jen 
the 
the 
ce. 
set 
ied 
2) 
the 
ZY 
nal 
set 
ny 
zy 
13) 
14) 
where F is some appropriate function. 
Practical membership functions for the fuzzy constraints in the 
PRED set can now be obtained by substituting for D and d 
based on the equations of the generic values defined in Table 3. 
For example to get the membership function for the fuzzy 
expression "much more than x", the width of the partition 
induced by the fuzzy restriction is computed from 
D = abs(x - x) (15) 
where x is a generic value lying on the boundary of the partition 
induced by much more than(x), and its value can be computed 
using the equations in Table 3 as 
ez n (E- VERYWIDE) qe 
From Table 3 VERYWIDE is defined as 
‘a VERYCLOSE 
VERYWIDE' = WIDE + ESAE (17) 
and after appropriate substitutions and simplification we get 
X 
[1-5 hen) ii 
The problem of determining the compatibility between y and 
much more than(x) is equivalent to the problem of finding the 
membership value of y in the fuzzy set induced by the fuzzy 
restriction "much more than" on x. From Eq. (14) an 
approximate value of the membership of y in the fuzzy relation 
R(x) is given by Eq. (19). 
X= 
abs(y - x) 
abs(x - x) ao 
Ur(y) = 
Assuming a square compatibility function for both much_more 
than(x) and much less than(x) their membership functions 
are given by Eqs. (20) and (21). The plot of the membership 
function for much less than (5) is shown in Figure 5. 
  
| 0, if x < x 
ios GO e LEES ifx>y>x (20) 
| 1. else where 
Rp ifx<y<x 
Uc<x(Y)={ 0, ify2x (21) 
| 1, elsewhere 
181 
  
d=y-x 
D=x -x; x=R(x) 
Figure 4: Construction of membership function 
from fuzzy partitions. 
After substituting the expression for x in Eq. (20) using Eq. 
(16) we get 
  
  
  
0, ifx<zx | 
| 
E 2 fx>y>x 
Ussx (¥) = 1 EX 
tan (I. E 
4 48 4 + log x 
| 
i, else where | 
(22) 
as the membership function for much more than(x). 
Using this approach membership functions can be constructed 
for all the fuzzy predicates in the PRED set. A Graphical 
representation of the fuzzy membership functions for about(5), 
more or less(5), and roughly equal to(5) are shown in 
Figures 7 to 8. It is clear by looking at the shapes of the 
  
0.8 4 
0.6 4 
0.4 4 
MEMBERSHIP VALUE 
0.2 4 
  
  
0.0 
  
" T T 
0 2 4 6 8 
DATABASE VALUE 
Figure 5: Membership Function for much less than 5