Information Technology Reference
In-Depth Information
Fig. 3.8
The 3-dimensional cube
I
3
; Fig. in [35]
What does a fuzzy set look like? A point in a cube. The set of all fuzzy subsets is
the unit hypercube
I
n
n
. A fuzzy set is a point in the cube
I
n
.” [35, p. 216]
The set of all fuzzy subsets is the unit hypercube
I
n
=[
0
,
1
]
n
. A fuzzy set is a
=[
,
]
0
1
point in the cube
I
n
.”
[35, p.
216] For illustrations see Figs 3.8 and 3.9.
“Ver-
tices of the cube
I
n
So the ordinary power set 2
X
,thesetof
are nonfuzzy sets.
all 2
n
nonfuzzy subsets of
X
, is the Boolean
n
-cube
B
n
:2
X
B
n
. Fuzzy sets fill
=
in the lattice
B
n
to produce the solid cube
I
n
2
X
I
n
.
:
F
(
)=
Therefore, fuzzy set
A
=
{
(
x
1
,
a
1
)
,...,
(
x
n
,
a
n
)
}
is represented by the
n
-dimensional vector
(
x
n
,
a
n
)
and
all
a
i
are elements in
[
0
,
1
]
. Consequently,
A
is a point in the
n
-dimensional unit
n
.
hypercube
[
0
,
1
]
3.5.1
The Fuzzy Hypercube and Structures
In 1989 Sadegh-Zadeh founded the international journal “Artificial Intelligence in
Medicine” where he published in the following year his article “Advances in fuzzy
theory” (Fig. 3.10 (a)) going back to Kosko's results. He pointed to the success
of Zadeh's fuzzy concepts and methods in Medicine that already “have become
standard in the application of fuzzy-theoretic tools to medical artificial intelligence
subjects”. However, by “Advances” he now turned to Kosko's new concepts of
the fuzzy hypercube, fuzzy set inclusion, equality, and similarity that “are of high
relevance to artificial intelligence in medicine research.” [64, p. 309]
We will present these concepts in the vein of this article and before that we will
reproduce Sadegh-Zadeh's definitions of
basic Zadeh structures
,
Zadeh structures
and
Zadeh spaces
. After that we will “complete” the fuzzy hypercube to be a
Zadeh
space
:
12
12
Here, “iff” means “if and only if”.
