Homework 2 Solutions W11

Homework 2 Solutions W11 - EECS 203 Homework 2 Solutions...

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Unformatted text preview: EECS 203: Homework 2 Solutions Section 1.5 1. (E) 4bd b) Disjunctive syllogism. d) Addition. 2. (M) 14bd b) Let r ( x ) = “ r is one of the five roommates listed”, d ( x ) = “ x has taken a course in discrete mathe- matics”, a ( x ) = “ x can take a course in algorithms”. 1. ∀ x ( r ( x ) → d ( x )) Premise 2. ∀ x ( d ( x ) → a ( x )) Premise 3. r ( y ) → d ( y ) Universal instantiation from (1) 4. d ( y ) → a ( y ) Universal instantiation from (2) 5. r ( y ) → a ( y ) Hypothetical syllogism from (3) and (4) 6. ∀ x ( r ( x ) → a ( x )) Universal generalization from (5) d) Let c ( x ) = “ x is in this class”, f ( x ) = “ x has been to France”, l ( x ) = “ x has visited the Louvre”. 1. ∃ x ( c ( x ) ∧ f ( x )) Premise. 2. ∀ x ( f ( x ) → l ( x )) Premise. 3. c ( y ) ∧ f ( y ) Existential instantiation from (1). 4. f ( y ) Simplification from (3). 5. c ( y ) Simplification from (3). 6. f ( y ) → l ( y ) Universal instantiation from (2). 7. l ( y ) Modus Ponens from (4) and (6). 8. c ( y ) ∧ l ( y ) Conjunction using (5) and (7). 9. ∃ x ( c ( x ) ∧ l ( x )) Existential generalization using (8) 3. (E) 24 Steps (3) and (5) are incorrect since simplification applies to conjunctions and not disjunctions. 4. (E) 28 We want to show that ∀ x ( ¬ R ( x ) → P ( x )) so let’s take an arbitrary a when instantiating 1. ∀ x ( P ( x ) ∨ Q ( x )) Premise. 2. ∀ x (( ¬ P ( x ) ∧ Q ( x )) → R ( x ) Premise. 3. P ( a ) ∨ Q ( a ) Universal instantiation from (1). 4. ( ¬ P ( a ) ∧ Q ( a )) → R ( a ) Universal instantiation from (2). 5. ¬ ( ¬ P ( a ) ∧ Q ( a )) ∨ R ( a ) Material Implication from(4). 6. P ( a ) ∨¬ Q ( a ) ∨ R ( a ) DeMorgan’s from (5). 7. R ( a ) ∨ P ( a ) Resolution from (3) and (6). 8. ¬ R ( a ) → P ( a ) Material Implication from (7). 9. ∀ x ( ¬ R ( x ) → P ( x )) Universal generalization using (8)....
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This note was uploaded on 02/12/2012 for the course EECS 203 taught by Professor Yaoyunshi during the Spring '07 term at University of Michigan.

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Homework 2 Solutions W11 - EECS 203 Homework 2 Solutions...

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