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10
Analysis of Fuzzy Logic Models
Beloslav Rie
Ā
an
M. Bel University, Banská Bystrica,
Matematický Ústav SAV, Bratislava
Slovakia
1. Introduction
One of the most important results of mathematics in the 20th century is the Kolmogorov model
of probability and statistics. It gave many impulses for research and develop so in theoretical
area as well as in applications in a large scale of subjects.
It is reasonable to ask why the Kolmogorov approach played so important role in the
probability theory and in mathematical statistics.
In disciplines which have been very
successfull for many centuries.
Of course, Kolmogorov stated probability and statistics on a new and very effective
foundation - set theory. For the first time in the history basic notions of probability theory
have been defined precisely but simply. So a random event has been defined as a subset of a
space, a random variable as a measurable function and its mean value as an integral. More
precisely, abstract Lebesgue integral. It is hopeful to wait some new stimuls from the fuzzy
generalization of the classical set theory.
The aim of the chapter is a presentation of some
results of the type.
2. Fuzzy systems and their algebraizations
Any subset
A
of a given space
Ω
can be identified with its characteristic function
χ
A
:
Ω
→{
0, 1
}
where
χ
A
(
ω
)=
1,
if
ω
∈
A
,
χ
A
(
ω
)=
0,
∈
if
ω
/
A
. From the mathematical point of view a fuzzy set is a natural generalization of
χ
A
(see
[73]). It is a function
ϕ
A
:
Ω
→
[
0, 1
]
.
Evidently any set (i.e. two-valued function on
Ω
,
χ
A
→{
0, 1
}
) is a special case of a fuzzy set
(multi-valued function),
ϕ
A
:
Ω
→
[
0, 1
]
.
