Graphics Reference
In-Depth Information
18.2
Basic Definitions
Definition.
Let I(
R
) denote the set of closed intervals in
R
.
By identifying the real number a with the interval [a,a] we shall consider
R
as a subset of I(
R
). Throughout this chapter we shall use capital letters to denote
intervals.
Definition.
If A = [a,b] ΠI(
R
), then define
()
=
()
=
lb A
a
,
and
ub A
b
.
Next, here are the basic arithmetic operators of addition, subtraction, multipli-
cation, and division on intervals.
Definition.
Let
*
Œ {+, -, ◊, /}. Let A, B Œ I(
R
). Define
{
}
A B
=
a b a
Œ
A and b
Œ
B
*
*
In the case of /, we shall always assume that 0 does not belong to B. At times we shall
abbreviate the product A ◊ B to AB.
18.2.1
Lemma.
If A = [a,b] and B = [c,d], then the following holds:
(1) A + B = [a + c,b + d]
(2) A ◊ B = [min(ac,ad,bc,bd), max(ac,ad,bc,bd)]
(3) A - B = [a - d,b - c] = A + [-1,-1] ◊ B
(4) A / B = [min(a/c,a/d,b/c,b/d), max(a/c,a/d,b/c,b/d)]
= [a,b] ◊ [1/d,1/c]
Proof.
This is an easy exercise.
18.2.2
Examples.
[-1,3] + [2,5] = [1,8]
[-1,3] - [2,5] = [-6,1]
[-1,3] ◊ [2,5] = [-5,15]
[-1,3] / [2,5] = [-1/2,3/2]
The next lemma summarizes some basic facts that, among other things, show that
the operations on I(
R
) act very much like they do on the reals
R
. The main fact that
keeps (I(
R
), +, ◊) from being a ring is that is does not have additive inverses.
18.2.3
Lemma.
Let A, B, C, D ΠI(
R
). Then
(1) (Commutativity) A + B = B + A and A ◊ B = B ◊ A.
(2) (Associativity) (A + B) + C = A + (B + C) and (A ◊ B) ◊ C = A ◊ (B ◊ C).
(3) (Identity) The intervals [0,0] and [1,1] are the unique additive and multi-
plicative identities, respectively. More precisely,
