The Real and Complex Number Systems
Ordered Sets
1.5 Definition
Let
S
be a set. An
order
on
S
is a relation, denoted by
<
, with the following two properties:
(i) If
x
∈
S
and
y
∈
S
then one and only one of the statements
x < y
,
x
=
y
,
y < x
is true.
(ii) If
x, y, z
∈
S
, if
x < y
and
y < z
, then
x < z
.
1.6 Definition
An
ordered set
is a set
S
in which an order is defined.
1.7 Definition
Suppose
S
is an ordered set, and
E
⊆
S
. If (
∃
β
∈
S
)(
∀
x
∈
E
)(
x
≤
β
), the
E
is
bounded
above
, and
β
is an
upper bound
of
E
. Lower bounds are defined in the same way (with
≥
in place of
≤
).
1.8 Definition
Suppose
S
is an ordered set,
E
⊆
S
, and
E
is bounded above.
Suppose there exists an
α
∈
S
with the following properties:
(i)
α
is an upper bound of
E
.
(ii) If
γ < α
then
γ
is not an upper bound of
E
.
Then
α
is called the
least upper bound of
E
or the
supremum of
E
, and we write
α
= sup
E
. The
greatest
lower bound
, or
infimum
, of a set
E
that is bounded below is defined in the same manner: The statement
α
= inf
E
means that
α
is a lower bound of
E
and that no
β
with
β > α
is a lower bound of
E
.
1.10 Definition
An ordered set
S
is said to have the
leastupperbound property
if the following is true:
If
E
⊆
S
,
E
is not empty, and
E
is bounded above, then sup
E
exists in
S
.
1.11 Theorem
Suppose
S
is an ordered set with the leastupperbound property,
B
⊆
S
,
B
is not empty,
and
B
is bounded below.
Let
L
be the set of all lower bounds of
B
.
Then
α
= sup
L
exists in
S
, and
α
= inf
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 following statements, Complex number, upper bound, RK

Click to edit the document details