A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 99

p(S 1 )andp 2 =p(S 2 )of two statementsS 1 andS 2 , then possible values of
p 1 &p 2 =p(S 1 &S 2 )form the interval

[max (p 1 +p 2 − 1 ,0),min (p 1 ,p 2 )]

As a numerical estimate, we can use a midpoint of this interval

p 1 &p 2 ≡^12 max (p 1 +p 2 − 1 ,0) +^12 min (p 1 ,p 2 )

or more generally, we can take a weighted average

p 1 &p 2 ≡αmax (p 1 +p 2 − 1 ,0) + (1−α)min(p 1 ,p 2 )

whereα∈(0,1). The corresponding OR operations are

p 1 (OR)p 2 ≡αmax (p 1 ,p 2 )+(1−α)min(p 1 +p 2 ,1)

Bouchon-Meunier, Kreinovich and Nguyen argue that these AND operations
explain the empirical law in psychology according to which a person can nor-
mally distinguish between no more than 7 ± 2 classes of objects. This is related
to the fact that in intelligent control, experts normally use≤ 9 different degrees
(such as ìsmall,î ìmedium,î etc.) to describe the value of each characteristic
(see [10]).

Negations The logical operationx^0 =NOTxsatisfies 10 =0and 00 =1
andx≤yimpliesx^0 ≥y^0. The operation is astrong negationif it also
satisfies(x^0 )^0 =x. Such an operation is also called aninvolutionor aduality.
Strong negations are characterized as continuous, strictly decreasing functions
η:[0,1]→[0,1]that satisfy

η(η(x)) = x

η(1) = 0

η(0) = 1

The word negation is often used to mean strong negation. Here are some com-
mon examples.

1. Standard: x^0 =1−x


2. Sugeno: x^0 =

1 −x

1+ax

,a>− 1

3. Yager: x^0 =(1−xa)

(^1) a
,a> 0

4. Exponential: x^0 =e

a

lnx,a> 0

5. Logarithmic: x^0 =loga(a−ax+1),a> 0 ,a 6 =1

6.x^0 =1−(1−(1−x)a)

(^1) a
,a> 0