27 Slides--More Quantifier Proofs

# 27 Slides--More Quantifier Proofs - CS103A HO 27 More...

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Unformatted text preview: CS103A HO# 27 More Quantifier Proofs 2/8/08 1 CS103A 2/8/08 Midterm Exam Tues., Feb. 12 7 - 9 pm Location: Gates B01 Open book (LPL), Open Notes, Crib Sheet Review Session Fri., Feb. 8 3:15 – 4:30 pm Location: Gates B01 Stanford Online, but not broadcast Today! ∀ x Cube(x) ∃ x Cube(x) ∀ x Cube(x) Cube(a) ∀ Elim 1 ∃ x Cube(x) ∃ Intro 2 1. 2. 3. ∀ x Cube(x) ∀ x Cube(x) → ∃ x Cube(x) ∃ xCube(x) → Elim 1, 5 1. 2. 3. 4. 5. 6. ∀ x Cube(x) ∀ x Cube(x) ∃ x Cube(x) ∀ x Cube(x) → ∃ x Cube(x) → Intro 2-4 ∃ xCube(x) → Elim 1, 5 a 1. 2. 3. 4. 5. 6. ∀ x Cube(x) ∀ x Cube(x) Cube(a) ∀ Elim 2 ∃ x Cube(x) ∃ Intro 3 ∀ x Cube(x) → ∃ x Cube(x) → Intro 2-4 ∃ xCube(x) → Elim 1, 5 a 1. 2. 3. 4. 5. 6. CS103A HO# 27 More Quantifier Proofs 2/8/08 2 ∀ x Cube(x) ∀ x Cube(x) Cube(a) ∀ Elim 2 ∃ x Cube(x) ∃ Intro 3 ∀ x Cube(x) → ∃ x Cube(x) → Intro 2-4 ∃ xCube(x) → Elim 1, 5 a 1. 2. 3. 4. 5. 6. ∀ x Cube(x) Cube(a) ∀ Elim 1 ∃ x Cube(x) ∃ Intro 2 1. 2. 3. ∀ x Cube(x) ∀ x Small(x) ∀ x(Cube(x) ∧ Small(x)) 13.3 ∀ x Cube(x) ∀ x Small(x) Cube(a) ∀ Elim 1 Small(a) ∀ Elim 2 Cube(a) ∧ Small(a) ∧ Intro 3,4 ∀ x(Cube(x) ∧ Small(x)) ∀ Intro 1. 2. 3. 4. 5. 6. 13.3 ∀ x Cube(x) ∀ x Small(x) Cube(a) ∀ Elim 1 Small(a) ∀ Elim 2 Cube(a) ∧ Small(a) ∧ Intro 3,4 This won't work! ∀ x(Cube(x) ∧ Small(x)) ∀ Intro You have to cite a subproof 1. 2. 3. 4. 5. 6. 13.3 ∀ x Cube(x) ∀ x Small(x) Cube(a) ∀ Elim 1 Small(a) ∀ Elim 2 Cube(a) ∧ Small(a) ∧ Intro 4,5 ∀ x(Cube(x) ∧ Small(x)) ∀ Intro 3 - 6 a 1. 2. 3. 4. 5. 6. 7. 13.3 ∀ x ¬ Cube(x) ∀ x ¬(Cube(x) ∧ Small(x)) CS103A HO# 27 More Quantifier Proofs 2/8/08 3 ∀ x ¬ Cube(x) ¬(Cube(a) ∧ Small(a)) ∀ x ¬(Cube(x) ∧ Small(x)) ∀ Intro 2-7 a 1. 2. 3. 4. 5. 6. 7. 8. ∀ x ¬ Cube(x) Cube(a) ∧ Small(a) ⊥ ¬(Cube(a) ∧ Small(a)) ¬ Intro 3-6 ∀ x ¬(Cube(x) ∧ Small(x)) ∀ Intro 2-7 a 1. 2. 3. 4. 5. 6. 7. 8. ∀ x ¬ Cube(x) Cube(a) ∧ Small(a) Cube(a) ∧ Elim 3 ¬ Cube(a) ∀ Elim 1 ⊥ ⊥ Intro 4,5 ¬(Cube(a) ∧ Small(a)) ¬ Intro 3-6 ∀ x ¬(Cube(x) ∧ Small(x)) ∀ Intro 2-7 a 1. 2. 3. 4. 5. 6. 7....
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27 Slides--More Quantifier Proofs - CS103A HO 27 More...

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