22. NOVEMBER 22ND: CARDINALITY IV
51
22. November 22nd: Cardinality IV
22.1
.
Prove that if
S
is uncountable, and
C
⊆
S
is countable, then

S
C

=

S

.
Answer:
Let
N
⊆
(
S
C
) be an infinite countable set, and let
C
=
C
∪
N
. The
union of two countable sets is countable (Example 21.7), so
C
is infinite countable:
C
∼
=
N
. Since we have
S
= (
S
C
)
∪
C
and
S
C
= (
S
C
)
∪
N
it follows that

S

=

S
C

, by Lemma 20.1.
22.2
.
Prove that the set of transcendental number has the cardinality of the con
tinuum.
Answer:
This is an immediate application of Exercise 22.1: Take
S
=
R
, and
C
=
A
, the set of algebraic numbers; then since
A
is countable, the result of Exercise 22.1
shows that the set
R
A
of transcendental numbers has the same cardinality as
R
.
22.3
.
Prove that the set
C
of complex numbers has the cardinality of the contin
uum.
Answer:
Complex numbers are numbers of the form
a
+
bi
, where
a
and
b
are real
numbers, and
i
is a square root of
−
1. Thus,
C
∼
=
R
×
R
: the knowledge of a complex
number is equivalent to the knowledge of its real part
a
and its imaginary part
b
. It
follows that

C

=

R
×
R
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 Fall '11
 Aluffi
 Natural number, Rational number, Countable set, Georg Cantor, Algebraic number

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