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between states that are consistent with a disease and states that are not. Therefore,
Sadegh-Zadeh expands this notion of disease to a notion of “disease to a certain
degree”:
Let's assume
H
D
H
to be a small set of human conditions. A fuzzy set
over
is
considered as a set of diseases only if there is a subset
{
D
1
,
D
2
,...,
D
n
}
of
H
and
there is a function
μ
D
so that:
μ
D
:
H →
[
0
,
1
]
with
⎨
1
,
ifH
i
\
D
n
},
called prototype disease.
X
∈{
D
1
,
D
2
,...,
ε
,
if there is a prototype disease
H
j
\
Y
μ
Δ
(
H
i
\
X
)=
with
p-similar
)=
ε
and no prototype disease
H
k
\
((
H
i
\
X
,
H
j
\
Y
.
(3.25)
⎩
Z
((
\
,
H
k
\
)
>
ε
with
p-similar
H
i
X
Z
Δ
=
{
(
,
μ
Δ
(
))
|
∈ H }
and
H
i
H
i
H
i
In this expanded definition a fuzzy set of following kind is created:
D
=
(
D
q
))
,
D
1
,
μ
D
(
D
1
))
,...,
(
D
q
,
μ
D
(
(3.26)
which consists of individual “prototypes” of diseases, which are all members of the
set
to different degrees.
The membership degree
D
. From this, we conclude that
a person may have a disease to a certain degree and that this person may have no
disease to a certain degree at the same time.
As already demonstrated, diseases can be classified by a set of symptoms. In
medicine, the study of classification of diseases is called
nosology
. Conventional
nosological systems classify a disease by cause (
etiology
), by genesis and develop-
ing of the disease (
pathogenesis
) or by the diseases' symptoms.
Today, the most common system is the
International Classification of Diseases
(ICD) that is also a billing system and classifies causes of death [33]. However, as
Sadegh-Zadeh argues, those conventional nosological systems pose problems and
need some improvements. He demands from nosological systems not only provid-
ing a database of classified diseases but also a clinical diagnosis. But due to the
fact that diseases are sets of symptoms that often go along with uncertainty, a one-
dimensional system isn't able to solve this problem as diseases with
n
-dimensional
sets can't be compared in only one dimension. Here is the point where the fuzzy
hypercube comes into play: Considering a disease with a set of criteria of length
n
,
this disease may be converted into a vector of length
n
and therefore displayed in
the hypercube [63].
Another disease with assimilable criteria can by displayed in the same hypercube
and thus, these diseases are not only classified but also comparable through their dis-
tance. The fuzzy sets' Hamming and Euclidean distances can be easily determined.
In doing so, one is able to make statements about relationships to other diseases and
possible diagnosis. This could be also an advantage if a disease is unknown. More-
over, it is possible to display the developing of a disease in the hypercube. Every
point in the hypercube would be the disease at a particular time, and achievements
of therapies, for example, could be reproduced. [63]
μ
D
(
Di
)
is of interval
[
0
,
1
]
