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CSE 20 –
Discrete
Mathematics
Dr. Cynthia Bailey Lee
Dr. Shachar Lovett
2
Today’s Topics:
Countably infinitely large sets
Uncountable sets
1.
2.

“To infinity, and beyond!” (really, we’re going to go beyond
infinity)
3
Set Theory and Sizes of Sets



How can we say that two sets are the
same size?
Easy for finite sets (count them)--what
about infinite sets?
Georg Cantor (1845-1918), who invented
Set Theory, proposed a way of
comparing the sizes of two sets that does
not involve counting how many things
are in each


Works for both finite and infinite
SET SIZE EQUALITY:


Two sets are the same size if there is a
bijective (injective and surjective) function
mapping from one to the other
Intuition: neither set has any element “left
over” in the mapping
4
Injective and Surjective
Natural
numbers
f is:
a)
b)
c)
d)
1
2
3
4
…
f
Injective
Surjective
Bijective (both (a) and (b))
Neither
a
aa
aaa
aaaa
…
Sequences
of a’s
5
Can you make a function that maps
from the domain Natural Numbers, to
the co-domain Positive Evens?
Natural
numbers
A.
B.
C.
1
2
3
4
…
2
4
6
8
Positive
evens
…
Yes and my function is bijective
Yes and my function is not bijective
No (explain why not)
6
Can you make a function that maps
from the domain Natural Numbers, to
the co-domain Positive Evens?
Natural
numbers
1
2
3
4
…
f
f(x)=2x
2
4
6
8
…
Positive
evens
7
Can you make a function that maps
from the domain Natural Numbers, to
the co-domain Positive Odds?
Natural
numbers
A.
B.
C.
1
2
3
4
…
1
3
5
7
Positive
odds
…
Yes and my function is bijective
Yes and my function is not bijective
No (explain why not)
8
Can you make a function that maps
from the domain Natural Numbers, to
the co-domain Positive Odds?
Natural
numbers
1
2
3
4
…
f
f(x)=2x-1
1
3
5
7
…
Positive
odds
9
Countably infinite size sets
 So
|ℕ| = |Even|, even though it seems
like it should be |ℕ| = 2|Even|
 Also, |ℕ| = |Odd|
 Another way of thinking about this is that
two times infinity is still infinity
 Does
that mean that all infinite size sets
are of equal size?
10
It gets even weirder:
Rational Numbers
(for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
ℚ+
=
𝑚
{ |𝑚, 𝑛
𝑛
∈ℕ}
1/1
1/2
1/3
1/4
1/5
1/6
…
2/1
2/2
2/3
2/4
2/5
2/6
…
3/1
3/2
3/3
3/4
3/5
3/6
…
4/1
4/2
4/3
4/4
4/5
4/6
…
5/1
5/2
5/3
5/4
5/5
5/6
...
6/1
6/2
6/3
6/4
6/5
6/6
…
…
…
…
…
…
…
11
It gets even weirder:
Rational Numbers
(for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
ℚ+
=
𝑚
{ |𝑚, 𝑛
𝑛
∈ℕ}
1/1
1/2
1/3
1/4
1/5
1/6
2/1
2/2
2/3
2/4
2/5
2/6
3/1
3/2
3/3
3/4
3/5
3/6
4/1
4/2
4/3
4/4
4/5
4/6
5/1
5/2
5/3
5/4
5/5
5/6
6/1
6/2
6/3
6/4
6/5
6/6
…
…
…
…
…
…
Is…
there a
bijection
from
…
the
…natural
numbers to
…Q+?
...
A. Yes
B. No
…
12
It gets even weirder:
Rational Numbers
(for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
ℚ+
=
𝑚
{ |𝑚, 𝑛
𝑛
∈ℕ}
1/1 1
1/2 2
1/3 4
1/4 6
1/5 10 1/6
…
2/1 3
2/2 x
2/3 7
2/4 x
2/5
2/6
…
3/1 5
3/2 8
3/3 x
3/4
3/5
3/6
…
4/1 9
4/2 x
4/3
4/4
4/5
4/6
…
5/1 11 5/2
5/3
5/4
5/5
5/6
...
6/1
6/2
6/3
6/4
6/5
6/6
…
…
…
…
…
…
…
13
Sizes of Infinite Sets

The number of Natural Numbers is equal to the
number of positive Even Numbers, even though
one is a proper subset of the other!


