Geoscience Reference
In-Depth Information
Convex fuzzy set
A
convex
fuzzy set is the set in which the membership function is monotonically
increasing or decreasing. Precisely, the membership function in a convex fuzzy set is
either (i) monotonically increasing, or (ii) monotonically decreasing, or (iii)
monotonically increasing and monotonically decreasing with increasing values of the
elements in the universe. For every real number, say
x
1
,
x
2
and
x
3
, with
x
1
<
x
2
<
x
3
"
A
(
x
2
)!min{
"
A
(
x
1
),
"
A
(
x
3
)}
(I.4)
Fuzzy number
A fuzzy set which is normal as well as convex is called a
fuzzy number
.
Fuzzy Extension Principle
Consider a function of several uncertain variables:
y
=
f
(
x
1
,…,
x
n
)
(I.5)
Let fuzzy sets
A
1
,…,
A
n
be defined on the universes
X
1
,
…
,X
n
such that
x
1
X
1
,…,
x
n
X
n
.
The mapping of these input sets can be defined as a fuzzy set
B:
B
=
f
(
A
1
,…,
A
n
)
(I.6)
where the membership function of the image
B
is given by
(I.7)
The Equation (I.7) is the mathematical expression for the Extension Principle (EP) of
fuzzy sets. The above equation is defined for a discrete-valued function
f
. If the function
f
is continuous-valued then the
max
operator is replaced by the
sup
(supremum) operator
(the supremum is the least upper bound).
Alpha-cut of a fuzzy set
An
!
-cut (alpha-cut) of a fuzzy set
A,
denoted as
A
"
is the set of elements
x
of a universe
of discourse
X
for which the membership function of
A
is greater than or equal to
a
. That
is,
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