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Unformatted text preview: Administration Midterm 1 is not early after all. We don’t think 3 weeks is enough material to merit a midterm. CS70: Lecture 3. Outline. 1. Proofs 2. Simple 3. Direct 4. by Contrapositive 5. by Cases 6. by Contradiction Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true I P is false. Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true I P is false. “ false = ⇒ ’anything’ ”, is true Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true I P is false. “ false = ⇒ ’anything’ ”, is true so “ P = ⇒ ’anything’ ” is true . Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true I P is false. “ false = ⇒ ’anything’ ”, is true so “ P = ⇒ ’anything’ ” is true . in particular P = ⇒ ( P ∨ Q ) is true . Simple theorem.. Theorem: P = ⇒ ( P ∨ Q ). Proof: I P is true. P ∨ Q is true “ ’anything’ = ⇒ true” is true so X = ⇒ ( P ∨ Q ) is true for all X , and in particular, P = ⇒ ( P ∨ Q ) is true I P is false. “ false = ⇒ ’anything’ ”, is true so “ P = ⇒ ’anything’ ” is true . in particular P = ⇒ ( P ∨ Q ) is true . More detailed but the “same” as truth table proof in some sense. Proof by truth table. Theorem: P = ⇒ ( P ∨ Q ). Proof by truth table. Theorem: P = ⇒ ( P ∨ Q ). Proof: P Q P ∨ Q T T T T F T F T T F F F Proof by truth table. Theorem: P = ⇒ ( P ∨ Q ). Proof: P Q P ∨ Q T T T T F T F T T F F F Look only at appropriate rows. Where theorem condition is true. Proof by truth table....
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 Fall '11
 Rau
 Prime number, Divisor, Mathematical terminology, Mathematical proof

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