Geoscience Reference
In-Depth Information
3.2 Fuzzy Sets and Fuzzy Logic
A process to derive probability distributions from punctual measurements has been
employed in the Theory of Fuzzy Sets since Zadeh (
1965
). The idea is to replace
exact numbers by intervals around them, with the pertinence of each point to the
interval decreasing as these points veer away from the initial measurement. Later, to
combine the probabilistic preferences, operations similar to those of Fuzzy Logic
(Zadeh
1978
) may be employed.
The concept of a fuzzy set was created by Lofti Zadeh in Zadeh (
1965
). Let X be
a space of points, with a generic element of X denoted by x. A fuzzy set A in X is
characterized by a membership function
µ
A
(x) which associates to each point in X a
real number in the interval [0, 1], with the values of
µ
A
(x) representing the degree
of membership of x to A. Thus, the nearer the value of
µ
A
(y) to unity, the higher the
degree of membership of y to A.
This de
nition of a set in the ordinary sense
of the term. The degrees of membership of 0 and 1 correspond to the two possi-
bilities of truth and false pertinence to an ordinary set, called a crisp set in the
Theory of Fuzzy Sets.
nition of a fuzzy set extends the de
3.2.1 Fuzzy Numbers
Membership functions for fuzzy sets can be de
ned in any form as long as they
follow the rules of the de
nition of a fuzzy set. The shape of the membership function
used de
nes the fuzzy set. For instance, a triangular membership function to deter-
mine a fuzzy interval with extremes a and b around a point M will have the form
8
<
0
x
a
;
x
a
M
a
;
a
x
M
l
A
x, a, M, b
ð
Þ
¼
b
x
b
M
;
:
M
x
b
0
b
x
;
Other way to represent a fuzzy set is by
α
-cuts. The membership function
l
A
:
R
!
½
0
1
of a fuzzy set A is determined by its family of
α
-cuts{A
α
}
α∈
[0, 1]
,
;
which are the intervals
;
A
a
¼ A
L
ðÞ
;
A
U
ðÞ
for
A
L
ðÞ
¼inf z
2
R
:
l
A
ðÞ
a
f
g
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