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Assignment 3 Jeff Rose
- Give a formal proof of the following theorems: (a) (( A ∧ ¬ B ) ∨ C ) → ( B → C )
- (^) ( A ∧ ¬ B ) ∨ C P
- B P
- (^) ¬ C P for IP
- (^) A ∧ ¬ B 1, 3, DS
- ¬ B 4, Simp.
- (^) ⊥ 2, 5, Conj.
- C 3, 6, IP
- B → C 2, 7, CP QED 1, 8, CP (b) ( A → B ) → (( A ∨ B ) ≡ B )
- (^) A → B P
- (^) ¬(( A ∨ B ) → B ) P for IP
- (^) ¬(¬( A ∨ B ) ∨ B ) 2, T, E
- (^) ( A ∨ B ) ∧ ¬ B 3, T, E
- (^) A ∨ B 4, Simp.
- (^) ¬ B 4, Simp.
- A 5, 6, DS
- B 1, 7, MP
- (^) ⊥ 6, 8, Conj.
- (^) ( A ∨ B ) → B 2, 9, IP
- ¬( B → ( A ∨ B )) P for IP
- (^) ¬(¬ B ∨ ( A ∨ B )) 11, T, E
- (^) B ∧ ¬( A ∨ B ) 12, T, E
- (^) B ∧ ¬ A ∧ ¬ B 13, T, E
- ⊥ 14, T, E
- (^) B → ( A ∨ B ) 11, 15, IP
- (^) (( A ∨ B ) → B ) ∧ ( B → ( A ∨ B )) 10, 16, Conj.
- (^) A ∨ B ≡ B 17, T, E QED 1, 18, CP
(c) (( A ∨ B ) ≡ B ) → ( A → B )
- (^) ( A ∨ B ) ≡ B P
- (^) (( A ∨ B ) → B ) ∧ ( B → ( A ∨ B )) 1, T, E
- (^) ( A ∨ B ) → B 2, Simp.
- A P
- (^) A ∨ B 4, Add
- B 3, 5, MP
- (^) A → B 4, 6, CP QED 1, 7, CP (d) B ∨ ((( A ∨ B ) → ( C ∧ D )) ∧ ( A ∨ C )) → ( A → B ) ∨ C
- (^) B ∨ ((( A ∨ B ) → ( C ∧ D )) ∧ ( A ∨ C )) P
- (^) B ∨ ((¬( A ∨ B ) ∨ ( C ∧ D )) ∧ ( A ∨ C )) 1, T, E
- B ∨ (((¬ A ∧ ¬ B ) ∨ ( C ∧ D )) ∧ ( A ∨ C )) 2, T, E
- (^) ¬(( A → B ) ∨ C ) P for IP
- (^) ¬( A → B ) ∧ ¬ C 4, T, E
- (^) ¬(¬ A ∨ B ) ∧ ¬ C 5, T, E
- A ∧ ¬ B ∧ ¬ C 6, T, E
- (^) ¬ B 7, Simp.
- (^) ((¬ A ∧ ¬ B ) ∨ ( C ∧ D )) ∧ ( A ∨ C ) 3, 8, DS
- (^) (¬ A ∧ ¬ B ) ∨ ( C ∧ D ) 9, Simp.
- A 7, Simp.
- (^) A ∨ B 11, Add.
- (^) ¬(¬ A ∧ ¬ B ) 12, T, E
- (^) C ∧ D 10, 13, DS
- C 14, Simp.
- (^) ¬ C 7, Simp.
- (^) ⊥ 15, 16, Conj.
