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call вЂњresemblanceвЂќ, might be as shown in Table 3.3.

Fuzzy relations may be given as matrices if the sets involved are discrete, or ana-

lytically if the sets are continuous, usually numbers from the real line. The members

of X and Y are often the members of fuzzy sets A and B; the fuzzy relation between

their members is often a function of their grades of membership in A and in B.

Theoretical fuzzy logical inference involves an important application of fuzzy

relations, in which the fuzzy relation is usually an implication, as discussed in

Section 3.1. (While this theory is important to fuzzy logicians, it is much less so

to builders of fuzzy expert systems, as we shall see below.) First, we must choose

an implication operator valid for classical logic. Suppose we picked the one given

in equation (3. 8). Next let A and B be two fuzzy numbers with membership func-

tions A(x) and B(x), respectively, shown in Figure 3.6. (Here X Вј Y Вј the set of

real numbers.) The fuzzy relation, from equation (3.8) is

R(x, y) Вј min(1, 1 ГЂ A(x) Гѕ B( y)) (3:25)

for x and y any real numbers. Then tv(A ! B) Вј R(x, y) in fuzzy logic. If A and B

are the two fuzzy numbers shown in Figure 3.6, the quadrant (x 0, y ! 0) of the

fuzzy relation R(x, y) is shown in Figure 3.7. (Other quadrants are not shown, since

they would obscure the graph.) R(x, y) is symmetric about the (y, R) plane, and is

everywhere 1 outside the region (24 , x , 4).

In fuzzy inference, we will need to compose fuzzy relations. If the fuzzy relation

exists in matrix form, the procedure in Section 3.4 may be followed. However, the

fuzzy relation may be given as a continuous function. Let R be a fuzzy relation on

X Г‚ Y and S another fuzzy relation on Y Г‚ Z. Then R(x, y) is a number in [0, 1]

for all x in X and all y in Y, and S(y, z) has its values in [0, 1] for all y in Y and

all z in Z. We compose R and S to get T, a fuzzy relation on X Г‚ Z. This is

written as R W S Вј T. We compute T as follows:

T(x, z) Вј supy{min{R(x, y), S(y, z)}} (3:26)

In equation (3.26), вЂњsupвЂќ stands for supremum, which must be used in place of max

for many inп¬Ѓnite sets. For example, the sup of x in [0, 1) Вј 1, but this interval has no

max. On the other hand, sup of x in [0, 1] Вј max of x in [0, 1] Вј 1. Other AND type

operators [eqs. (3.2) вЂ“ (3.3)] may be used in place of min in equation (3.26).

TABLE 3.3 A Fuzzy Relation

Fred Mike Sam

John 0.2 0.8 0.5

Jim 0.9 0.3 0.0

Bill 0.6 0.4 0.7

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

Figure 3.6 Membership functions of two fuzzy numbers A and B.

As an example of composition, suppose the universe X has three members, and

the universe Y has two members. Let the relation between X and Y be given in

(3.27) as matrix R:

2 3

0:4 0:8

4 0:2 0:9 5

RВј (3:27)

1 0

Figure 3.7 Part of one quadrant of fuzzy relation A ! B between fuzzy numbers A(x) and

B( y) shown in Figure 3.6. Quadrant shown is x , 0, y . 0.

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47

3.5 TRUTH VALUE OF FUZZY PROPOSITIONS

Now suppose that a fuzzy set A, a fuzzy subset of X, has grades of membership

A Вј (0:5 0:8 0:2) (3:28)

and we wish to compose fuzzy set A with R to get the grades of membership in fuzzy

set B, a fuzzy subset of Y. Then

Bj Вј max(min(A1 , Ri, j ), min(A2 , Ri, j ) . . . (3:29)

Applying (3.29) to the data in (3.27) and (3.28), we obtain

B Вј (0:4 0:8) (3:30)

3.5 TRUTH VALUE OF FUZZY PROPOSITIONS

A general form of a common simple fuzzy proposition is

(A (comparison operator) B) (3:31)

in which A and B are compatible data items and the comparison operator is compa-

tible with the data types of A and B. Not all data types are compatible; for example,

we cannot compare an integer to a character string. The data types also restrict the

comparison operators that may be used with them; for example, we may only use

Boolean comparisons between scalar numbers, but may use fuzzy (approximate)

comparisons if one of the operands is fuzzy. Below we will discuss the most import-

ant fuzzy propositions: comparison of single-valued data, including members of

discrete fuzzy sets; and comparison of fuzzy numbers.

3.5.1 Comparing Single-Valued Data

Suppose a fuzzy proposition A has the form

A: (x Вј y) (3:32)

where x and y are single-valued data, such as integers. Say x has value 3 and truth

value 0.8; y has value 3 and truth value 0.7. What is the truth value of proposition A?

It would seem fairly obvious that A cannot have a truth value greater than the

truth value of either of its components. The truth value of x is 0.8; the truth value

of the comparison is 1.0, since the value of x equals precisely the value of y; and

the truth value of y is 0.7. The truth value of A is then

tv(A) Вј min(0:8, 1:0, 0:7) Вј 0:7 (3:33)

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

This is easily generalized to

tv(x (comparison operator) y) Вј min(tv(x), tv(comparison), tv( y)) (3:34)

giving us a general way for evaluating the truth value of fuzzy proposi-

tions, which involve single-value data. Note that for Boolean comparison opera-

tors (,, ,Вј, Вј, .Вј, ,, ,.) the truth value of the comparison will always be

0 or 1.)

Members of discrete fuzzy sets may be tested in the same way. Consider

proposition B:

B Вј (size is Small) (3:35)

where size is a discrete fuzzy set, of which Small is a member. The truth value of the

fuzzy set size is one. Since Small is a member of size, the truth value of вЂњisвЂќ is also 1.

The truth value of Small is its grade of membership in size, say 0.75. Then the truth

value of B is given by

tv(B) Вј min(1, 1, (tv(Small)) Вј 0:75 (3:36)

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