Information Technology Reference
In-Depth Information
Definition 21.
If A is a fuzzy set of arbitrary length and if X is a part of A, then we
write A
\
X to indicate that A is a fuzzy set with X being part of it.
With this notation Sadegh-Zadeh arranged in [66] human conditions, like heart at-
tack and stomach ulcer, “as
uniform fuzzy sets
with respect to their comparable
criteria
{
C
1
,
C
2
,...,
C
m
}
:
•
myocardial_infarction
\{
(
C
1
,
a
1
)
,
(
C
2
,
a
2
)
,...,
(
C
m
,
a
m
)
}
•
gastric_ulcer
\{
(
C
1
,
b
1
)
,
(
C
2
,
b
2
)
,...,
(
C
m
,
b
m
)
}
as, for example
•
myocardial_infarction
\{
bodily_lesion
,
1
)
,
(
pain
,
0
.
7
)
,
(
distress
,
0
.
8
)
}
•
gastric_ulcer
\{
(
bodily_lesion
,
1
)
,
(
pain
,
0
.
3
)
,
(
distress
,
0
.
5
)
}
and to compare them with respect to their uniform, terminal criteria segments:
•
X
=
{
(
C
1
,
a
1
)
,
(
C
2
,
a
2
)
,....,
(
C
m
,
a
m
)
}
•
Y
=
{
(
C
1
,
b
1
)
,
(
C
2
,
b
2
)
,...,
(
C
m
,
b
m
)
}
”
Obviously there are many other features that are not considered in these segments,
e.g. blood pressure, bacterial infections etc. because they may not be comparable.
Due to the fact that this arrangement allows just
partial
comparisions of human
conditions Sadegh-Zadeh defined “partial similarity, symbolized by p-similar
(
A
\
X
,
B
\
Y
)
, according to the following definition [66, p. 623f.]:
Definition 22.
p-similar
(
A
\
X
,
B
\
Y
)=
r,
if and only if similar
(
X
,
Y
)=
r.”
With this definition and [theorem 2 in the list at the end of the subsection 3.5.3] we
can calculate the partial similarity of myocardial_infarction and gastric_ulcer in the
example above as follows:
1
+
0
.
3
+
0
.
5
1
.
8
(
\
,
\
)=
=
=
.
p-similar
myocardial_infaction
X
gastric_ulcer
Y
0
72.
1
+
0
.
7
+
0
.
8
2
.
5
Let's assume that
{
D
1
,
D
2
,...,
D
n
}
would be a small set of human conditions, be-
cause of a set of criteria
which these conditions have in common.
Each of these conditions is interpreted in a certain society as a disease. For this so-
ciety there is an agreement of degree
{
C
1
,
C
2
,...,
C
m
}
ε
of partial similarity. This degree is “a pillar
of the construction” [66, p. 623f.]:
Definition 23.
1. Any element of the base set
{
D
1
,
D
2
,...,
D
n
}
is a disease.
2. A human condition H
\
X is a disease, if there is a disease D
i
\
Y
∈{
D
1
,...,
D
n
}
and an
ε
>
0
such that p-similar
(
H
\
X
,
D
i
\
Y
)
≥
ε
.
According to this definition a proper choice of
is
chosen, the more diseases a society accepts and vice versa. However, the value of
ε
ε
is essential: The smaller the
ε
is not chosen by a physician but by society. Anyway, this notion of disease is a
notion that can be comprised in binary logic, an explicit difference is made between
