Cardinality Part I
Original Notes adopted from December 3, 2001 (Week 13)
c
±
P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics
S
and
T
have the same
cardinality
if there exists
f
:
S
→
T
onetoone onto (i.e. a
“pairing” ) or
onetoone correspondence.
We showed that

N

=

E

=

Q
+


S

=

N

iﬀ
S
is an inﬁnite set whose elements can be listed. We call such sets
“countably inﬁnite”, or say they have cardinality
ℵ
0
.

S

=
ℵ
0
means

S

=

N

.

[0
,
1]
 6
=
ℵ
0
Proof
We’ll show no list can contain all numbers in [0,1].
a
ij
∈ {
0
,
1
,
2
,
3
,
4
,....
9
}
Suppose we have a list
c
1
,c
2
,c
3
,
···
, write them as
c
1
=
.a
11
a
12
a
13
a
14
a
15
···
c
2
=
.a
21
a
22
a
23
a
24
a
25
···
c
3
=
.a
31
a
32
a
33
a
34
a
35
···
······
In ambiguous cases, pick representation with all 9’s. e.g.
.
34999
···
=
.
3500000.
Let
x
=
.b
1
b
2
b
3
b
4
···
where
b
j
any digit other than 0, 9 or
a
jj
Then
x
isn’t among numbers listed for it diﬀers from the
k
th number listed in its
k
th
place.
Therefore

[0
,
1]
 6
=
ℵ
0
We say [0,1] has the cardinality of the continuum, or

[0
,
1]

=
c
Deﬁnition.

S

6

T

(“The cardinality of
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.
 Spring '10
 Applebaugh
 Math, Finite set, Cardinal number, C. Theorem, S

Click to edit the document details