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 lets 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 ) DeMorgans 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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Homework 2 Solutions W11 - EECS 203: Homework 2 Solutions...

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