Geoscience Reference
In-Depth Information
one. Fuzzy sets are perceived as generalization of classical crisp sets which are their
speci
c case. Quality
“
to be fuzzy
”
is often expressed as ambiguity, not as inac-
curacy or uncertainty.
De
nition of fuzzy set using the characteristic function.
Let
X
be a universe set (crisp set). A fuzzy set A of the universe
X
is de
ned by a
characteristic function called membership function
such that
l
A
:
X
!
hi
0
1
μ
A
;
where
μ
A
(x) is the membership value of x in A.
The membership value assigns a degree of membership to a fuzzy set to any
element.
μ
A
(x) = 1
element x belongs to a fuzzy set for sure
μ
A
(x) = 0
element x doesn
'
t belong to a fuzzy set for sure
0
\
l
A
ð
x
Þ
\
1 we aren
'
t sure if element x belongs to a fuzzy set.
Each function
X
!
hi
0
1
determines any fuzzy set de
nitely.
;
ed by mathematical functions [
3
].
We usually compose the membership functions of elementary linear functions.
These are trapezoidal, triangular, S-shaped and L-shaped membership functions.
We often use more complicated rounded functions, too as Gaussian function, bell-
shaped function, sinusoidal function etc.
The membership degree to the fuzzy set is speci
2 Operations on Fuzzy Sets and Fuzzy Logic
Operations complement, union and intersection on fuzzy sets are de
ned in similar
way as on crisp sets [
4
].
The standard intersection of two fuzzy sets A and B is a fuzzy set with the
membership function de
ned by
l
A
\
B
ð
x
Þ¼
min
ð
l
A
ð
x
Þ
;
l
B
ð
x
ÞÞ
:
Zadeh's intersection
The standard union of two fuzzy sets A and B is a fuzzy set with the membership
function de
ned by
l
A
[
B
ð
x
Þ¼
max
ð
l
A
ð
x
Þ
;
l
B
ð
x
ÞÞ
Zadeh's union
The standard complement of fuzzy set A is a fuzzy set with the membership
function de
ned by
l
A
ð
x
Þ¼
1
l
A
ð
x
Þ
Zadeh's complement
Functions for modelling fuzzy conjunction are called triangular norms (t-norms),
for fuzzy disjunction triangular conorms (t-conorms). They are assumed as func-
tions of two variables de
ned on a unit square.
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