26+Review+session+solutions - Handout#26 CS103 Robert...

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Handout #26 CS103 April 19, 2011 Robert Plummer CS103 Review Session Solutions Logic Give a formal proof for the following. Use numbered steps and refer to those numbers in the justification you give for each step. 1. ¬ x (R(x) S(x)) 2. y (S(y) M(y) L(y)) 3. x ¬(R(x) S(x)) De Morgan's, 1 4. Let c be an arbitrary object 5. ¬(R(c) S(c) Universal instantiation, 3 6. ¬R(c) ¬S(c) De Morgan's, 5 7. S(c) M(c) L(c) Universal instantiation, 2 8. ¬R(c) M(c) L(c) Resolution, 6, 7 9. ¬R(c) (L(c) M(c)) Associativity, Commutativity, 8 10. R(c) (L(c) _M(c)) Equivalence in table, 9 11. x (R(x) (L(x) M(x))) Universal generalization, 4—10
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2 Translations Translate each of the following sentences into first-order logic. The questions describe events occurring at a Halloween party. The domain of discourse includes exactly the people who attended the party. You may only use the following predicates: M(x) x wore a mask to the party. S(x, y) x scared y at the party. F(x, y) x and y are friends Notes: F should be considered to be symmetric, i.e. F(x, y) F(y, x). It is possible to scare yourself. It is possible to be friends with yourself. 1) Nobody who didn’t wear a mask scared a friend. x y((S(x, y) F(x, y)) M(x))  x y( M(x) F(x, y) S(x, y)) 2) A person who wore a mask and who had no friends scared everybody else at the party. x( M(x)  yF(x, y) z(z x S(x, z) ) 3) No two people wearing masks scared each other. x y( (M(x) M(y) y x) (S(x, y) S(y, x))  x y( M(x) M(y) y x S(x, y) S(x, y)) Common Mistake: For the first translation, writing ( S(x, y) S(y, x)). The original sentence allows that a person x wearing a mask scared another person y in a mask, as long as y did not also scare x. But this translation would not allow for that possibility. 4) Everybody with at least two friends scared somebody.
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26+Review+session+solutions - Handout#26 CS103 Robert...

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