- (^) ( A → B ) ∨ C 4, 17, IP QED 1, 18, CP
The Fifth Trial The same rules apply, and here are the signs: 1 AT LEAST ONE ROOM CONTAINS A LADY
THE OTHER ROOM
CONTAINS A LADY
What should the prisoner do? (L 1 ≡ S 1 ) ∧ (L 2 ≡ ¬S 2 ) ∧ (S 1 ≡ (L 1 ∨ L 2 )) ∧ (S 2 ≡ L 1 )
- L 1 ≡ S 1 P
- (^) L 2 ≡ ¬S 2 P
- (^) S 1 ≡ (L 1 ∨ L 2 ) P
- S 2 ≡ L 1 P
- (^) ¬L 1 P for IP
- (^) (L 1 → S 1 ) ∧ (S 1 → L 1 ) 1, T, E
- (^) S 1 → L 1 6, Simp.
- (^) ¬S 1 5, 7, MT
- (^) (S 1 → (L 1 ∨ L 2 )) ∧ ((L 1 ∨ L 2 ) → S 1 ) 3, T, E
- (^) (L 1 ∨ L 2 ) → S 1 9, Simp.
- (^) ¬(L 1 ∨ L 2 ) 8, 10, MT
- (^) ¬L 1 ∧ ¬L 2 11, T, E
- (^) ¬L 2 12, Simp.
- (^) (L 2 → ¬S 2 ) ∧ (¬S 2 → L 2 ) 2, T, E
- (^) ¬S 2 → L 2 14, Simp.
- (^) S 2 13, 15, MT
- (^) (S 2 → L 1 ) ∧ (L 1 → S 2 ) 4, T, E
- (^) S 2 → L 1 17, Simp.
- (^) L 1 16, 18, MP
- (^) ⊥ 5, 19, Conj.
- L 1 5, 20, IP The prisoner should pick the first room.
The Sixth Trial The king was particularly fond of this puzzle, and the next one too. Here are the signs: 1 IT MAKES NO DIFFERENCE WHICH ROOM YOU PICK
THERE IS A LADY
IN THE OTHER ROOM
What should the prisoner do? (L 1 ≡ S 1 ) ∧ (L 2 ≡ ¬S 2 ) ∧ (S 1 ≡ (L 1 ≡ L 2 )) ∧ (S 2 ≡ L 1 )
- L 1 ≡ S 1 P
- (^) L 2 ≡ ¬S 2 P
- S 1 ≡ (L 1 ≡ L 2 ) P
- S 2 ≡ L 1 P
- (^) ¬L 2 P for IP
- (^) (L 2 → ¬S 2 ) ∧ (¬S 2 → L 2 ) 2, T, E
- (^) ¬S 2 → L 2 6, Simp.
- (^) S 2 5, 7, MT
- (^) (S 2 → L 1 ) ∧ (L 1 → S 2 ) 4, T, E
- (^) S 2 → L 1 9, Simp.
- (^) L 1 8, 10, MP
- (^) (L 1 → S 1 ) ∧ (S 1 → L 1 ) 1, T, E
- (^) L 1 → S 1 12, Simp.
- (^) S 1 11, 13, MP
- (^) (S 1 → (L 1 ≡ L 2 )) ∧ ((L 1 ≡ L 2 ) → S 1 ) 3, T, E
- (^) S 1 → (L 1 ≡ L 2 ) 15, Simp.
- (^) L 1 ≡ L 2 14, 16, MP
- (^) (L 1 → L 2 ) ∧ (L 2 → L 1 ) 17, T, E
- (^) L 1 → L 2 18, Simp.
- (^) L 2 11, 19, MP
- (^) ⊥ 5, 20, Conj.
- L 2 5, 21, IP The prisoner should pick the second room.
