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CSE 20 DISCRETE MATH Prof. Shachar Lovett Clicker frequency: CA

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Todays topics Countably infinitely large sets 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 ( ), 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

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f is: a) one-to-one b) onto c) Bijective (both (a) and (b)) d) Neither Sequences of a’s Natural numbers 1234…1234… a aa aaa aaaa … f one-to-one and onto

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A. Yes and my function is bijective B. Yes and my function is not bijective C. No (explain why not) 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?

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f(x)=2x 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 Evens?

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A. Yes and my function is bijective B. Yes and my function is not bijective C. No (explain why not) 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?

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f(x)=2x-1 Natural numbers 1234…1234… 1357…1357… f Positive odds Can you make a function that maps from the domain Natural Numbers, to the co-domain 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) 1/11/21/31/41/51/6… 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 …………………

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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) 1/11/21/31/41/51/6… 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

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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 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 …………………

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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

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Not all infinite sets have the same size

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Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1]. Natural numbers 1234…1234… ???z?…???z?… f Real numbers in [0,1]

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nf(n) … … … …… What is x in this example? a).244… b).134… c).031… d).245… Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1].

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nf(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 x? 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. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1].

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nf(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 x cannot be f(n) for any n How do we know that x ≠ f(n) for any n? a)We can’t know if x = f(n) without knowing what f is and what n is b)Because x’s n th digit differs from n‘s n th digit c)Because x’s n th digit differs from f(n)’s n th digit Thm. |[0,1]| != |N| Proof by contradiction: Assume |[0,1]| = |N|, so a bijective function f exists between N and [0,1].

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Thm. |[0,1]| != |N|

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Diagonalization 20 nf(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 … ……

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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 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!

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Some infinities are more infinite than other infinities |N| is called א 0 o |E+| = |Q| = א 0 |[0,1]| is maybe א 1 o 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]| = 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

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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 “The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” –David Hilbert

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Next class Final review

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