Class 19: Uncountable Sets

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 19: Uncountable Sets cs2102: Discrete Mathematics | F16 uvacs2102.github.io 0 David Evans University of Virginia
2. Plan Recap: how many tsils? Uncountable Sets! 1 PS8 Due Friday Exam 2 Review Tuesday (if requested)
3. Countably Infinite 2 A set is countable if and only if there exists a surjective function from ℕ to . A set is infinite, if there is no bijection between and any ℕ% . A set is countably infinite if it is countable and it is infinite. A set is countably infinite if there exists a bijection between and ℕ. Is the set of Tsils countable?
4. Tsil Definition (PS8) Base: the empty tsil () Constructor: for any tsil, , and object, , postpend(, ) is a tsil 3 How many tsils?
5. Tsil- ∅ Definition Base: the empty tsil () Constructor: for any tsil, , postpend(, ∅) is a tsil 4
6. Challenge: How many Trees? (PS8 Definition) 5 null: Tree node: Tree×ℕ×Tree → Tree
7. Power Sets 6 The power set of a set is the set of all subsets of . ∈ ⟺ ⊆ What is the cardinality of ()?
8. 7 For all finite sets , = 2|G|.
9. 8 For all sets , > | |.
10. 9 For all sets , > | |. Bogus non-proof: Proof by induction: ∷= ∀ sets where = . > . Base case: 0 : = ∅. = {∅}. = 1 > = 0. Inductive case: ∀ ∈ ℕ. ⇒ + 1 . for all sets where = + 1, ∃ where = , ∉ . = ∪ ⇒ > ⇒ + 1 > + 1 Since () includes all elements of () and include , this means > ⇒ ( + 1). Therefore, always holds, so we can conclude for all sets , > | |.
11. 10 For all sets , > | |. Bogus non-proof: Proof by induction: ∷= ∀ sets where = . > . Base case: 0 : = ∅. = {∅}. = 1 > = 0. Inductive case: ∀ ∈ ℕ. ⇒ + 1 . for all sets where = + 1, ∃ where = , ∉ . = ∪ ⇒ > ⇒ + 1 > + 1 Since () includes all elements of () and include , this means > ⇒ ( + 1). Therefore, always holds, so we can conclude for all sets , > | |. 1 2 3 4 5 6 7 8 9 10 11 Which step is wrong?
12. Georg Cantor’s Shocking Proof (~1874) 11 Historical note: this isn’t actually Cantor’s proof, although it is commonly called Cantor’s Theorem. Cantor came up with the diagonalization argument we will see next; this proof is believed to have been first done by Hessenberg (1906).
13. 12 For all sets , > | |.
14. 13 For all sets , > | |.
15. Is ℕ countable? 14
16. Countable Strings? 15 BitStrings = ∀ ∈ ℕ . {0, 1}\
17. Countable Infinite Strings? 16 InfiniteBitStrings = {0, 1}]
18. Ordinals vs. Cardinals 17 size position
19. Smallest Infinite Ordinal 18 = smallest infinite ordinal number ∞ + 1 = + 1 >
20. Smallest Infinite Ordinal 19 = smallest infinite ordinal number ∞ + 1 = + 1 > ∀ ∈ ℕ . 2 > +
21. Countable Infinite Strings? 20 InfiniteBitStrings = {0, 1}]
22. Cantor’s Diagonalization Argument 21
24. Charge Infinities are Spooky, Strange, and Surprising PS8 due Friday Exam 1 next Thursday Send review requests by Monday 23