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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
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 Spring '10
 Applebaugh
 Math

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