The number of Rational Numbers is equal to the
number of Natural Numbers



|ℕ| = |E+|, not |ℕ| = 2|E+|
|ℕ| = |ℚ+|, not |ℚ+| ≈ |ℕ|2
But it gets even weirder than that:
It might seem like Cantor’s definition of “same size”
for sets is overly broad, so that any two sets of
infinite size could be proven to be the “same size”

Actually, this is not so
14
Thm. |ℝ| != |ℕ|
Proof by contradiction: Assume |ℝ| = |ℕ|, so a
bijective function f exists between ℕ and ℝ.
• Want to show: no matter how f is designed (we don’t know
how it is designed so we can’t assume anything about
that), it cannot work correctly.
• Specifically, we will show a number z in ℝ that can never be
f(n) for any n, no matter how f is designed.
• Therefore f is not surjective, a contradiction.
Natural
numbers
1
2
3
4
…
f
?
?
?
z
?
…
Real
numbers
Thm. |ℝ| != |ℕ|
15
Proof by contradiction: Assume |ℝ| = |ℕ|, so a
bijective function f exists between ℕ and ℝ.
• We construct z as follows:
• z’s nth digit is the nth digit of f(n), PLUS ONE*
(*wrap to 1 if the digit is 9)
• Below is an example f
n
f(n)
1
.100000…
2
.333333…
3
.314159…
…
…
What is z in this
example?
a) .244…
b) .134…
c) .031…
d) .245…
16
Thm. |ℝ| != |ℕ|
Proof by contradiction: Assume |ℝ| = |ℕ|, so a
bijective function f exists between ℕ and ℝ.
• We construct z as follows:
• z’s nth digit is the nth digit of f(n), PLUS ONE*
(*wrap to 1 if the digit is 9)
• Below is a generalized f
n
f(n)
1
.d11d12d13d14…
2
3
…
.d21d22d23d24…
.d31d32d33d34…
…
What is z?
a) .d11d12d13…
b) .d11d22d33 …
c) .[d11+1] [d22+1] [d33+1] …
d) .[d11+1] [d21+1] [d31+1] …
Thm. |ℝ| != |ℕ|
17
Proof by contradiction: Assume |ℝ| = |ℕ|, so a
bijective function f exists between ℕ and ℝ.
• How do we reach a contradiction?
• Must show that z cannot be f(n) for any n
• How do we know that z ≠ f(n) for any n?
n
f(n)
1
.d11d12d13d14…
2
3
…
.d21d22d23d24…
.d31d32d33d34…
…
a) We can’t know if z = f(n)
without knowing what f
is and what n is
b) Because z’s nth digit
differs from n‘s nth digit
c) Because z’s nth digit
differs from f(n)’s nth
digit
Thm. |ℝ| != |ℕ|
• Proof by contradiction: Assume |ℝ| = |ℕ|, so
a correspondence f exists between N and ℝ.
• Want to show: f cannot work correctly.
• Let z = [z’s nth digit = (nth digit of f(n)) + 1].
• Note that z∈ℝ, but ∀n∈ℕ, z != f(n).
• Therefore f is not surjective, a contradiction.
• So |ℝ| ≠ |ℕ|
• |ℝ| > |ℕ|
Diagonalization
n
f(n)
1
.d11d12d13d14d15d16d17d18d19…
2
.d21d22d23d24d25d26d27d28d29…
3
.d31d32d33d34d35d36d37d38d39…
4
.d41d42d43d44d45d46d47d48d49…
5
.d51d52d53d54d55d56d57d58d59…
6
.d61d62d63d64d65d66d67d68d69…
7
.d71d72d73d74d75d76d77d78d79…
8
.d81d82d83d84d85d86d87d88d89…
9
.d91d92d93d94d95d96d97d98d99…
…
…
19
20
Some infinities are more infinite
than other infinities
•
Natural numbers are called countable
•
•
•
Real numbers are uncountable
•
•
Any set that can be put in correspondence with ℕ is
called countable (ex: E+, ℚ+).
Equivalently, any set whose elements can be
enumerated in an (infinite) sequence a1,a2, a3,…
Any set for which cannot be enumerated by a
sequence a1,a2,a3,… is called “uncountable”
But it gets even weirder…
•
There are more than two categories!
21
Some infinities are more infinite
than other infinities
|ℕ| is called ‫א‬0
o |E+| = |ℚ| = ‫א‬0
|ℝ| is maybe ‫א‬1
o
o
Although we just proved that |ℕ| < |ℝ|, and
nobody has ever found a different infinity
between |ℕ| and |ℝ|, mathematicians haven’t
proved that there are not other infinities between
|ℕ| and |ℝ|, making |ℝ| = ‫א‬2 or greater
In fact, it can be proved that such theorems can
never be proven…
Sets exist whose size is ‫א‬0, ‫א‬1, ‫א‬2, ‫א‬3…
An infinite number of aleph numbers!
o
An infinite number of different infinities
22
Famous People:
Georg Cantor (1845-1918)

His theory of set size, in particular
transfinite numbers (different infinities)
was so strange that many of his
contemporaries hated it





Just like many CSE 20 students!
“scientific charlatan” “renegade”
“corrupter of youth”
“utter nonsense” “laughable” “wrong”
“disease”
“I see it, but I don't believe it!” –Georg
Cantor himself
“The finest product of mathematical genius and one of
the supreme achievements of purely intellectual human
activity.” –David Hilbert