Information Technology Reference
In-Depth Information
Let
I
j
I
[
a
,
b
]
) be a set containing iterated revisions of [
a
,
b
] upto
j
-th iterations.
ↆ
[
a
,
b
]
(
Definition 3.5.
C
I
is a choice function over
{
I
j
U
}
such that
≥
0
,
[
a
,
b
]
∈
[
a
,
b
]
:
j
(i)
C
I
(
I
j
I
j
≤
I
[
c
,
d
] implies
C
I
(
I
j
≤
I
C
I
(
I
j
[
a
,
b
]
)
∈
[
a
,
b
]
and (ii) [
a
,
b
]
[
a
,
b
]
)
[
c
,
d
]
).
Let us present one such case of iterative-revision of intervals below.
>
0,
C
ʵ
:
U
→
U
such that
C
ʵ
([
a
,
b
])=[
a
+ ʵ
,
b
] if
a
+ ʵ
<
b
= [
b
,
b
],otherwise.
Definition 3.6.
For
ʵ
C
ʵ
:
U
→
U
such that
C
ʵ
([
a
,
b
])=[
a
,
b
−
ʵ] if
a
<
b
−
ʵ
= [
a
,
a
],otherwise.
C
ʵ
:
U
→
U
such that
C
ʵ
([
a
,
b
])=
C
ʵ
([
a
,
b
])
∩
C
ʵ
([
a
,
b
]).
b
−
a
n
and ap-
plying
C
ʵ
finite number of times on [
a
,
b
], one instance of
I
[
a
,
b
]
, we call
I
[
a
,
b
]
, can be
obtained in the following way.
Let us choose an arbitrarily fixed number
n
.Given[
a
,
b
],fixing
ʵ
≥
Definition 3.7.
I
[
a
,
b
]
=
{
C
i
0
,
C
0
([
a
,
b
])=[
a
,
b
]
,
C
i
−
1
([
a
,
b
]):
i
≥
([
a
,
b
]) is non-degenerate
}
.
ʵ
ʵ
ʵ
we have
I
2
,
I
2
1
2
Note 3.2.
Tak ing
ʵ
=
[
.
3
,.
7]
=
{
[
.
3
,.
7]
,
[
.
3
,.
3]
}
[
.
1
,.
9]
=
{
[
.
1
,.
9]
,
[
.
4
,.
4]
}
.Itis
to be noted that as
I
2
[
.
3
,.
7]
contains only two iterations
I
2
[
.
3
,.
7]
=
I
2
,
j
[
.
3
,.
7]
=
{
[
.
3
,.
7]
,
[
.
3
,.
3]
}
2, where
I
2
,
j
for any number of iterations
j
≥
[
.
3
,.
7]
contains intervals upto
j
-th iterations.
Definition 3.8.
C
I
(
I
ʵ
,
j
I
ʵ
,
j
[
a
,
b
]
)=
∩
[
a
,
b
]
,where
j
denotes the number of iterations.
Note 3.3.
As
I
ʵ
is obtained by finitely many iterations,
C
I
(
I
[
a
,
b
]
)=
∩
I
[
a
,
b
]
,whichis
adegenerate interval. Clearly,
C
I
(of Definition 3.8) satisfies condition (i) of the Defi-
nition 3.5 as
C
I
(
I
ʵ
,
j
[
a
,
b
]
)=
C
j
I
[
a
,
b
]
. To check that
C
I
also satisfies condition (ii)
of Definition 3.5 we need to prove a series of results below.
([
a
,
b
])
∈
ʵ
Proposition 3.5.
C
ʵ
([
a
,
b
])
ↆ
e
[
a
,
b
].
Proof.
Two cases arise. (i)
a
+
<
b
i.e.
a
<
b
−
ʵ
(ii) otherwise.
For all these cases the result is straightforward from the definitions of
ʵ
ↆ
∩
e
and
.
Theorem 3.5.
[
a
,
b
]
≤
I
[
c
,
d
] implies
C
ʵ
([
a
,
b
])
≤
I
C
ʵ
([
c
,
d
]).
Proof.
Let [
a
,
b
]
≤
I
[
c
,
d
],then
a
≤
c
,
b
≤
d
,and
a
≤
b
,
c
≤
d
.
Now (i)
C
([
a
,
b
])=[
a
+
ʵ
,
b
] or (ii)
C
([
a
,
b
])=[
b
,
b
].
ʵ
ʵ
(i) Let
C
([
a
,
b
])=[
a
+
ʵ
,
b
] i.e.
a
+
ʵ
<
b
≤
d
.Also
a
+
ʵ
≤
c
+
ʵ
.
ʵ
Now either
c
+
ʵ
<
d
or
d
≤
c
+
ʵ
. Both the cases yield
C
([
a
,
b
])
≤
I
C
([
c
,
d
]), as
ʵ
ʵ
for
c
+
ʵ
<
d
,
C
([
c
,
d
])=[
c
+
ʵ
,
d
]. i.e., [
a
+
ʵ
,
b
]
≤
I
[
c
+
ʵ
,
d
],and
ʵ
for
d
≤
c
+
ʵ
,
C
([
c
,
d
])=[
d
,
d
] i.e., [
a
+
ʵ
,
b
]
≤
I
[
d
,
d
],since
a
+
ʵ
<
b
≤
d
.
ʵ
(ii) Let
C
([
a
,
b
])=[
b
,
b
],i.e.
b
≤
a
+
ʵ
≤
c
+
ʵ
and also
b
≤
d
.
ʵ