The Eighth Trial “There are no signs above the doors!” exclaimed the prisoner. “Quite true,” said the king. “The signs were just made, and I haven't had time to put them up yet.” “Then how do you expect me to choose?” demanded the prisoner. “Well, here are the signs,” replied the king. THIS ROOM CONTAINS A TIGER BOTH ROOMS CONTAIN TIGERS “That's all well and good,” said the prisoner anxiously, “but which sign goes on which door?” The king thought a while. “I needn't tell you,” he said. “You can solve this problem without that information.” “Only remember, of course,” he added, “that a lady in the lefthand room means the sign which should be on that door is true and a tiger in it means the sign should be false, and that the reverse is true for the righthand room.” What is the solution? (L 1 ≡ S 1 ) ∧ (L 2 ≡ ¬S 2 ) ∧ ((S 1 ≡ ¬L 1 ) ∨ (S 1 ≡ (¬L 1 ∧ ¬L 2 ))) ∧ ((S 1 ≡ ¬L 1 ) ≡ (S 2 ≡ (¬L 1 ∧ ¬L 2 ))) ∧ ((S 1 ≡ (¬L 1 ∧ ¬L 2 )) ≡ (S 2 ≡ ¬L 2 ))
- L 1 ≡ S 1 P
- (^) L 2 ≡ ¬S 2 P
- (^) (S 1 ≡ ¬L 1 ) ∨ (S 1 ≡ (¬L 1 ∧ ¬L 2 )) P
- (^) (S 1 ≡ ¬L 1 ) ≡ (S 2 ≡ (¬L 1 ∧ ¬L 2 )) P
- (^) (S 1 ≡ (¬L 1 ∧ ¬L 2 )) ≡ (S 2 ≡ ¬L 2 ) P
- (^) (L 1 → S 1 ) ∧ (S 1 → L 1 ) 1, T, E
- (^) L 1 → S 1 6, Simp.
- (^) S 1 → L 1 6, Simp.
- (^) S 1 ≡ ¬L 1 P for IP
- (^) (S 1 → ¬L 1 ) ∧ (¬L 1 → S 1 ) 9, T, E
- (^) S 1 → ¬L 1 10, Simp.
- (^) ¬L 1 → S 1 10, Simp.
- (^) ¬S 1 P for IP
- (^) L 1 12, 13, MT
- (^) ¬L 1 7, 13, MT
- (^) ⊥ 14, 15, Conj.
- (^) S 1 13, 16, IP
- ¬L 1 11, 17, MP
- (^) L 1 8, 17, MP
- (^) ⊥ 18, 19, Conj.
- (^) ¬(S 1 ≡ ¬L 1 ) 9, 20, IP
- (^) S 1 ≡ (¬L 1 ∧ ¬L 2 ) 3, 21, DS
- (^) (S 1 → (¬L 1 ∧ ¬L 2 )) ∧ ((¬L 1 ∧ ¬L 2 ) → S 1 ) 22, T, E
- S 1 → (¬L 1 ∧ ¬L 2 ) 23, Simp.
- (^) (¬L 1 ∧ ¬L 2 ) → S 1 23, Simp.
- (^) ((S 1 ≡ (¬L 1 ∧ ¬L 2 )) → (S 2 ≡ ¬L 2 )) ∧ ((S 2 ≡ ¬L 2 ) → (S 1 ≡ (¬L 1 ∧ ¬L 2 ))) 5, T, E
- (^) (S 1 ≡ (¬L 1 ∧ ¬L 2 )) → (S 2 ≡ ¬L 2 ) 26, Simp.
- S 2 ≡ ¬L 2 22, 27, MP
- (^) ¬L 2 P for IP
- (^) (S 2 → ¬L 2 ) ∧ (¬L 2 → S 2 ) 28, T, E
- (^) ¬L 2 → S 2 30, Simp.
- S 2 29, 31, MP
- (^) L 1 P for IP
- (^) S 1 7, 33, MP
- (^) ¬L 1 ∧ ¬L 2 24, 34, MP
- ¬L 1 35, Simp.
- (^) ⊥ 33, 36, Conj.
- (^) ¬L 1 33, 37, IP
- (^) ¬L 1 ∧ ¬L 2 29, 38, Conj.
- (^) S 1 25, 39, MP
- (^) L 1 8, 40, MP
- (^) ⊥ 38, 41, Conj.
- L 2 29, 42, IP The prisoner should pick the second room.