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6.825 Exercise Solutions: Week 3

Exercise Solutions: Week 3 SolutionsSeptember 27, 2004 Converting to CNFC onvert the following sentences to conjunctive normal (A B) CAnswer: ( A B) C(A B) C(A C) ( B C) (B C)Answer: A B C3. (A B) (B A)Answer:( A B) ( B A)T rue4. ( P (P Q))Answer: P ( P Q)P P QT rue5. (P (Q R)) (P (R Q))Answer: ( P Q R) ( P R Q)(P Q R) ( P R Q)(P P R Q) (Q P R Q) ( R P R Q) P Q R6. (P Q) ((Q R) (P R))Answer: ( P Q) ( ( Q R) ( P R))(P Q) ((Q R) ( P R))(P Q) ((Q P R) ( R P R))(P Q) Q P R(P P Q R) ( P Q Q R)T rue1 First Order Logic SentencesFor each of the following English sentences, write a corresponding sentence in The only good extraterrestrial is a drunk extraterrestrial. (x) Good(x) Drunk(x)2. The Barber of Seville shaves all men who do not shave themselves. x. Shaves(x, x) Shaves(BarberOf Seville, x)3. There are at least two mountains in England.

There is no interpretation that makes the first sentence true and the second sentence false. Reason: For the second sentence to be false, ∀x.f(x) has to be true, and g(A) has to be false. With these two requirements, we can see that the first sentence cannot be true because f(x) is true for ∀x, and g(A) is false. 2

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