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CSE 20 – Discrete Mathematics

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CSE 20 – Discrete Mathematics

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  1. 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. CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett

  2. Today’s Topics: • Countably infinitely large sets • Uncountable sets • “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 1 2 3 4 … a aa aaa aaaa … f f is: • Injective • Surjective • Bijective (both (a) and (b)) • Neither Sequences of a’s Natural numbers

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

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

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

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

  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+ℕ

  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+ℕ Is there a bijection from the natural numbers to Q+? Yes 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+ℕ

  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! • |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

  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. 1 2 3 4 … ? ? ? z ? … f Real numbers Natural numbers

  15. 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 an example f What is z in this example? .244… .134… .031… .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 generalizedf What is z? .d11d12d13… .d11d22d33… .[d11+1] [d22+1] [d33+1]… .[d11+1] [d21+1] [d31+1]…

  17. Thm. |ℝ| != |ℕ|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? We can’t know if z = f(n) without knowing what f is and what n is Because z’s nth digit differs from n‘s nth digit Because z’s nth digit differs from f(n)’s nth digit

  18. 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 nthdigit = (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|

  19. Diagonalization

  20. Some infinities are more infinite than other infinities • Natural numbers are called countable • Any set that can be put in correspondence with ℕis 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!

  21. Some infinities are more infinite than other infinities • |ℕ| is called א0 • |E+| = |ℚ| = א0 • |ℝ| is maybeא1 • 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! • 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

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