Database Reference
In-Depth Information
ii. A single number (e.g. Age=22),
iii. Set of scalars (e.g. Aptitude={average, good}),
iv. Set of numbers (e.g. {20, 21, 25}),
v. A possibilistic distribution of scalar domain values (e.g. Age={0.4/average, 0.7/good}),
vi. A possibilistic distribution of scalar domain values (e.g. Age={0.4/23, 1.0/24, 0.8/25}),
vii. A real number from [0,1] (e.g. Heavy=0.9),
viii. A designated null value (e.g. Age=unknown).
Arithmetic Operations
Arithmetic operations on different fuzzy data types are already discussed in Sharma, Goswami & Gupta,
(2008).
Fuzzy Comparison Operators
FTS relational Model is designed to support the different data types as proposed by Rundensteiner et al
(1989) for fuzzy relational representations that correspond to the approach of Zemankova and Kandel
(1984, 1985). To supports queries that may contain qualifications involving imprecise and uncertain
values, FTS relational model is equipped with fuzzy comparison operators. These operators (
EQ,NEQ
)
and (
GT,GOE,LT,LOE
) are defined as follows:
Definition:
A resemblance relation, EQ of
U
is a fuzzy binary relation on
U
×
U
, that fulfills the fol-
lowing properties ∀
x y U
,
∈
, where
U
is the universe of discourse.
i. Reflexive:
m
EQ
( , )
=
1
,
x x
ii. Symmetric:
m
( , )
x y
=
m
( , )
y x
EQ
EQ
Lemma:
Let
EQ
be a resemblance relation on a set
U
. For all α with 0<α≤10, α-
level
sets
EQ
∧
are
tolerance relation on U
.
The concept of an α-resemblance was introduced by Rundensteiner et al (1989).
Definition:
Given a set
U
with a resemblance relation
EQ
as previously defined. Then,
U E
, is
called a resemblance space. An α-level set
EQ
∧
induced by E
Q
is termed as an α-resemblance set. Define
the relationship of two values x
,
y
∧
U
that resemble each other with a degree larger than or equal to α
(i.e.
µ
( , )
³ ) as α-resemblant. The following notation is proposed for the notion of two values
x,y
being α-resemblant:
xEQ y
a
x y
α
EQ
. A set
P
Í is called an α-preclass on
U E
, , if
x y P
, Î ,
x
and
y
are α-resemblant (i.e.
xEQ
a
holds).
To define fuzzy relations GREATER THAN (
GT
) and LESS THAN (
LT
), let us consider a proximity
relation
P
defined as given below:
Definition:
A proximity relation
P
over a universe of discourse
U
is reflexive, symmetric and transi-
tive fuzzy relation with
m
P
u u
1
,
Î (Kandel, 1986).
Definition:
Let
P
1
is a proximity relation defined over
U
. Fuzzy relational operator
GT
is defined to
be a fuzzy subset of
U
×
U
, where
μ
GT
satisfies the following properties ∀
(
,
)
1
Î , where
u u
[ ,
]
U
1
2
u u
,
U
∈
:
1
2
