similarly for R = much less than we have: 
much less than ; = {(x,y)|x € X, y € X, x << y} for all 
(xy) € XxX. (2) 
where, >> and << have the usual meaning, much greater than, 
and much less than, respectively. 
Definitions of all the relations in the PRED set can be produced 
by a similar process. For practical purposes it is instructive to 
represent the partitions in Eqs. (1) and (2) geometrically. This is 
achieved by transforming the definitions in Eqs. (1) and (2) into 
their polar counterpart. Figure 3 defines the important 
geometrical parameters needed for this. Thus referring to Figure 
3, if the partition generated by the general comparison operator, 
R, is denoted by Pp it can be defined as: 
Pg = ([0,,04] | 0t, € [0, 1/2], Ot € [0, 1/2], Ot 2 Oy} 3) 
Now substituting the operator equal , for R, in Eq. (3) we get 
the following definition for the equality operator (Eq. 4): 
P= lim 
Œy>7/4,0/—>7/4 
( [0.01] ) 2 [1/4, 1/4] (4) 
The equality of the upper and lower bounds in Eq. (4) means 
that the equality operator partitions the search space diagonally. 
This definition is therefore equivalent to the one given in 
Dowsing et al(1986). The partitions induced by the operators 
greater and less are defined by Eqs. (5) and (6) below. 
P = lim 
00,0 —7T/4- 
([0.,04]) - [0, x/4-] (5) 
P.= lim 
y >7/4+,0 > 7/2 
{[œu,0u]} = [x/4+, x/2] (6) 
where z/4- and 7/44 mean infinitesimally smaller than, and 
greater than, respectively. 
The above strategy can be used to define the partitions 
corresponding to the fuzzy restrictions "much greater than" and 
"much less than" as given in Eqs. (7) and (8) below. 
P,,= lim ([0,04]) (7) 
0,90, 
& 7/4 - VERYWIDE' 
Pec = lim ([0,,04]) (8) 
Q,—7/4 4 VERYWIDE', 
oy 7/2 
178 
  
  
  
GG:x- Xgg = x/tan(tan(n1/4 - aj) for much more than(x) 
E: y =x =x, for equal to(x) 
LL: x = xj = x/tan(nv4 a) for much less than (x) 
Oy, 01 : 
Xy = X = x /cos(T1/4)cos(T1/4 - ji), 
X] = X = x/cos(x1/4)cos(x1/4 + a), for all other cases. 
Figure 3: Geometrical parameters of the partitions 
induced by fuzzy expressions. 
In Eqs. (7) and (8) the terms WIDE, VERYWIDE, and others 
listed in Table 1, represent a fuzzy constant whose value may be 
subjectively assigned to reflect individual conception of the 
vague expressions "verywide", "wide", etc. They may be 
interpreted as generic band width of the fuzzy intervals 
associated with the fuzzy expressions. Based on these 
parameters, generic band widths associated with all fuzzy 
expressions in PRED can be assigned as shown in Table 2. 
When assigning values to the fuzzy constants in Tables 1 and 2 
two important criteria must be considered: 
1. Itis important for the generated partitions to be 
compatible with common sense in accordance with the 
criteria for suitable characteristics of membership functions 
(Magrez and Smets, 1989). 
2. The usefulness of fuzzy sets in modelling fuzzy 
concepts, class, or linguistic variables depends on the 
appropriateness of the selected membership 
functions(Kandel, 1986). 
Because in this experiment membership functions are 
constructed from subjectively assigned partitions, the second 
criteria will be used to test the validity of the assigned partitions. 
Accordingly the partitions assigned to fuzzy variables will be 
deemed proper if membership functions constructed from them 
are similar to or comparable with those assigned by 
conventional fuzzy sets methods.