Download Infinite Sets

Infinite Sets
A set which contains n elements for some natural number, n, is said to be a finite set. An
infinite set is one that is not finite.
Example A = 1, 2, 3 is a finite set. B = n ∈ ℕ : n > 3 is an infinite set.
A set A is a subset of the set B if every element of A is also an element of B. The set A is a
proper subset of B is every element of A is an element of B but there are elements of B that
are not elements of A.
Example E = n = 2m : m ∈ ℕ is a proper subset of ℕ. In particular, E consists of all the
even positive integers.
An alternative definition for an infinite set asserts that a set is infinite if it can be put in one
to one correspondence with a proper subset of itself.
Example The set of positive integers, ℕ, can be put into one to one correspondence with E,
the proper subset of even integers as follows:
1 2 3 ⋯
n
↕ ↕ ↕
↕
2 4 6
2n
⋯
This establishes ℕ as an infinite set. More precisely, we say that ℕ is countably infinite.
Other sets that are countably infinite include ℤ, the set of all integers including zero. The
following correspondence establishes the countability of ℤ.
1 2
3
4
5
6 ⋯ 2n 2n + 1
↕ ↕
↕
↕
↕
↕
0 1 −1 2 −2 3 ⋯
↕
↕
n
−n
Less obvious is the fact that the rational numbers in 0, 1, ℚ ∩ 0, 1, is countable. To see
this, list these rationals in the following array:
1
2
3
3
4
⋮
1
2
2
5
3
5
1
3
2
7
3
7
1
4
2
9
3
8
1
5
2
11
3
10
1
6
2
13
3
11
1
7
2
15
3
13
⋯
⋯
⋯
Note that in each row, fractions are reduced to lowest terms so as to not repeat fractions
1
from an earlier row. Now count the fractions along diagonals as follows:
1
2
3
4
5
6
7
8
↕
↕
1
2
↕
2
3
↕
1
3
↕
2
5
↕
3
4
↕
1
4
↕
2
7
1
⋯
⋯
In this way each of the fractions in the list can be seen to correspond to one and only one
positive integer; i.e., the set of rationals between 0 and 1 is countably infinite. The
correspondence displayed in the array above is called an enumeration of the rationals in
0, 1. This is not the only possible enumeration of this set.
To illustrate how counterintuitive infinite sets can be, we can now show that in spite of the
fact that there are an infinite number of rational numbers in the interval 0, 1, these
numbers take up very little room in this interval. To see this, let  denote an arbitrarily small
positive number and let r k : k ∈ ℕ denote an enumeration of the rationals in 0, 1. Then
each rational is contained in an inteval I k = r k − 10 −2k , r k + 10 −2k  of length 210 −2k and the
set of all these rationals is covered by the union of these intervals. Although the intervals I k
⋃ Ik
are not mutually disjoint, we can estimate,
, the length of the union, by the following
k
sum,
⋃ Ik
k
∞
≤
−2
∑ 210 −2k = 2 1 +1010 −2
≤ 2
k=1
This means that the rational numbers in 0, 1 can be covered by a family of open intervals
the sum of whose lengths can be made arbitrarily small. That is to say, "most" of the room
in 0, 1 is taken up by irrational numbers.
The set of irrational numbers in 0, 1 is an infinite set that is not countable. To see this, we
suppose that the set is countable and show that this assumption leads to a contradiction.
Each irrational number in 0, 1 has a decimal representation of the form 0. d 1 d 2 … and if the
set is countable then there is a enumeration of the form
z 1 = d 11 d 12 d 13 …
z 2 = d 21 d 22 d 23 …
z 3 = d 311 d 32 d 33 …
z 4 = d 41 d 42 d 43 …
⋮
Now let z = z 1 z 2 z 3 … denote a number in 0, 1 where
zi =
5 if
d ii ≠ 5
0 if
d ii = 5
Then z differs from z i in the i-th place for every i and it follows that this number z is nowhere
2
on the list in the enumeration. But the list was assumed to contain all the irrationals in 0, 1
and we conclude that no such enumeration is possible.
Theorem A countable union of countable sets is countable
To prove this result, let A i : i ∈ ℕ denote a countable collection of sets, where each
A i = a ij : j ∈ ℕ is countably infinite. Then we can arrange all the elements of each set in
an array as follows
a 11 a 12 a 13 a 14 a 15 ⋯
a 21 a 22 a 23 a 24 a 25 ⋯
a 31 a 32 a 33 a 34 ⋯
⋮
⋮
⋯
Then we can count along the diagonals of the array, as we did for the rationals, to get the
following enumeration
1
2
3
4
5
6
7
8
↕
↕
↕
↕
↕
↕
↕
↕
⋯
a 11 a 12 a 21 a 13 a 22 a 31 a 14 a 23 ⋯
This establishes that the elements of the union ⋃ A k is a countable set.
k
Since ℚ ∩ 0, 1 is countable, it follows that ℚ ∩ k, k + 1 is countable for every k ∈ ℕ. Then
⋃ ℚ ∩ k, k + 1 = ℚ is also countable.
k
3