CS103
HO#18
SlidesRelations
April 16, 2010
1
Infinite Sets
How about the rationals?
Countably infinite—same size as
N
How about the reals?
We will use a different kind of diagonal argument to show that
the real numbers are not denumerable.
In fact, we will show
that we cannot count the reals in the open interval (0,1).
R
(0,1)
= {x  (x
∈
R)
∧
(0 < x < 1)}
Suppose we have an enumeration of R
(0,1)
, i.e., a list of all reals
in the interval (0, 1).
Z
+
R
(0,1)
10
.
d
11
d
12
d
13
...
20
.
d
21
d
22
d
23
...
30
.
d
31
d
32
d
33
...
...
i0
.
d
i1
d
i2
d
i3
...
...
where d
ij
∈
{0, 1, 2, .
..,9}
Suppose we have an enumeration of R
(0,1)
, i.e., a list of all reals
in the interval (0, 1).
Z
+
R
(0,1)
.
d
11
d
12
d
13
...
.
d
21
d
22
d
23
...
.
d
31
d
32
d
33
...
...
.
d
i1
d
i2
d
i3
...
...
where d
ij
∈
{0, 1, 2, .
..,9}
Now we construct a new
number x = 0.p
1
p
2
p
3
...
where
pi = (d
i
+ 1) mod 10
(we will not use 0 and 9
so that we do not produce
duplicates).
x is not in the list, since it
differs from every number
along the diagonal!
We conclude that it is not
possible to produce such
a list.
Infinite Sets
How about the rationals?
Countably infinite—same size as
N

N
 =
ℵ
0
How about the reals?
Uncountable—bigger than
N
Every real number in (0, 1) can be represented by an infinite fraction:
0.10000000.
..
0.17917917.
..
0.31415926.
..
This is true regardless of the base of the number system used.
For example:
0.1 (decimal) = 0.0
0011
0011
0011
0011 . . . (base 2)
The fractional part has a 1
st
, 2
nd
, 3
rd
, .
.. digit
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View Full DocumentCS103
HO#18
SlidesRelations
April 16, 2010
2
Every real number in (0, 1) can be represented by an infinite fraction:
0.10000000.
..
0.17917917.
..
0.31415926.
..
This is true regardless of the base of the number system used.
For example:
0.1 (decimal) = 0.0
0011
0011
0011
0011 . . . (base 2)
The fractional part has a 1
st
, 2
nd
, 3
rd
, .
.. digit
So the digits of the fraction can be put in 1to1 correspondence with
N
So each fraction has 
N
 digits, i.e.,
ℵ
0
digits
Every real number in (0, 1) can be represented by an infinite fraction:
0.10000000.
..
0.17917917.
..
0.31415926.
..
This is true regardless of the base of the number system used.
For example:
0.1 (decimal) = 0.0
0011
0011
0011
0011 . . . (base 2)
The fractional part has a 1
st
, 2
nd
, 3
rd
, .
.. digit
So the digits of the fraction can be put in 1to1 correspondence with
N
So each fraction has 
N
 digits, i.e.,
ℵ
0
digits
We could form a subset of
N
by including the positions of the 1 digits
For the above, this would be {4, 5, 8, 9, 12, 13, 16, 17, .
..}
So every real in (0, 1) corresponds to a particular subset, and
every subset corresponds to a particular real number
Every real number in (0, 1) can be represented by an infinite fraction:
0.10000000.
..
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 Fall '09
 Set Theory, Order theory, Binary relation, Transitive relation

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