CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. 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 Permissions beyond the scope of this license may be available at LeeCreative Commons Attribution- NonCommercial-ShareAlike 4.0 International Licensehttp://peerinstruction4cs.org

Today’s Topics: 1. Countably infinitely large sets 2. Uncountable sets  “To infinity, and beyond!” (really, we’re going to go beyond infinity) 2

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 ( ), 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 3

Injective and Surjective f is: a) Injective b) Surjective c) Bijective (both (a) and (b)) d) Neither 4 Sequences of a’s Natural numbers 1234…1234… a aa aaa aaaa … f

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens? A. Yes and my function is bijective B. Yes and my function is not bijective C. No (explain why not) 5 Positive evens Natural numbers 1234…1234… 2468…2468…

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Evens? f(x)=2x 6 Natural numbers 1234…1234… 2468…2468… f Positive evens

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds? A. Yes and my function is bijective B. Yes and my function is not bijective C. No (explain why not) 7 Positive odds Natural numbers 1234…1234… 1357…1357…

Can you make a function that maps from the domain Natural Numbers, to the co-domain Positive Odds? f(x)=2x-1 8 Natural numbers 1234…1234… 1357…1357… f Positive odds

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? 9

It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 10 1/11/21/31/41/51/6… 2/12/22/32/42/52/6… 3/13/23/33/43/53/6… 4/14/24/34/44/54/6… 5/15/25/35/45/55/6... 6/16/26/36/46/56/6 …………………

It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 11 1/11/21/31/41/51/6… 2/12/22/32/42/52/6… 3/13/23/33/43/53/6… 4/14/24/34/44/54/6… 5/15/25/35/45/55/6... 6/16/26/36/46/56/6 ………………… Is there a bijection from the natural numbers to Q + ? A.Yes B.No

It gets even weirder: Rational Numbers (for simplicity we’ll do ratios of natural numbers, but the same is true for all Q) 12 1/1 11/2 21/3 41/4 61/5 101/6… 2/1 32/2 x2/3 72/4 x2/52/6… 3/1 53/2 83/3 x3/43/53/6… 4/1 94/2 x4/34/44/54/6… 5/1 115/25/35/45/55/6... 6/16/26/36/46/56/6 …………………

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 13

Thm. | ℝ | != | ℕ | Proof by contradiction: Assume | ℝ | = | ℕ |, so a bijective function f exists between ℕ and ℝ. 14 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…1234… ???z?…???z?… f Real numbers

15 n f (n) … … … …… We construct z as follows: z’s n th digit is the n th 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.

16 n f (n) 1.d 1 1 d 1 2 d 1 3 d 1 4 … 2.d 2 1 d 2 2 d 2 3 d 2 4 … 3.d 3 1 d 3 2 d 3 3 d 3 4 … …… What is z? a).d 1 1 d 1 2 d 1 3 … b).d 1 1 d 2 2 d 3 3 … c).[d ] [d ] [d ] … d).[d ] [d ] [d ] … 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 n th digit is the n th digit of f(n), PLUS ONE* (*wrap to 1 if the digit is 9) Below is a generalized f

17 n f (n) 1.d 1 1 d 1 2 d 1 3 d 1 4 … 2.d 2 1 d 2 2 d 2 3 d 2 4 … 3.d 3 1 d 3 2 d 3 3 d 3 4 … …… 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 n th digit differs from n‘s n th digit c)Because z’s n th digit differs from f(n)’s n th digit Thm. | ℝ | != | ℕ | Proof by contradiction: Assume | ℝ | = | ℕ |, so a bijective function f exists between ℕ and ℝ.

Thm. | ℝ | != | ℕ |

Diagonalization 19 n f (n) 1.d 1 1 d 1 2 d 1 3 d 1 4 d 1 5 d 1 6 d 1 7 d 1 8 d 1 9 … 2.d 2 1 d 2 2 d 2 3 d 2 4 d 2 5 d 2 6 d 2 7 d 2 8 d 2 9 … 3.d 3 1 d 3 2 d 3 3 d 3 4 d 3 5 d 3 6 d 3 7 d 3 8 d 3 9 … 4.d 4 1 d 4 2 d 4 3 d 4 4 d 4 5 d 4 6 d 4 7 d 4 8 d 4 9 … 5.d 5 1 d 5 2 d 5 3 d 5 4 d 5 5 d 5 6 d 5 7 d 5 8 d 5 9 … 6.d 6 1 d 6 2 d 6 3 d 6 4 d 6 5 d 6 6 d 6 7 d 6 8 d 6 9 … 7.d 7 1 d 7 2 d 7 3 d 7 4 d 7 5 d 7 6 d 7 7 d 7 8 d 7 9 … 8.d 8 1 d 8 2 d 8 3 d 8 4 d 8 5 d 8 6 d 8 7 d 8 8 d 8 9 … 9.d 9 1 d 9 2 d 9 3 d 9 4 d 9 5 d 9 6 d 9 7 d 9 8 d 9 9 … ……

Some infinities are more infinite than other infinities 20 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 a 1,a 2, a 3,… Real numbers are uncountable Any set for which cannot be enumerated by a sequence a 1,a 2,a 3,… is called “uncountable” But it gets even weirder… There are more than two categories!

Some infinities are more infinite than other infinities 21 | ℕ | is called א 0 o |E+| = | ℚ | = א 0 | ℝ | is maybe א 1 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 | = א 2 or greater o In fact, it can be proved that such theorems can never be proven… Sets exist whose size is א 0, א 1, א 2, א 3 … An infinite number of aleph numbers! o An infinite number of different infinities

Famous People: Georg Cantor ( )  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 22 “The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” –David Hilbert