The meaning of the trigonometric expressions appearing in Eqs. 
(10) and (11) is obvious from Figure 3. The condition in Eq. 
(9) may now be used by the query processor for the 
approximate selection of database objects which satisfy the 
fuzzy query. 
  
  
Xu= «B cos s - CLOSE) (10) 
Xi = cosa] cos [r + CLOSE) (11) 
Thus by means of the concept of fuzzy partitions and generic 
values the original fuzzy query is transformed into an interval 
comparison problem in which the interval bounds correspond to 
the cut-out points, peak points, turnover points, or any other 
desirable characteristic points of the membership function of the 
fuzzy object. It must however be noted that manipulation of 
fuzzy numeric data by interval arithmetic can only be tolerated 
for low accuracy requirements (Kandel, 1986, Klir and Folger, 
1988). 
3.2 Construction of Fuzzy Membership Functions 
From Geometric Fuzzy Partitions. 
Construction of membership functions is a necessary step 
towards verification and validation of the assigned partitions in 
line with the second criteria in section 2.1. In addition the 
membership functions are useful in themselves, as theoretically 
well founded tools for representing and querying fuzzy 
knowledge(Schmucker, 1984; Dubois and Prade, 1988; 
Kandel, 1986). 
The construction of membership functions must be done subject 
to certain desirable characteristics of membership functions 
(Zadeh et al. 1975; Kandel, 1986; Klir and Folger, 1988): 
(i) The membership function must map the set of objects in 
the universe of discourse into the interval [0,1]. 
(ii) The membership function must satisfy necessary fuzzy 
set theoretic properties with respect to fuzzy union, 
intersection, complementation etc. 
Let R(x) represent a general fuzzy predicate or fuzzy comparison 
operator in the PRED set, where xe X is some crisp numeric 
value in the real numbers universe. If the generic value induced 
by R on x is denoted by x, and x is selected such that it is a 
bounding value, then it lies on the boundary of the partition 
generated by R in the universe of discourse. 
Denote the amount by which R "stretches" x by D. Then 
D =|x- x | represents the width of the partition generated by R 
in X x X. The set {x |x = R(x)}, for all xe X, defines the 
partition induced by R. Let y represent some crisp value (or the 
crisp generic value of a fuzzy number) in the search space. 
Denote the difference between y and x by d. Then by set 
membership definition (Klir and Folger, 1988), y is contained 
in the partition induced by R(x) if the condition(see Figure 4): 
dely-d<D (12) 
is satisfied. This condition means that y falls within the 
"stretch" of R(x). 
To provide a fuzzy set theoretic basis for the comparison of 
fuzzy values, membership functions for the general fuzzy 
restriction, R(x), may be constructed by the following 
procedure(Figure 4): 
1. Setthe width D of the partition generated by R(x) on the 
v-axis (horizontal axis) as shown in Figure 4. 
Draw a line of unit length along the u-axis(vertical axis). 
Link the end of the unit line with point D on the v-axis. 
Plot the distance d, of y from x, along the v-axis. 
Mirror project d perpendicularly on to the u-axis and 
denote its image by Up. 
Un B UN 
6. The distance, Up of the projection point is proportional 
to the strength of the membership of x; in the fuzzy set 
represented by R(x). It may therefore be regarded as a 
first approximation to its fuzzy membership value. 
7. Letting y cover the range of all values in X modify Up 
by applying intensification, dilation, normalization, 
concentration (Schmucker, 1984; Kandel, 1986), or any 
other fuzzy set theoretic transformation function, F, to 
arrive at a suitable shape of the membership function. 
From Figure 4 an approximate formula for computing the fuzzy 
membership value is obtained(Eq. 13). 
Ur = $ (13) 
To enforce the condition that fuzzy membership values must be 
in the range [0,1] Eq.(13) is rewritten as 
ee | 0, ify#RG) | 
| Fld) if y = R(x) | qo 
Table 3: COMPUTATION OF GENERIC VALUES FROM GENERIC BAND 
WIDTH. 
Fuzzy Predicate Generic Value Equation 
much greater than(x) 
much less, than(x) 
slightly more than(x) 
slightly less than(x) 
x" = x/tan(pi/4 - VERYWIDE") 
x" = x/tan(pi/4 + VERYWIDE" 
x" = x/cos(pi/4)cos(pi/4 - VERYCLOSE') 
x" = x/cos(pi/4)cos(pi/4 + VERYCLOSE) 
more. or less(x) x" = x/cos(pi/4 +- CLOSE1") 
about(x) x" = x/cos(pi/4 +- CLOSE) 
roughly_equal_to(x) 
x" = x/cos(pi/4 +- CLOSE") 
more_than(x) x" =x +D; D >0 real number 
less_than(x) x" =x +D; D <0 real number 
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