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1.3 Fuzzy Sets
Though the concept of a set plays a fundamental role not only in the mathematics
science, it has no rigorous definition. It is considered that this word is commonly
understood as a quantity of roughly homogeneous (in any sense) elements.
Within this set it is possible to define subsets and it is necessary to do this
strictly and explicitly. Let us have a set X consisting of elements x . Its subset A
can be defined, for example, by means of characteristic function
1
x
A
;
()
h A
x
=
0
x
A
.
Thus, characteristic function allows mapping of a set to another set of two
elements: 0 and 1. It can be written as follows:
()
{ }
h A
So, for any element of set X there are two possibilities: it can either belong or not
belong to a set A .
If to consider a set of all subsets, it is possible to apply, in a certain way,
operations of intersection, union and complement to this set, and those can be
expressed as operations to corresponding characteristic functions.
Using sets, we can define various concepts. Let us explain with the following
example.
x
:
X
0
.
Example 1.1. Formalization of the “normal functioning of an object” concept on the
basis of the classical theory of sets. Let some parameter be defined at a universal set
and takes values from X to X . Then, based on the substantial sense of a problem,
the concept of “normal functioning of an object” can be defined by means of a set A
the characteristic function of which is shown in Fig. 1.2.а.
Under this formalization it is assumed that all us ers unam big uously understand
the given concept and agree with boundary values x and x . This assumption
is met, for example, when boundary values are computed on the basis of an
approved mathematical model of the process of an investigated object functioning.
Operations of intersection, union and complement can be interpreted as logical
connectives "and", "or" and "not”, accordingly. In this case, we mean Boolean
(two-value, binary) logic. With expressions of some concepts
in the
a
,
a
,
...,
a
1
2
n
form of sets
, 21 available, it is possible to find the sets corresponding to
these concepts, in the form of logic functions
A
A
,
...,
A
(
)
, 21 .
The theory of sets and corresponding Boolean logic makes the foundation of
classical mathematics and everything based on it, up to advanced computer
processors. Models of complicated engineering and physical systems, and chemical
processes were well described with this language and successfully implemented
using computers. Insufficient speed, memory size, and also some other engineering
complexities of implementation of these models were the only problem.
f
a
a
,
...,
a
n
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