CSE 20 – Discrete MathematicsDr. Cynthia Bailey LeeDr. Shachar Lovett
Peer Instruction in Discrete Mathematics by Cynthia Leeis licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.Based on a work at http://peerinstruction4cs.org.Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org.
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Today’s Topics:1. Countably infinitely large sets2. Uncountable sets
“To infinity, and beyond!” (really, we’re going to go beyond infinity)
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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
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Injective and Surjective
f is:a) Injectiveb) Surjectivec) Bijective (both (a) and (b))d) Neither
Sequences of a’sNatural
numbers
1234…
aaaaaaaaaa
…
f
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Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens?
A. Yes and my function is bijectiveB. Yes and my function is not bijectiveC. No (explain why not)
Positive evens
Natural numbers
1234…
2468…
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Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens?
f(x)=2x
Natural numbers
1234…
2468…
f
Positive evens
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Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds?
A. Yes and my function is bijectiveB. Yes and my function is not bijectiveC. No (explain why not)
Positive odds
Natural numbers
1234…
1357…
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Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds?
f(x)=2x-1
Natural numbers
1234…
1357…
f
Positive odds
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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?
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It gets even weirder:Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
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… … … … … … …
ℚ+ ℕ
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It gets even weirder:Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
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… … … … … … …
ℚ+ ℕ
Is there a bijection from the natural
numbers to Q+?
A. YesB. No
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It gets even weirder:Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q)
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… … … … … … …
ℚ+ ℕ
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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! |ℕ| = |E+|, not |ℕ| = 2|E+|
The number of Rational Numbers is equal to the number of Natural Numbers |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
14Thm. |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
1234…
???z?…
f
Real numbers
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n f(n)1 .100000…2 .333333…3 .314159…… …
• 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
What is z in this example? a) .244…b) .134…c) .031…d) .245…
Thm. |ℝ| != |ℕ|Proof by contradiction: Assume |ℝ| = |ℕ|, so a bijective function f exists between ℕ and ℝ.
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n f(n)1 .d11
d12d13
d14…
2 .d21d22
d23d24
…3 .d31
d32d33
d34…
… …
What is z?a) .d11
d12d13
…b) .d11
d22d33
…c) .[d11
+1] [d22+1] [d33
+1]
…d) .[d11
+1] [d21+1] [d31
+1]
…
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
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n f(n)1 .d11
d12d13
d14…
2 .d21d22
d23d24
…3 .d31
d32d33
d34…
… …
• 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?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 bijective function f exists between ℕ and ℝ.
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 zℝ, but nℕ, z != f(n).
• Therefore f is not surjective, a contradiction.
• So |ℝ| ≠ |ℕ|• |ℝ| > |ℕ|
Diagonalization
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n f(n)1 .d11
d12d13
d14d15
d16d17
d18d19
…2 .d21
d22d23
d24d25
d26d27
d28d29
…3 .d31
d32d33
d34d35
d36d37
d38d39
…4 .d41
d42d43
d44d45
d46d47
d48d49
…5 .d51
d52d53
d54d55
d56d57
d58d59
…6 .d61
d62d63
d64d65
d66d67
d68d69
…7 .d71
d72d73
d74d75
d76d77
d78d79
…8 .d81
d82d83
d84d85
d86d87
d88d89
…9 .d91
d92d93
d94d95
d96d97
d98d99
…… …
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Some infinities are more infinite than other infinities• Natural numbers are called countable
• 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,…
• Real numbers are uncountable• 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!
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Some infinities are more infinite than other infinities|ℕ| is called 0א
o |E+| = |ℚ| = 0א
|ℝ| is maybe 1אo 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| = 2א or greater
o In fact, it can be proved that such theorems can never be proven…
Sets exist whose size is 3א, 2א, 1א, 0א …An infinite number of aleph numbers!
o An infinite number of different infinities
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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