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39

3.3 FUZZY SETS

(i.e., the real line), or a subset of the real line such as all non-negative numbers. In

this sense, the term “membership function” means speci¬cally a function de¬ned on

numbers from the real line. Examples of such membership functions will be given in

Sections 3.3.2 and 3.3.3. Grades of membership for universes other than the real

numbers are normally calculated by the ¬ring of rules, estimated by an observer

or some other method, rather than by membership functions.

At this point, we shall not be concerned with the important topic of how the mem-

bership functions are to be determined. We will discuss in later sections and chapters

methods of determining memberships and of specifying membership functions. Klir

and Yuan (1995) devote a whole chapter to construction of fuzzy sets and operations

on fuzzy sets.

There are two very special fuzzy sets needed in fuzzy expert systems: (1) discrete

fuzzy sets; and (2) fuzzy numbers. We will now discuss both of these fuzzy sets in detail.

3.3.1 Discrete Fuzzy Sets

If X ¼ is ¬nite, the simplest discrete fuzzy set D is just a fuzzy subset of X. We can

write D as

m1 m2 m

,..., n

, (3:16)

D¼

x1 x2 xn

where the membership value of x1 in D is m1. Also, if X is not ¬nite but D(x) = 0 for

only x ¼ fx1, x2 , . . . , xng we write D as in equation (3.16). Conventionally, the truth

value of a member in a fuzzy set is called its grade of membership.

An example might be a fuzzy set Diagnosis of psychiatric diagnoses, shown in

(3.17). This is a non-numeric fuzzy set, since its members describe a non-numeric

quantity.

m1 m2 m3

, ,

Diagnosis ¼ (3:17)

depression bipolar disorder schizophrenia

Discrete fuzzy sets for fuzzy expert systems may be numeric or non-numeric,

depending on whether their members describe numeric or non-numeric quantities.

(3.17) shows a non-numeric fuzzy set. As an example of a discrete numeric fuzzy

set consider (3.18),

m1 m2 m3

, , (3:18)

Size ¼

small medium large

whose members describe a numeric quantity, Size. Members of a numeric discrete

fuzzy set always describe a numeric quantity. Such discrete fuzzy sets are called

linguistic variables, with members linguistic terms, and are discussed in Section

3.3.3.

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40 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: I

3.3.2 Fuzzy Numbers

Fuzzy numbers represent a number of whose value we are somewhat uncertain.

They are a special kind of fuzzy set whose members are numbers from the real

line, and hence are in¬nite in extent. The function relating member number to its

grade of membership is called a membership function, and is best visualized by a

graph such as Figure 3.1. The membership of a number x from the real line is

often denoted by m(x). Fuzzy numbers may be of almost any shape (though conven-

tionally they are required to be convex and to have ¬nite area), but frequently they

will be triangular (piecewise linear), s-shape (piecewise quadratic) or normal (bell-

shaped). Fuzzy numbers may also be basically trapezoidal, with an interval within

which the membership is 1; such numbers are called fuzzy intervals. Fuzzy intervals

may have linear, s-shape or normal “tails”, the increasing and decreasing slopes.

Figures 3.1“ 3.4 illustrate fuzzy numbers with these shapes.

Assume that triangular and s-shaped fuzzy numbers start rising from zero at

x ¼ a; reach a maximum of 1 at x ¼ b; and decline to zero at x ¼ c. Then the mem-

bership function m(x) of a triangular fuzzy number is given by

m(x) ¼ 0, x a

a,x

¼ (x À a)=(b À a), b

(3:19)

¼ (c À x)=(c À b), b , x c

¼ 0, x . c

For trapezoids, similar formulas are used employing b1 and b2 instead of b.

A piecewise quadratic fuzzy number is a graph of quadratics m(x) ¼ c0 þ c1 x þ

c2 x2 passing through the pairs of points (a, 0), ((a þ b)/2, 0.5); ((a þ b)/2, 0.5),

(b, 1); (b, 1), ((b þ c)/2, 0.5); and ((b þ c)/2, 0.5), (b, 0). At the points a, b and

c, d(m(x))/dx ¼ 0. For x , a and x . c, m(x) ¼ 0.

To derive simple formulas for the quadratic fuzzy numbers, it is best to consider

an effective translation of the x axis to the points x ¼ a for the ¬rst region, x ¼ b for

Figure 3.1 Membership function for a triangular fuzzy 1. Membership of 22 is 0.25.

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41

3.3 FUZZY SETS

An s-shape fuzzy 1. Membership of 22 is about 0.1.

Figure 3.2

the second and third regions, and x ¼ c for the fourth region. At these three points

the ¬rst derivatives are 0, and the membership functions are given by

aþb

; m(x) ¼ k1 (x À a)2

For a x

2

aþb

b; m(x) ¼ 1 À k1 (b À x)2

For x

2

(3:20)

bþc

; m(x) ¼ 1 À k2 °x À bÞ2

For b x

2

bþc

b; m(x) ¼ k2 °c À xÞ2

For x

2

The constants k1 and k2 are easily evaluated by realizing that at x ¼ (a þ b)=2 and

x ¼ (b þ c)=2, the memberships are 0.5.

Figure 3.3 A normal fuzzy 1. Membership of 22 is 0.2.

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