Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA CSE 20 Discrete math Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/

Todays topics Countably infinitely large sets Uncountable sets “To infinity, and beyond!” (really, we’re going to go beyond infinity)

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 (one-to-one and onto) function mapping from one to the other Intuition: neither set has any element “left over” in the mapping

one-to-one and onto f is: one-to-one onto Bijective (both (a) and (b)) 1 2 3 4 … a aa aaa aaaa … f Sequences of a’s Natural numbers f is: one-to-one onto Bijective (both (a) and (b)) Neither

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens? 1 2 3 4 … 2 4 6 8 … Natural numbers Positive evens Yes and my function is bijective Yes and my function is not bijective No (explain why not)

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens? 1 2 3 4 … 2 4 6 8 … f Natural numbers Positive evens f(x)=2x

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds? 1 2 3 4 … 1 3 5 7 … Natural numbers Positive odds Yes and my function is bijective Yes and my function is not bijective No (explain why not)

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds? 1 2 3 4 … 1 3 5 7 … f Natural numbers Positive odds f(x)=2x-1

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?

It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 𝑄 + = 𝑛 𝑚 :𝑛,𝑚∈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+? It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 𝑄 + = 𝑛 𝑚 :𝑛,𝑚∈𝑁 Is there a bijection from the natural numbers to Q+? Yes No 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

It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 𝑄 + = 𝑛 𝑚 :𝑛,𝑚∈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

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

Not all infinite sets have the same size We will prove a very powerful theorem: the natural numbers and real numbers have different sizes In fact, it’s enough to consider real numbers in [0,1] 𝑁 ≠|[0,1]| The proof will use a new idea: diagonalization

Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. 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 real number 𝑥∈[0,1] that can never be f(n) for any n Therefore f is not onto, a contradiction. 1 2 3 4 … ? z … f Natural numbers Real numbers in [0,1]

Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. We construct 𝑥∈[0,1] as follows: The nth digit of x is the nth digit of f(n), PLUS ONE* (*wrap to 1 if the digit is 9, ie compute MOD 10) Below is an example f What is x in this example? .244… .134… .031… .245… n f(n) 1 .100000… 2 .333333… 3 .314159… …

Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. We construct 𝑥∈[0,1] as follows: The nth digit of x is the nth digit of f(n), PLUS ONE* (*wrap to 1 if the digit is 9, ie compute MOD 10) Below is a generalized f n f(n) 1 .d11d12d13d14… 2 .d21d22d23d24… 3 .d31d32d33d34… … What is x? .d11d12d13… .d11d22d33 … .[d11+1] [d22+1] [d33+1] … .[d11+1] [d21+1] [d31+1] …

Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. How do we reach a contradiction? Must show that x cannot be f(n) for any n How do we know that x ≠ f(n) for any n? We can’t know if x = f(n) without knowing what f is and what n is Because x’s nth digit differs from n‘s nth digit Because x’s nth digit differs from f(n)’s nth digit n f(n) 1 .d11d12d13d14… 2 .d21d22d23d24… 3 .d31d32d33d34… …

Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. Want to show: f cannot work correctly. Define 𝑥∈[0,1] as follows: x’s nth digit = (nth digit of f(n)) + 1 (mod 10). We have: ∀𝑛∈𝑁, 𝑥≠𝑓(𝑛) Therefore f is not onto, a contradiction. So |[0,1]| ≠ |N| In fact: |[0,1]| > |N|

Diagonalization n f(n) 1 .d11d12d13d14d15d16d17d18d19… 2

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!

Some infinities are more infinite than other infinities |N| is called א0 |E+| = |Q| = א0 |[0,1]| is maybe א1 Although we just proved that |N| < |[0,1]|, and nobody has ever found a different infinity between |N| and |[0,1]|, mathematicians haven’t proved that there are not other infinities between |N| and |[0,1]|, making |[0,1]| = 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! An infinite number of different infinities

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

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