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In-Depth Information
Definition 8.
ξ
is a Zadeh space
iff there are X , Y , Z, and d such that:
1.
ξ
=
X
,
Y
,
Z
,
d
;
2.
is a Zadeh structure,
3. d is metric over Z.
X
,
Y
,
Z
,
d
To pass to identify the fuzzy hypercube as a particular Zadeh space, Sadegh-Zadeh
defined some more concepts and - for simplicity's sake he also confined himself to
finite sets only
Definition 9.
A set X with n elements has
2
n
subsets. We name the set of all these
subsets the
powerset of
X and it is denoted by
2
X
.
Definition 10.
The
fuzzy powerset of
X , is the set of all fuzzy subsets in X and it is
denoted by F
2
X
(
)
.
2
X
2
X
2
X
. Partic-
(
)
(
)
⊆
We emphasize that
F
is uncountably infinite and we have
F
2
X
(
)
ularly we mention that
F
is not a fuzzy set!
2
X
Definition 11.
A Zadeh structure
ξ
=
X
,
Y
,
Z
,
is called
complete
if Z
=
F
(
)
.
Finally, Sadegh-Zadeh brought it to the point:
“In a complete Zadeh structure
2
X
2
X
n
. The sin-
X
,
Y
,
F
(
)
, the fuzzy powerset
F
(
)
forms a unit hypercube
[
0
,
1
]
of 2
X
2
X
are the
n
coordinates of the cube. Thus, the 2
n
members
gletons
{
x
i
}
⊆
F
(
)
of the ordinary powerset 2
X
inhabit the 2
n
corners of the cube. The rest of the fuzzy
2
X
n
powerset
F
(
)
fills in the lattice to produce the solid cube. The cube
[
0
,
1
]
may
therefore be termed a
fuzzy hypercube
.” [64, p. 313]
Definition 12.
A Zadeh space with metric d is complete..
Using Minkowski metrics in the fuzzy hypercubes, we can calculate distances be-
tween fuzzy sets. For that we define
c
n
i
=
1
as the sum of the mem-
bership values of the corresponding fuzzy set
A
(
fuzzy set cardinality
or
fuzzy set
count
). However, this is the Hamming distance of
a
to the empty set 0 at the origin
of the hypercube:
(
A
)=
∑
μ
A
(
x
i
)
n
i
=
1
μ
A
(
x
i
)=
n
i
=
1
|
μ
A
(
x
i
)
−
0
|
=
n
i
=
1
|
μ
A
(
x
i
)
−
μ
0
(
x
i
)
|
=
l
1
c
(
A
)=
(
A
,
0
)
(3.15)
3.5.2
Fuzzy Entropy
“The amount of vagueness and indeterminacy a set carries within itself is referred
to as its fuzziness or
fuzzy entropy
” Sadegh-Zadeh wrote starting the section “Fuzzi-
ness and clarity” in [64]. To the word “
fuzzy entropy
” he pinned a footnote saying
“This terminology is due to relationsships between fuzziness and probabilistic Shan-
non entropy. The latter one is a special cse of fuzzy entropy” and then he referred
to the works of Kosko that already have been cited, but first of all to a pioneering
article by Aldo de Luca and Settimo Termini [21].
