Figure 4.16 Truth value of proposition P ¼ (A ,¼ B). Shown are A, B, triangular fuzzy

numbers, and C, ,¼ B. Truth value tv(A ,¼ B) of proposition (A ,¼ B) is 0.7.

If M and N are two fuzzy subsets of the real numbers ht(M, N) stands for the

height of their intersection. More formally,

ht(M, N) ¼ supx { min (M(x), N(x)}

Now, we may de¬ne v( ), shown in Figure 4.16.

Let us now return to the fuzzy proposition in Example 4 in Section 4.5. Suppose

pulse ¼ 56 and “around” 60 produces the triangular fuzzy number A with base on

the interval [45, 75], vertex at x ¼ 60, and it has straight-line segment sides. Then

the truth value of this fuzzy proposition is v(56, (,A)) ¼ (,A)(56), the membership

value of 56 in the fuzzy set (,A). The truth value is evaluated to be 0.27.

4.7 FUZZY PROPOSITIONS

Propositions in a fuzzy expert system are found in the antecedent of rules, where

their truth values are combined to yield the antecedent truth value.

Among the simpler propositions are those that test whether a datum exists, ignor-

ing its value. The proposition

x (4:46)

simply asserts that x exists; that is, it tests whether x has been assigned a value. The

truth of that proposition is the truth value of x. If, for example, x had been assigned a

value of 36 with truth value 0.75, the truth value of (4.31) would be 0.75. If x does

not exist, the proposition™s truth value is 0.

Most propositions test the value of a datum against some speci¬ed value. Prop-

ositions involving single-valued data have been discussed in Chapter 3, Section 3.5.

Here, we brie¬‚y review that discussion.

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79

4.7 FUZZY PROPOSITIONS

A proposition involving only single-valued data (integers, ¬‚oats, strings) is of the

form

P ¼ (value of datum)(comparison operator)(comparison value) (4:47)

and its truth value is

tv(P) ¼ tv(datum) AND tv(comparison) AND tv(comparison value) (4:48)

Another proposition tests the grade of membership of a member of a discrete fuzzy

set. For example,

size is Small (4:49)

where size is a discrete fuzzy set of which Small is a member. size is, of course, mul-

tivalued, having several members, of which Small is 1. The truth value of the entire

discrete fuzzy set size is 1; the truth value of the comparison is 1, since Small is a

member of size; and the truth value of Small is the grade of membership of Small in

size, say 0.523. The truth value of the proposition is then min(1, 1, 0.523) ¼ 0.523.

A more complex proposition is frequently used in fuzzy control:

x is fast (4:50)

where x is a scalar number and fast is a fuzzy number. The truth value of x might be

0.765 (although in the real world it is more likely to be 1); the truth value of the com-

parison is the grade of membership of x in the fuzzy number fast, say 0.654; and the

truth value of the fuzzy number itself is 1. The truth of the proposition is then

min(0.765, 0.654, 1) ¼ 0.654.

These comparisons may involve hedges. For example, (4.34) could be modi¬ed

to read

x is very fast (4:51)

When evaluating (4.35), we must include the effect of the hedge very. Suppose that

very has been de¬ned as squaring membership values. very fast is another fuzzy

number, obtained from fuzzy number fast by squaring its grades of membership.

Since the grade of membership of x in fast was 0.654, the grade of membership

of x in very fast is 0.6542 ¼ 0.428. The truth value of (4.35) is then min(0.765,

0.428, 1) ¼ 0.428.

Similarly, if (4.33) had read

size is somewhat Small (4:52)

and somewhat were de¬ned ¬¬¬¬¬¬¬¬¬¬¬ a square-root transformation, the truth value of

pto be

(4.52) would be min (1, 1, 0:523 ¼ 0:723) ¼ 0:723.

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80 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

Fuzzy numbers may be speci¬ed as literals using hedges. For example, prop-

osition (4.50) could have been written

x is about 70 (4:53)

in which the hedge about changes 70 into a fuzzy number with dispersion say þ/2

10% around 70.

Propositions may include approximate comparisons; for example, in

speed , ¼ roughly 30 (4:54)

the truth value of the comparison is obtained by comparing speed with a fuzzy

number ,¼ about 30, as discussed in Section 4.7. Suppose speed is 35, with

truth value 1. The membership of 35 in (, ¼ roughly 30) might be 0.7, and

the truth value of the fuzzy number itself is 1. The truth value of the proposition

is then min(1, 0.7, 1) or 0.7.

Propositions may also involve truth values directly. For example, suppose we

have a discrete fuzzy set size that has a member Large. We can test the grade of

membership of this discrete fuzzy set member directly, by using the notation size.-

Large to represent Large™s grade of membership, in proposition

size.Large ,¼ 0:250 (4:55)

The truth value of the grade of membership itself is 1, as are the truth values of

all truth values. If size.Large is 0, the truth value of the comparison is 1, as is the

truth value of the literal 0.250. The truth value of the proposition is then

min(1, 1, 1) ¼ 1.

A similar proposition is

x.cf ¼ 0 (4:56)

which tests whether the truth value of x is 0. Truth values so tested have themselves a

truth value of 1. The truth value of the proposition (x.cf ¼ 0) is then the fuzzy AND

of the truth value of x.cf, one; the truth value of the Boolean comparison ¼ 1 or 0;

and the truth value of 0, set to one by default since it is a literal.

The truth value of complex logical propositions is obtained simply by using the

logical connectives AND, OR, and NOT as indicated in Section 4.2.1, evaluating

from left to right, using parentheses if necessary. For example, suppose the

complex logical proposition P is given by

P ¼ A AND B OR NOT C (4:57)

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81

4.7 FUZZY PROPOSITIONS