Information Technology Reference
In-Depth Information
When Zadeh established three years later the theory of Fuzzys Sets and Systems
he proposed a solution for the problem: In his seminal article on Fuzzy Sets he
introduced new mathematical entities that “are not classes or sets in the usual sense
of these terms, since they do not dichotomize all objects into those that belong to the
class and those that do not”. He established a “way of dealing with classes in which
there may be intermediate grades of membership.” “... there may be a continuous
infinity of grades of membership, with the grade of membership of an object
x
in a
fuzzy set
A
represented by a number
f
A
(
x
)
in the interval
[
0
,
1
]
.” [103].
(a)
(b)
Fig. 3.4
(a): Lotfi A. Zadeh in the 1960s.; (b) Headline of his article in the year 1965 [104].
These new concepts provided a “convenient way of defining abstraction - a pro-
cess which plays a basic role in human thinking and communication.” To generalize
various concepts of ordinary set theory, he defined equality, containment, comple-
mentation, intersection, and union relating to fuzzy sets
A
,
B
in any universe of
discourse
X
as follows (for all
x
∈
X
):
•
A
=
B
if and only if
A
(
x
)=
B
(
x
)
,
•
A
⊆
B
if and only if
A
(
x
)
⊆
B
(
x
)
,
•¬
A
is the complement of
A
if and only if
μ
¬
A
(
x
)=
1
−
μ
A
(
x
)
,
•
∪
(
)=
(
μ
(
)
,
μ
(
))
A
B
if and only if
μ
x
max
x
x
,
A
∪
B
A
B
•
∩
(
)=
(
μ
(
)
,
μ
(
))
A
B
if and only if
μ
x
min
x
x
.
A
∩
B
A
B
3.3.2
The Fuzzification of Systems
In a talk at the Symposium on System Theory occurred in the Polytechnic Institute
in Brooklyn in the same year Zadeh presented “A New View on System Theory”,
“which provide a way of treating fuzziness in a quantitative manner”. In the sym-
posium's proceedings there is a shortened manuscript version of the talk with the
heading “Fuzzy Sets and Systems” ([104, p. 29], 3.4 (b)) where he introduced the
concept of fuzzy systems for the first time:
