Goals
We will
define finite and infinite,
define a particular infinity,
unsuccessfully attempt to construct something bigger, and
eventually find a bigger set.
How big is it? Sooooo Big
What does “infinite” mean to you? Write down a personal
explanation or definition.
How big is it? Sooooo Big
N = {n ∈ Z|n > 0}, and Nk = {1, 2, 3, . . . , k}.
Definition
A set A is finite iff it is equivalent (in size) to Nk for some integer
k.
Definition
A set A is infinite iff it is not finite.
The First Foray
Definition
A set A is denumerable iff it is equivalent (in size) to N.
The cardinality of denumerable sets is written ℵ0 .
Definition
A set A is countable iff it is finite or denumerable.
The First Foray
Prove Z is denumerable.
Prove N × N is denumerable.
How to Build a Bigger Set
Theorem
If A is denumerable, A ∪ {x} is denumerable.
Proof: If x ∈ A, then A ∪ {x} = A, which is denumerable.
Suppose x 6∈ A. Because A is denumerable there exists a
one-to-one and onto function f : N → A. Consider the function
x if n = 1
g (n) =
f (n − 1) if n > 1
We show g is one-to-one. Suppose g (n) = g (m).
Case 1: n = m = 1. n = m as desired.
Case 2: n = 1 and m 6= 1. Then g (n) = x and g (m) = a ∈ A and
x 6= a. This contradicts that g (n) = g (m), so this is not possible.
Case 3: n 6= 1 and m 6= 1. Then g (n) = f (n − 1) and
g (m) = f (m − 1). Because f is one-to-one, f (n − 1) = f (m − 1)
implies n − 1 = m − 1 or n = m.
How to Build a Bigger Set
We show g is onto. Let y ∈ A ∪ {x}.
Case 1: y = x. By definition of g g (1) = x, so this is covered.
Case 2: y 6= x. Because f is onto there exists n ∈ N such that
f (n) = y . Thus g (n + 1) = f ([n + 1] − 1) = f (n) = y , and y is
covered.
Because g is one-to-one and onto, A ∪ {x} is denumerable.
How to Build a Bigger Set
Theorem
If A is denumerable and B is finite, then A ∪ B is denumerable.
Theorem
If A and B are disjoint denumerable sets, then A ∪ B is
denumerable.
Proof: Because A and B are denumerable there exist one-to-one
and onto functions f : N → A and g : N → B. Consider
f n+1
2
if n is odd
h=
n
g 2 if n is even
Finish the proof.
How to Build a Bigger Set
Prove the following theorem.
Theorem
For a collection of n pairwise disjoint, denumerable sets Ai , ∪ni=1 Ai
is denumerable.
Theorem
For a collection of n denumerable sets Ai , ∪ni=1 Ai is denumerable.
How would we prove this one?
How to Build a Bigger Set
How big are the rationals Q? Consider the diagram on page 252.
Note this defines the following mapping
N 1
Q 11
2
3
4
2
1
1
2
3
1
5 ...
1
3 ...
This is one-to-one and onto. Thus Q is denumerable.
How to Build a Bigger Set
Theorem
If A and B are denumerable sets, then A × B is denumerable.
Have we built a bigger set yet?
How to Build a Bigger Set
Theorem
The interval (0, 1) is uncountable.
Proof: We show first that the interval is not finite. Consider the
function s : N → (0, 1) defined by s(n) = 1/2n . This is a
one-to-one function, so kRk ≥ kNk which is infinite.
The following is a proof by contradiction that (0, 1) is not
denumerable. Suppose (0, 1) is denumerable. Thus we can write
f (1) = 0.a1,1 a1,2 a1,3 a1,4 a1,5 . . .
f (2) = 0.a2,1 a2,2 a2,3 a2,4 a2,5 . . .
f (3) = 0.a3,1 a3,2 a3,3 a3,4 a3,5 . . .
..
.
5 if ai,i 6= 5
Consider the number b ∈ (0, 1) defined by bi =
3 ai,i = 5
Note that b 6= f (n) for any n, because it differs from n in digit
an,n . This is a contradiction. Thus (0, 1) is not denumerable.
How to Build a Bigger Set
Is the set of all, finite integer sequences denumerable? (e.g.,
{3, 1, 4, 1, 5, 9, 2, 6, 5, 3})
What about the set of all integer sequences?
What about the set of all rational sequences?