Database Reference
In-Depth Information
GT:
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 ∀
u u
,
∈
U
:
1
2
LT:
Let
P
2
is a proximity relation defined over a universe of discourse
U
. The fuzzy relational operator
LT
is defined to be a fuzzy subset of
U
×
U
, where
μ
LT
satisfies the following properties
∀
0
if
u
≤
u
1
2
m
(
u u
,
)
=
m
(
u u
,
)
otherwise
.
GT
1
2
P
1
2
1
u u
,
∈
U
:
1
2
NEQ, GOE, LOE:
Membership functions of fuzzy relations `NOT EQUAL' (
NEQ
), `GREATER
THAN OR EQUAL' (
GOE
) and `LESS THAN OR EQUAL' (
LOE
) can be defined based on that of
EQ
,
0
if
u
≥
u
1
2
m
(
u u
,
)
=
LT
1
2
m
(
u u
,
)
otherwise
.
P
1
2
1
(
u u
u u
u u
,
)
[
1
(
u u
,
)]
m
m
m
=
=
=
−
u u
m
NEQ
1
2
EQ
1
2
GT
and
LT
as follows:
m
m m
α-Cut:
Given a fuzzy set
A
defined on
U
and any number α
∧
[0,1], the α-cut
α
A
, and the strong α-cut,
(
,
)
max[
m
(
,
),
(
u u
,
)]
GOE
1
2
GT
1
2
EQ
1
2
(
,
)
min[
(
u u
,
),
(
u u
,
)]
LOE
1
2
LT
1
2
EQ
1
2
α
{ | ( ) }
{ | ( ) }
α-Resemblance:
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
A
=
u
µ
u
≥
α
α+
A
, are the crisp sets
A
α
+
A
=
u
µ
u
>
α
A
( , )
³ ) as α-resemblant. The following notation is proposed for the notion of two val-
ues
x
,
y
being α-resemblant:
xEQ
∧
y
.
A set P
∧
U
is called an α-preclass on
U E
, , if
x y P
x y
α
EQ
, Î ,
x
and
y
are α-resemblant (i.e.
xEQ
∧
y
holds).
