fuzzy set A on the same universe X: the shaded boundary represents the boundary regionof A. In ∼ ∼ the central (unshaded) region of the fuzzy set, point a is clearly a full member of the set.
Fuzzy Logic with Engineering Applications, Second Edition T. J. Ross  2004 John Wiley & Sons, Ltd ISBNs: 0-470-86074-X (HB); 0-470-86075-8 (PB)

CLASSICAL SETS
X (Universe of discourse) X (Universe of discourse)

25

c A a a A ~

b

b

(a)

(b)

FIGURE 2.1 Diagrams for (a)crisp set boundary and (b) fuzzy set boundary.

Outside the boundary region of the fuzzy set, point b is clearly not a member of the fuzzy set. However, the membership of point c, which is on the boundary region, is ambiguous. If complete membership in a set (such as point a in Fig. 2.1b) is represented by the number 1, and no-membership in a set (such as point b in Fig. 2.1b) is represented by0, then point c in Fig. 2.1b must have some intermediate value of membership (partial membership in fuzzy set A) on the interval [0,1]. Presumably the membership of point c in A approaches ∼ ∼ a value of 1 as it moves closer to the central (unshaded) region in Fig. 2.1b of A, and ∼ the membership of point c in A approaches a value of 0 as it moves closer to leaving the ∼ boundary region of A. ∼ Inthis chapter, the precepts and operations of fuzzy sets are compared with those of classical sets. Several good books are available for reviewing this basic material [see for example, Dubois and Prade, 1980; Klir and Folger, 1988; Zimmermann, 1991; Klir and Yuan, 1995]. Fuzzy sets embrace virtually all (with one exception, as will be seen) of the definitions, precepts, and axioms that defineclassical sets. As indicated in Chapter 1, crisp sets are a special form of fuzzy sets; they are sets without ambiguity in their membership (i.e., they are sets with unambiguous boundaries). It will be shown that fuzzy set theory is a mathematically rigorous and comprehensive set theory useful in characterizing concepts (sets) with natural ambiguity. It is instructive to introduce fuzzy sets by first...