Graphics Reference
In-Depth Information
(4) |xA| = |x||A|.
(5) |AB| = |A||B|.
(6) d(A,B) = |A - B|.
(7) d(xA,xB) = |x| d(A,B).
(8) d(AB,AC) £ |A| d(B,C).
Proof.
Straightforward.
Definition.
Let A = [a,b] ΠI(
R
). Define the
width
of the interval A, w(A), and the
midpoint
of A, mid(A), by
()
=-
()
=+
(
)
2.
wA
b a
and
mid A
a
b
18.2.13
Lemma.
Let A, B ΠI(
R
).
(1) w(A) =
max
xy
-
.
xy A
Œ
(2) If A Õ B, then w(A) £ w(B).
(3) w(A ± B) = w(A) + w(B).
Proof.
Easy.
We generalize to
R
n
.
I(
R
n
) = I(
R
)
n
.
Definition.
The elements of I(
R
n
) are products of intervals in
R
and clearly have the form
[
]
¥
[
]
¥¥
[
]
ab
,
a b
,
...
a
nn
,
,
11
2 2
where a
i
£ b
i
. The elements of I(
R
n
) will be called
intervals
of
R
n
. We extend the
standard interval operations +, -, ◊, and / to the intervals of
R
n
in a coordinate-wise
manner.
If
X
Õ
R
n
, then
Definition.
()
=Œ
(
{
)
}
.
n
I
X
A
I
R
A
Õ
X
Definition.
Let A = A
1
¥ A
2
¥ ...¥ A
n
be an interval in
R
n
. Define the
width
of A,
w(A), and the
midpoint
of A, mid(A), by
()
=
(
()( )
( )
)
wA
max
wA
,
wA
,...,
wA
n
1
2
and
()
=
(
()
( )
( )
)
mid A
mid A
,
mid A
,...,
mid A
n
.
1
2
Define the
absolute value
of A, |A|, by
