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)
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Prove: Always Ends
Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ+, ends.
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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..
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Using Induction
Using Strong Induction
strong
P(0),..P(n)
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Strong Induction Rule / Principle
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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.
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Proved using Well-Ordering Principle in Class 3
Proof using
Strong Induction
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Which proof method is “more useful”?
1. Well Ordering Principle
2. Induction
3. Strong Induction
Slack..
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They are equally powerful!
Each principle implies the other two..
We keep all 3 of them to use whichever is more convenient.
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Review…
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Proof Methods:
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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.
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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))
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Relations Functions
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Binary Relation R between A,B
(from A to B) with graph ⊆ ×
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Example 0:
Suppose relation : → has the property that:
∀, . ¬()
Which properties below must have as well?
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Example 1:
Suppose relation : → has the property that:
∀, 1
, 2
. 1
∧ 2
→ 1
= 2
Which properties below must have as well?
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How to convert formulas into:
DNF, CNF, 3CNF
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DNF
Example: → ↔
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CNF
Example: → ↔
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3CNF
Example1: → ↔
Example2: = ∨ ∧ ∨ ∨ ∨
Write a 3CNF such that is satisfiable if and only if is satisfiable