Class 11: Strong Induction

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

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

Transcript

1. Class 11: Stronger Induction cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia Exam 1 is in class, Thursday, October 6. See today’s notes for details an preparation advice. Before 6:59pm Wednesday, send me topics you would like to review (read directions on notes carefully, feel free to collude to hit optimal result).
2. Plan: All about Induction Induction Practice Induction in Practice “Strong” Induction Time permitting: resolve the chicken-egg conundrum! Exam 1 is in class, Thursday, October 6. See today’s notes for details an preparation advice. Before 6:59pm Wednesday, send me topics you would like to review (read directions on notes carefully, feel free to collude to hit optimal result).
3. What’s on Exam 1? MCS Chapters 1-5 Classes 1-11 and Notes
4. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1).
5. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Started last class: prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷=
6. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Started last class: prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷= all subsets of ℕ of size + 1 have a minimum element. More naturally: ∀ ∈ ℕ1. ∷= all subsets of ℕ of size have a minimum element.
7. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Started last class: prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷= all subsets of ℕ of size + 1 have a minimum element. More naturally: ∀ ∈ ℕ1. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ1. : 1. Prove (1). 2. Prove ∀ ∈ ℕ1. ⟹ ( + 1). We can extend the induction principle to any well-ordered set with a “+ 1” operation that covers all the elements!
8. ∀ ∈ ℕ1. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ1. : 1. Prove (1). 2. Prove ∀ ∈ ℕ1. ⟹ ( + 1). 1. Prove 1 . “ ”
9. ∀ ∈ ℕ1. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ1. : 1. Prove (1). 2. Prove ∀ ∈ ℕ1. ⟹ ( + 1). 2. Prove ∀ ∈ ℕ1. ⟹ ( + 1).“ ”
11. Take-Away Game Start with = 16 sticks Each turn: player must remove 1, 2, or 3 sticks Winner is player who takes the last stick
12. Do you want to be Player 1 or Player 2? Start with = 16 sticks Each turn: player must remove 1, 2, or 3 sticks Player who takes last stick wins
14. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, , always ends.
15. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, , always ends.
16. Prove: Winning Strategy Theorem. For a Take-Away game with any initial number of sticks, , either Player 1 has a winning strategy or Player 2 does.
17. What about Checkers?
19. Theorem. Player 1 has a winning strategy for a Take-Away game with sticks, ∀ ∈ ℕ. ≠ 4. Player 2 has a winning strategy ∀ ∈ ℕ. = 4. Left for exercise, but we can do this Thursday.
20. Induction in Practice
21. Prove list_sum is correct:
23. Strong Induction Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Strong Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. (∀ ∈ ℕ, ≤ . ) ⟹ ( + 1).
24. Example. A tree with nodes has − 1 edges.
25. Do we really need “strong induction”?
26. Restating ∷= any predicate on Strong Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. (∀ ∈ ℕ, ≤ . ) ⟹ ( + 1).
27. Restating Strong Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. (∀ ∈ ℕ, ≤ . ) ⟹ ( + 1). ∷= any predicate on G ∷= ∀ ∈ ℕ, < . () Any proof using strong induction, can be rewritten to use “weak” induction by just making P(n) stronger.
28. Chicken vs. Egg
29. Theorem: There are no chickens.
30. Definition (Wikipedia). A species is defined as the largest group of organisms in which two individuals are capable of reproducing fertile offspring. Definition. A chicken is an organism of the species Gallus gallus domesticus. Reasonable Assumption. Assume time is quantized into the smallest possible time quanta, so at each time tick at most a single new organism is born.
31. Proof by Induction: ∀ ∈ ℕ. ∷= the number of chickens at time is 0.
32. Base case: (0). At time 0, there were no chickens. The temperature of the universe was 1028 ℃, and even fried chicken cannot exist under those conditions.
33. Proof by Induction: ∀ ∈ ℕ. ∷= the number of chickens at time is 0. Inductive case: ⟹ ( + 1). Case 1: no new organism is both at time + 1. There are still no chickens. Reasonable Assumption. Assume time is quantized into the smallest possible time quanta, so at each time tick at most a single new organism is born.
34. Proof by Induction: ∀ ∈ ℕ. ∷= the number of chickens at time is 0. Case 2: a new organism, , is born at time + 1. Definition (Wikipedia). A species is defined as the largest group of organisms in which two individuals are capable of reproducing fertile offspring.
35. Proof by Induction: ∀ ∈ ℕ. ∷= the number of chickens at time is 0. Case 2: a new organism, , is born at time + 1. - Case 2a: can produce viable offspring with an existing organism, in which case it is not a chicken since it is part of a species that existed at time . - Case 2b: cannot produce viable offspring with any existing organisms. It might be a “chicken”, but it can’t reproduce, so will eventually die without any offspring.
36. But…chickens appear to exist!
37. If a “proof” leads to an apparent falsehood, what went wrong?
38. Salamander Speciation in California There was no “first” chicken. But, some group of pre-chickens became the origin of the “chicken” species when they were reproductively isolated from other pre-chickens.
39. Charge PS5 Due Friday (6:29pm) Exam 1 Oct 6: See notes about requesting review