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6. Definition of Zadeh's fuzzy logic theory
Mathematical theory of fuzzy sets by
Zadeh
has allowed since more than a quarter of a
century ago up to now to describe fuzzy definitions and knowledge and operate with this
knowledge as well as make fuzzy outputs. (Пивкин , Бакулин и др)
The universal set is indicated by  E
Elements of a set are indicated by  X
R 
Indicates several properties, and
x
 indicates element
E
.
Ordinary (certain) Subset
A
of Universal Set
E
, whose elements
content the Property R, is indicated as the set of Ordered Pairs
A
= {m
A
(
х
)/
х
}
, where
m
A
(
х
)

is a characteristic function,
accepting
Value 1, if
x
contents Property
R,
and vice versa
0
.
Difference of a fuzzy subset from a certain one is that there is not a single valued answer
“yesno” for the elements of
x
of E, regarding Property
R
.
In this respect, Fuzzy set A of Universal set
E
is indicated as the set of Ordered Pairs
A
=
{m
A
(
х
of Subset
A
of Universal Set
E
.
7. Comparison of psychological theories by Piaget with mathematical
theories of Zadeh
Let us understand the term “a universal set” by Zadeh as “groups” and “groupings” or the
vocabulary of a language and sign it though  E. Then, the “elements of a set” by Zadeh and
“groups” by Piaget will be considered as the parts of speech. Let us sign these elements by x.