Class 11: Induction Practice

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

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

Transcript

1. Exam 1 is in class, Thursday, October 5. See today’s notes for details an preparation advice. Thursday’s class may be an exam review - read directions on notes carefully, feel free to collude to hit optimal result. Class 11: Induction Practice cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans | University of Virginia
2. Plan: All about Induction Induction Practice Induction in Practice Exam 1 is in class, Thursday, October 5. See today’s notes for details an preparation advice. Before 6:59pm Wednesday, send 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). = 2|5|
5. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). To fit into exact induction principle: ∀ ∈ ℕ. ∷=
6. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Prove ∀ ∈ ℕ. pow ℕ: = 2:
8. Prove by Induction: sum of first positive integers is :(:;<) = Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1).
9. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷=
10. Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). To fit into exact induction principle: ∀ ∈ ℕ. ∷= all subsets of ℕ of size + 1 have a minimum element. More naturally: ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Prove that all non-empty finite subsets of ℕ have a minimum element.
11. 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: ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). We can extend the induction principle to any well-ordered set with a “+ 1” operation that covers all the elements!
12. ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). 1. Prove 1 . “ ”
13. ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1).“ ”
14. 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 Do you want to be Player 1 or Player 2?
16. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ;, ends.
17. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ;, ends.
18. Charge Look at today’s notes (already posted) – Send requests for exam review by tomorrow PS5 Due Friday Exam 1 next Thursday (Oct 5)