Database Reference
In-Depth Information
Table 3. Excerpt of the BookDB relation
ID
ISBN
Vendor
Price
b
b
b
b
4
0679726691
0679726691
0679726691
0062059041
BooksForLess
LowestPrices
QualityBooks
BooksForLess
14.75
13.50
18.80
7.30
2.3.2.1 Fuzzy Sets Theory
Fuzzy set theory, introduced by Zadeh (Zadeh, 1956; Zadeh, 1975, Zadeh, 1999), is a mathematical tool
for translating user's perception of the domain (often formulated in a natural language) into computable
entities. Such entities are called fuzzy terms (linguistic labels). Fuzzy terms are represented as fuzzy
sets and may be fuzzy values (e.g.,
young
), fuzzy comparison operators (e.g.,
much greater than
), fuzzy
modifiers (e.g.,
very
,
really
) or fuzzy quantifiers (e.g.,
most
).
Mathematically, a fuzzy set
F
of a universe of discourse
xiv
U
is characterized by a membership func-
tion
m
F
given by:
→
m
:
U
u
0,1
F
→
m
(u)
F
where
m
F
(u)
, for each
uU
, denotes the degree of membership of
u
in the fuzzy set
F
. An element u U is
said to be in the fuzzy set
F
if and only if
m m
F
(u)
0 and to be a full member if and only if
m
F
(u) =
1. We
call support and kernel of the fuzzy set
F
respectively the sets:
{
}
{
}
support
( )
F = u U |
∈
m
(u)
0 and kernel
( )
F = u U |
∈
m
(u) =
1
.
F
F
Furthermore, if
m
fuzzy sets
F
1
,
F
2
, … and
F
m
are defined over
U
such that ∀
i = m ,F
1,
≠
j
,F U
≠
i
i
and ∀ ∈
∑
m
, the set
{F
1
,…, F
m
}
is called a fuzzy partition (Ruspini, 1969) of
U
.
For example, consider the attribute Salary with domain
D
Salary
= [0, 110] K€. A typical fuzzy partition
of the universe of discourse
D
Salary
(i.e., the employees' salaries) is shown in Figure 8, where the fuzzy
sets (values)
none
,
miserable
,
modest
,
reasonable
,
comfortable
,
enormous
and
outrageous
are defined.
Here, the crisp value 60K€ has a grade of membership of 0.5 for both the
reasonable
and the
comfort-
Table
fuzzy sets, i.e.,
μ
reasonable
(60K€) =
μ
comfortable
(60K€) = 0.5.
u U,
(u)
F
i
2.3.2.2 Practical Extensions of SQL
In the literature, several extensions of SQL have been proposed to allow the use of fuzzy terms in data-
base queries. Examples include the work of Tahani (Tahani, 1977), FQUERY (Zadrozny & Kacprzyk,
1996) and SQLf (Bosc & Pivert, 1995).
