Information Technology Reference
In-Depth Information
Definition 4.
ξ
is a
basic Zadeh structure
iff there are X , f , and A such that:
ξ
=
,
,
,
1.
;
2. X is a non-empty set;
3.
X
f
A
f is a function such that f
:
X
→
[
0
,
1
]
;
4. A
=
{
(
x
,
f
(
x
))
|
x
∈
X
}
.
Definition 5.
A is a fuzzy set iff there are X and f such that
ξ
=
X
,
f
,
A
,
is a basic
Zadeh structure.
However, Sadegh-Zadeh refered here to the more general concept of a ('non-basic')
Zadeh structure
that he had already defined in the “Advances”-article. We notice that
the following definitions of
Zadeh structures
and
Zadeh spaces
are generalizations
of the
basic fuzzy structures
:
14
Definition 6.
ξ
is a
Zadeh structure
iff there are X , Y , and Z such that:
1.
;
2. X is a non-empty set;
3. Y
ξ
=
X
,
Y
,
Z
,
=
{
μ
1
,
μ
2
,...}
is a finite or infinte set of functions;
4. Z
=
{
A
1
,
A
2
,...}
is a finite or infinite family of sets;
5. Each
μ
i
∈
Y maps X to the unit interval
[
0
,
1
]
;
6. A
i
=
{
(
x
,
μ
i
(
x
))
|
x
∈
X
}
for every A
i
∈
Z with i
≥
1
.
To pass on to
Zadeh spaces
we need the concept of
metric spaces
:
Ω
,
,
Definition 7.
The pair
d
is a metric space
iff there are
Ω
, and d such that:
1.
is a non-empty set;
2. d is a binary function from
Ω
Ω
×
Ω
to
R
such that for all x
,
y
,
z
∈
Ω
:
•
d
(
x
,
y
)
≥
0
(non-negativity),
•
d
(
x
,
y
)=
0
iff x
=
y
(identification property),
•
d
(
x
,
y
)=
d
(
y
,
x
)
(symmetry),
•
d
(
x
,
y
)+
d
(
y
,
z
)=
d
(
x
,
z
)
(triangel property),
n
,thesetof
d
is called a
metric
or a
distance function
over
Ω
and if, e.g.,
Ω
= R
all
n
-dimensional real vectors (
n
≥
1), then we get the most well-known class of
metrics, the
1
/
p
for
p
i
p
Minkowski metrics
:
l
p
•
(
x
,
y
)=(
∑
|
a
i
−
b
i
|
)
≥
1
=
1
Special cases of this metrics class are the following distance functions:
n
i
Hamming distance:
l
1
•
(
x
,
y
)=
∑
|
a
i
−
b
i
|
,
(
p
=
1).
=
1
2
1
/
2
Euclidean distance:
l
2
n
i
=
1
•
(
x
,
y
)=
∑
|
a
i
−
b
i
|
,
p
=
2).
14
Moreover, in Sadegh-Zadeh's own contribution to this topic (chapter 2) we have a defini-
tion of his concept of a
fuzzy structure
.
