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Exploring Countable and Uncountable Sets in Discrete Mathematics

This educational resource delves into the fascinating world of set theory, focusing on countably and uncountably infinite sets. Learn how Georg Cantor's revolutionary ideas allow us to compare the sizes of infinite sets without counting. Discover the definition of size equality through bijections, and explore various functions mapping natural numbers to even and odd numbers. Uncover the surprising equality of the set of natural numbers and rational numbers while proving the theorem that the set of real numbers is a different size altogether, showcasing the oddities of infinity in mathematics.

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Exploring Countable and Uncountable Sets in 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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