Class 12

Discrete Math

Transcript

1. Class 12: Strong induction Review cs2102: Discrete Mathematics David Evans, Mohammad Mahmoody | University of Virginia
2. Plan 1. Strong Induction 2. Review: • Proof methods • Functions/relations • CNF,DNF,3CNF
3. 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)
4. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ+, ends.
5. Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ+, ends. Is this proof by induction? Seems similar but still different..
6. Using Induction Using Strong Induction strong P(0),..P(n)
7. Strong Induction Rule / Principle
8. Theorem 2. Player 1 has a winning strategy for a Take-Away game with sticks, ∀ ∈ ℕ. ≠ 4. Player 2 has a winning strategy ∀ ∈ ℕ. = 4. Theorem 1 . A Take-Away game with any initial number of sticks, ∈ ℕ+, ends.
9. Proved using Well-Ordering Principle in Class 3 Proof using Strong Induction
10. Which proof method is “more useful”? 1. Well Ordering Principle 2. Induction 3. Strong Induction Slack..
11. They are equally powerful! Each principle implies the other two.. We keep all 3 of them to use whichever is more convenient.
12. Review…
13. Proof Methods:
14. Proof by contra-positive Recall → is equivalent to ¬ → ¬ (check truth table) So, when we want to prove → instead we prove ¬ → ¬ Example: Suppose is a real number. Prove that if is irrational, then √ is also irrational.
15. Proof by contradiction Example: Proving that is not well ordered. Usually useful when we want to prove something like ∀. () or ∀ ¬() (i.e. there is no x P(x))
16. Relations Functions
17. Binary Relation R between A,B (from A to B) with graph ⊆ ×
18. Example 0: Suppose relation : → has the property that: ∀, . ¬() Which properties below must have as well?
19. Example 1: Suppose relation : → has the property that: ∀, 1 , 2 . 1 ∧ 2 → 1 = 2 Which properties below must have as well?
20. How to convert formulas into: DNF, CNF, 3CNF
21. DNF Example: → ↔
22. CNF Example: → ↔
23. 3CNF Example1: → ↔ Example2: = ∨ ∧ ∨ ∨ ∨ Write a 3CNF such that is satisfiable if and only if is satisfiable