Graphics Reference
In-Depth Information
()
=
[
]
AXAAX
AYAAY
=+=+
for all X in I
R
R
if and only if X
00
11
,.
,.
()
=
[]
=◊
=◊
for all Y in I
if and only if Y
(4) I(
R
) has no zero divisors.
(5) The only elements [a,b] in I(
R
) that have an additive or multiplicative inverse
are those for which a = b. However, we do have
Œ
()
0
Œ-
A
A
and
1
Œ
A A
for all A
I
R
.
(6) A ◊ (B + C) + A ◊ B + A ◊ C (subdistributivity)
a ◊ (B + C) = a ◊ B + a ◊ C, a Œ
R
A ◊ (B + C) = A ◊ B + A ◊ C if bc ≥ 0 for all b Œ B and c Œ C
(7) If A Õ C and B Õ D, then A
*
B Õ C
*
D, for
*
Œ {+, -, ◊, /}. This is often expressed
by saying that the standard interval arithmetic operators are
inclusion isotone
or
inclu-
sion monotonic
.
Proof.
This is fairly straightforward. See [AleH83].
18.2.4
Example.
The distributive law fails:
[
◊
()
+- -
[
(
)
]
=
[
]
[
]
◊
[]
+
[
◊- -
[
]
=-
[
]
12
,
11
,
1 1
,
00
,
but
12
,
11
,
12
,
1 1
,
11
, .
Property (7) in Lemma 18.2.3 is particularly important. In the context of compu-
tations, it tells us that as new errors creep into computations we can keep control of
them.
It is possible to define a metric on I(
R
).
Definition.
Define
()
¥
()
Æ
dI
:
RRR
I
by
(
[
] [
]
)
=
(
)
dab cd
,
,
,
max
a c b d
-
,
-
.
18.2.5
Lemma.
The function d defines a metric on I(
R
).
Proof.
This is easy to show directly, but actually it is the well-known Hausdorff
metric.
18.2.6
Example.
d([-1,3],[2,5]) = max(|-1 - 2|,|3 - 5|) = 3.
18.2.7
Theorem.
(1) (I(
R
), d) is a complete metric space.
(2) The operations of addition, subtraction, multiplication, and division defined
on I(
R
) are continuous.
