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2.2
Representation of Fuzzy Sets
The principal role of
ʱ
-cuts and strong
ʱ
-cuts in fuzzy set theory is their capability to
represent fuzzy sets. To represent fuzzy sets by its
ʱ
-cuts it is needed to convert each of
the
ʱ
-cuts to a special fuzzy set
ʱ
A
, defined as follow
ʱ
A
(
x
)
ʱ
A
(
x
)=
ʱ
·
(3)
This representation is usually referred to as decomposition of
A
. In the following,
three decomposition theorems of fuzzy sets are shown [15].
Theorem 21
(First Decomposition Theorem).
For every
A ∈ F
(
X
)
,
A
=
ʱA∈
[0
,
1]
ʱ
A,
where
ʱ
A
is defined by equation 3 and
∪
denotes the standard fuzzy union.
2.3
Extension Principle for Fuzzy Sets
It is said that a crisp function
Y
(4)
is fuzzified when it is extended to act on fuzzy sets defined on
X
and
Y
.Thatis,the
fuzzified function, for which the same symbol
f
is usually used, has the form:
f
:
X
ₒ
f
:
F
(
X
)
ₒ
F
(
Y
)
(5)
Then, the ExtensionPrinciple allows us to compute fuzzy functions from crisp ones:
[
f
(
A
)] (
y
)= sup
x| y
=
f
(
x
)
A
(
x
)
(6)
2.4
Interval Type-2 Fuzzy Sets
A Type-2 fuzzy set is a collection of infinite Type-1 fuzzy sets into a single fuzzy set.
It is defined by two membership functions (upper and lower MFs) which shape the
footprint of uncertainty (FOU). Mendel [16] define the FOU as the union of all primary
membership degrees, the upper membership function (UMF) as a subset that has the
maximum membership degree of the footprint of uncertainty and the lower membership
function (LMF) as a subset that has the minimum membership degree of the footprint
of uncertainty. An IT2FS is described as:
A
=
x∈X
1
/
(
x, u
)=
x∈X
1
/u
/x,
(7)
u∈J
x
u∈J
x
where
x
is the
primary variable
,
J
x
an interval in [0,1], is the
primary membership
of
x
,
u
is the secondary variable, and
u∈J
x
1
/
(
x, u
) is the
secondary membership function
(MF) at
x
. Equation (7) means that
A
:
X
−ₒ {
[
a, b
]:0
≤
a
≤
b
≤
1
}
. Uncertainty
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