Class 18: Spooky Infinities

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

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

Transcript

1. Class 18: Spooky Infinities cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans University of Virginia
2. Plan Review: Countably Infinite Sets Any Bigger Sets? Preparation for Monday Uncountable Sets Problem Set 7 is due Friday Exam 2 is in two weeks (Nov 9)
3. Countably Infinite 3 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.
4. Example: ℤ is countably infinite 4
5. Example: (ℕ×ℕ) is countably infinite 5
6. Example: ℚ is countably infinite 6
7. Tsil Definition (PS7) Base: the empty tsil () Constructor: for any tsil, , and object, postpend(, ) is a tsil 7 How many tsils?
8. Tsil-∅ Definition Base: the empty tsil () Constructor: for any tsil, postpend(, ∅) is a tsil 8
9. Power Sets 9 The power set of a set is the set of all subsets of . ∈ ⟺ ⊆ What is the cardinality of ()?
10. 10 For all finite sets , > ?
11. 11 For all finite sets , = 2|D|. Proven in Class 11
12. 12 For all finite sets , > ||.
13. 13 For all sets , is > ?
14. 14 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 includes , this means > ⇒ ( + 1). Therefore, always holds, so we can conclude for all sets , > | |.
15. 15 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 includes , 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?
16. Georg Cantor’s Shocking Proof (~1874) 16 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).
17. 17 For all sets , > | |.
18. 18 For all sets , > | |.
19. Is ℕ countable? 19
20. Trick-or-Treat Protocols 20
21. “Trick or Treat” 21 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked
22. Illogical Threat 22 ∨ ∧ ⟹
23. “Trick or Treat” 23 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked Tricker must convince Victim that she poses a credible threat: prove she is a qualified tricker
24. Trick-or-Treat Trickers? Victim 24 Any problems with this?
25. Proof without Disclosure How can the tricker prove their trickability, without allowing the victim to now impersonate a tricker? 25
26. Challenge-Response Protocol 26 Prover: proves knowledge of by revealing (, ) . Verifier: convinces prover knows , but learns nothing useful about . Verifier: picks random . Need a one-way function: hard to invert, but easy to compute.
27. Example: RSA 27 Ee (M ) = Me mod n Dd (C ) = Cd mod n Correctness property: Ee (Dd ()) =
28. Trick-or-Treat Trickers? Victim 28
29. Trick-or-Treat Trickers? Victim 29 How does victim know e and n? Verify: j = jmod =
30. 30 “Elsa #253224”, = 3482..., = 1234... signed by Tricker’s Buroo Verify: j = jmod = Verify Tricker’s Buroo signature on certificate
31. 31 “virginia.edu”, = … = ... signed by Certificate Authority Verify and Decrypt: l j () = Verify signature on certificate Server
32. 32 “virginia.edu”, = … = ... signed by Certificate Authority Verify and Decrypt: l j () = Verify signature on certificate Server
34. Countable Strings? 34 BitStrings = ∀ ∈ ℕ . {0, 1}m
35. Countable Infinite Strings? 35 InfiniteBitStrings = {0, 1}n
36. 36 Class 9
37. Smallest Infinite Ordinal 37 = smallest infinite ordinal number ∞ + 1 = + 1 >
38. Smallest Infinite Ordinal 38 = smallest infinite ordinal number ∞ + 1 = ∞ = ∞ - 1 + 1 > ∀ ∈ ℕ . 2 > +
39. Countable Infinite Strings? 39 InfiniteBitStrings = {0, 1}n
40. Cantor’s Diagonalization Argument 40
42. Charge Enjoy Halloween But don’t be victimized by any unsubstantiated threats or spooky infinities! 42 Problem Set 7 due Friday Really spooky question: is there any set bigger than (ℕ) ?