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Unformatted text preview: EECS 203: DISCRETE MATHEMATICS Homework 3 Solutions 1. Chapter 1.5, Problem 24 From the statement “ P ( c ) ∨ Q ( c )” one cannot conclude that “ P ( c )” is true. (The simplification inference says that given “ P ( c ) ∧ Q ( c )” one can conlclude that “ P ( c )” is true.) 2. Chapter 1.5, Problem 28 1. ∀ x ( P ( x ) ∨ Q ( x )) premise 2. ∀ x (( ¬ P ( x ) ∧ Q ( x )) → R ( x )) premise 3. ∀ x ( P ( x ) ∨ ¬ Q ( x ) ∨ R ( x )) DeMorgan, defn. of → (line 2) 4. P ( c ) ∨ Q ( c ) universal instantiation (line 1) 5. ( P ( c ) ∨ R ( c )) ∨ ¬ Q ( c ) universal instantiation (line 3) 6. P ( c ) ∨ R ( c ) resolution, idempotency of ∨ (lines 4,5) 7. ¬ R ( c ) → P ( c ) defn. of → (line 6) 8. ∀ x ( R ( x ) → P ( x )) universal generalization (lines 4,5,7) 3. Kevin Bacon is an actor. Kevin Bacon has a Bacon number of 0 and is the only person with this Bacon number. Anyone who has appeared in the same movie as another person with a Bacon number has a Bacon number. A person has Bacon number i +1 if and only if he or she has appeared in a movie with someone whose Bacon number is i , but never with a person whose Bacon number is less than i . Use set notation to define B i : the set of people with bacon number i . (Use the predicate A ( p,m ), “person p has appeared in movie m ”.) Start with B and B 1 ....
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This note was uploaded on 12/20/2010 for the course EECS 203 taught by Professor Yaoyunshi during the Fall '07 term at University of Michigan.
- Fall '07