Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 141

x belongs to F partially, then the degree of membership of x in fuzzy set F can have
any intermediate value, such as 0.5, 0.9, etc ., within 0 and 1. Mathematically, this
is stated as

-

1, if xF (completely)

°

P x 0,1 , if x F

xF

(partially)

®
°

F

0, if totally non-member

¯

1.0 1.0

Short Tall

0.5

Tall

Short

Height

1.8m 1.801m

a

2.1m 1.5m 2.1m Height

1.8m
b

Figure 4.1. (a) Crisp (ordinary) set; (b) fuzzy set

Figure 4.1(a) shows an example of two crisp sets, “short” and “tall”, where it is
shown that even if the height of a person is 1.7999 m then that person definitely
belongs to the “short” category only. This is because the crisp set “short” includes
heights up to 1.8 m. In contrast, if the height of the same person had been just
1.8011 m, as per the same Figure 4.1(a), then the person would belong to the
category “tall”, as in this case the height is 0.0001 m greater than 1.801 m and that
categorizes the person into the crisp set “tall”. This is obviously quite impractical.
Similarly, Figure 4.1(b) shows the example of two fuzzy sets, “short” and
“tall”, where it is shown that if the height of a person is less than or equal to 1.5 m,
then the person belongs to the category “short”, whereas if the height is say 1.8 m
then the person belongs to the category “short” with a degree of membership 0.5
and at the same time the person is considered as “tall” with a degree of
membership equal to 0.5.
In order to explain the importance of fuzzy sets or membership functions,
Boyle's law, as a practical example, is considered, that states that the pressure ( P )
of a given mass of gas varies inversely proportional to the volume ( V ) of the gas,
provided the gas temperature ( T ) remains constant.
Using the fuzzy linguistic rules, Boyle's law can be stated as:

Rule-1: IF pressure is high and temperature is constant THEN volume is

low

Rule-2: IF pressure is medium and temperature is constant THEN volume is

medium

Rule-3: IF pressure is low and temperature is constant THEN volume is

high .