Page 28 - Advanced Linear Algebra

12 Advanced Linear Algebra

Theorem 0.10 (Zorn's lemma ) If is a partially ordered set in which every
7
7
chain has an upper bound, then has a maximal element.
We will use Zorn's lemma to prove that every vector space has a basis. Zorn's
lemma is equivalent to the famous axiom of choice. As such, it is not subject to
proof from the other axioms of ordinary (ZF) set theory. Zorn's lemma has many
important equivalancies, one of which is the well-ordering principle . A well
ordering on a nonempty set is a total order on with the property that every
?
?
nonempty subset of has a least element.
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Theorem 0.11 (Well-ordering principle ) Every nonempty set has a well
ordering.
Cardinality

Two sets and have the same cardinality , written
;
:
;
((:~ ( (
(
)
if there is a bijective function a one-to-one correspondence between the sets.
The reader is probably aware of the fact that
o
o
( ( ~ { (( and ( ( ~ r ((
where o denotes the natural numbers, the integers and r { the rational
numbers.
If is in one-to-one correspondence with a subset of , we write : ; ((  ( ( . If
:
;
; is in one-to-one correspondence with a proper
: subset of but not all of ,
;
then we write ((: ( ( . The second condition is necessary, since, for instance,
;
o { is in one-to-one correspondence with a proper subset of and yet is also in
o
{
one-to-one correspondence with itself. Hence, o (( ~ ( { . (
This is not the place to enter into a detailed discussion of cardinal numbers. The
intention here is that the cardinality of a set, whatever that is, represents the
“size” of the set. It is actually easier to talk about two sets having the same, or
)
different, size cardinality than it is to explicitly define the size cardinality of
(
)
(
a given set.
Be that as it may, we associate to each set a cardinal number, denoted by : ((
:
or card²:³ , that is intended to measure the size of the set. Actually, cardinal
numbers are just very special types of sets. However, we can simply think of
them as vague amorphous objects that measure the size of sets.
Definition
1 A set is finite if it can be put in one-to-one correspondence with a set of the
)
form { ~ ¸ Á Á Ã Á c ¹ , for some nonnegative integer . A set that is