HW3-Solutions - EECS 203 DISCRETE MATHEMATICS Homework 3...

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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 0 and B 1 . Let M ( a, b ) be the predicate that persons a and b have appeared in a common movie: M ( a, b ) ≡ ∃ m ( A ( a, m ) A ( b, m )) From the definition of Bacon numbers we have: B 0 = { KevinBacon } B 1 = { a | M ( a, KevinBacon ) a = KevinBacon }
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