Civil Engineering Reference
In-Depth Information
and
1
1
l
=
A
L
(
t
)
λ
(
t
)
d
t
,
l
f
=
A
L
(
t
)
f
(
t
)
λ
(
t
)
d
t
0
0
1
1
u
=
A
U
(
t
)
λ
(
t
)
d
t
,
u
f
=
A
U
(
t
)
f
(
t
)
λ
(
t
)
d
t
0
0
b
2
,
,
≥
−
≥
.
Note that
a
b
c
0 and that by applying Cauchy inequality we have
ac
0
b
2
=
(
)
−
>
Since
f
f
x
is not constant, we have
ac
0, which implies the matrixes
⎡
⎤
⎡
⎤
ab
00
bc
00
00
ab
00
bc
2
abb
bc
0
b
⎣
⎦
,
⎣
⎦
φ
:
=
ψ
:
=
4
0
c
are invertible. It is easy to verify that
⎡
⎣
⎤
⎦
,
⎡
⎣
⎤
⎦
c
−
b
00
−
ba
00
00
c
c
2
−
bc
−
bc
1
(
ac
−
b
2
1
2
c
(
ac
−
b
2
φ
−
1
ψ
−
1
−
bc
2
ac
−
b
2
b
2
=
=
)
2
−
b
)
b
2
2
ac
−
b
2
−
bc
00
−
bc
A
A
Also, we define real functions
s
i
=
s
i
(
)
and
t
i
=
t
i
(
)
by
(
s
1
,
s
2
,
s
3
,
s
4
)
:
=
u
f
)
φ
−
1
(
l
,
l
f
,
u
,
u
f
)
ψ
−
1
.
Now, let's define four subsets of fuzzy numbers as follows:
and
(
t
1
,
t
2
,
t
3
)
:
=(
l
+
u
,
l
f
,
Lemma 1.
The four subsets
Γ
i
,
1
≤
i
≤
4
, are disjoint and form a partition of fuzzy
numbers.
A
A
Theorem 2.
Let T
(
)
∈
F
(
R
)
and let T
(
)
be its general f-trapezoidal approxima-
A
tion. Then, T
(
)
can be computed in the following cases: If, then
If A
A
∈
Γ
1
,thenT
(
)=[
s
1
+
s
2
f
(
a
)
,
s
3
+
s
4
f
(
a
)]
.
If A
A
∈
Γ
2
,thenT
(
)=[
t
1
+
t
2
f
(
a
)
,
t
1
+
t
3
f
(
a
)]
.
2
ab
bc
−
1
If A
A
∈
Γ
3
,thenT
(
)=[
x
,
x
+
yf
(
a
)]
,where
(
x
,
y
)=(
l
+
u
,
u
f
)
.
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