REAL ANALYSIS
CARDINAL NUMBERS
We use
S
for the cardinal number of a set
S
.
I
S
≤
T
(or
T
≥
S
) is to mean “
∃
a 11 correspondence between
S
and a
subset of
T
” (not necessarily a proper subset).
II
S
=
T
is to mean “
∃
a 11 correspondence between
S
and
T
.
[
<
is to mean
≤
but not =]
We have that:
(i)
The definitions are reasonable when applied to finite sets.
(ii) (a)
≤
is transitive, i.e.
X
≤
Y
Y
≤
Z
⇒
X
≤
Z
(b)
= is transitive
X
=
Y
Y
=
Z
⇒
X
=
Z
= is symmetric
S
=
T
⇔
T
=
S
= is reflexive
S
=
S
(iii)
(Bernstein’s Lemma)
S
≤
T
T
≤
S
⇒
S
=
T
(iv)
For any two sets either
S
≤
T
or
T
≤
S
.
A set
S
is said to be enumerable (denumerable, countable)
⇔ ∃
a 11 corre
spondence between
S
and the set of all natural numbers.
χ
0
is called the cardinal number of the set of all natural numbers.
1. If
S
≤
χ
0
either
S
is finite or
S
=
χ
0
2. If
S
=
χ
0
S
can be put in 11 correspondence with proper subset of
itself.
3. Any infinite subset contains an enumerable subset.
1
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4. If
S
=
χ
0
and
T
is infinite then
S
∪
T
=
T
.
5. A set is infinite
⇔
it can be put into 11 correspondence with a proper
subset of itself.
Proof of A
Suppose
U
and
V
are such that
U
=
V
=
χ
0
.
Then
U
=
u
1
u
2
u
3
. . .
V
=
v
1
v
2
v
3
. . .
U
∪
V
=
u
1
v
2
u
2
v
2
. . .
=
W
=
w
1
w
2
w
3
w
4
. . .
therefore
U
∪
V
=
χ
0
.
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 Spring '98
 pfitz

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