CMSC 250 0201 0202 Homework 2 Fall 2005 Solutions You must write the solutions to the problems single sided on your own lined paper with all sheets stapled together and with all answers written in sequential order or you will lose points 1 Give the negations of the following statements using DeMorgan s Law a Fords and Chevys are good cars Either Fords are not good cars OR Chevys are not good cars b Either Sue won t go to the wedding or Fred will Sue will go to the wedding AND Fred won t go to the wedding c Neither the Sun nor the Moon are heavenly bodies Either the Sun is a heavenly body OR the Moon is a heavenly body 2 For each of the following statements give its inverse converse and contrapositive a If it rains in Spain then it is May Inverse If it doesn t rain in Spain then it is not May Converse If it is May then it rains in Spain Contrapositive If it isn t May then it doesn t rain in Spain b Good times will follow if you go to a party Inverse If you do not go to a party then good times will not follow Converse If good times followed then you went to a party Contrapositive If good times do not follow then you didn t go to a party c If trees are blooming then it s spring Inverse If trees are not blooming then it isn t spring Converse If it s spring then trees are blooming Contrapositive If it isn t spring then trees are not blooming d You will pass only if you work hard Inverse You will not pass only if you do not work hard Converse If you passed then you worked hard Contrapositive If you did not pass then you did not work hard 1 3 Write the complete truth table for each of the following a p q r p p q r r p q 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 r p 0 1 0 1 0 0 0 0 b a b c a b c c 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 a b c 0 1 0 0 1 0 1 1 b c 0 1 0 0 0 1 0 0 p q r p 1 1 0 1 1 1 1 1 4 For each of the following state the one single rule from Theorem 1 1 1 page 14 or Table 1 3 1 Page 39 that could be used to go directly from the statement s to the conclusion or none if there is no such rule a statement p q p conclusion q None b statements p q p conclusion q Modus ponens c statements a b a c conclusion b c None item statements a b a c conclusion c b None 2 5 Use a truth table to determine if the following argument is valid or not State whether or not it is valid and give your reasons why it is valid or not P1 P2 P3 Therefore p 1 1 1 1 0 0 0 0 q 1 1 0 0 1 1 0 0 r 1 0 1 0 1 0 1 0 p q r q p r p Premise p q 1 1 1 1 1 1 0 0 Premise r q 1 1 0 1 1 1 0 1 p 0 0 0 0 1 1 1 1 r 0 1 0 1 0 1 0 1 Premise p r 1 1 1 1 0 1 0 1 Conclusion p 0 0 0 0 1 1 1 1 Critical Critical Critical Critical Since the premises are all true in the first row and the conclusion is false in the first row the argument is invalid 6 Use the rules of inference you were given to prove the following P1 P2 P3 P4 a P5 P6 Therefore Number 1 2 3 4 5 6 r s y s y w p r w s y w p x Statement s r r p p p x Reason Dilemma proof by division into cases Disjunctive syllogism Double negation law Modus tollens Double negation law Disjunctive addition 3 Lines Used P2 P5 and P6 P1 and 1 2 P4 and 3 4 5 P1 P2 b P3 P4 Therefore Number 1 2 3 4 5 6 7 8 9 10 11 P1 P2 P3 Therefore Number 1 2 3 4 c 5 6 7 8 9 a b m a b g d a m e f f d h e h Statement a b a m a m e f e e e h a b a b g d d h d h e h Reason Distributive law Conjunctive Simplification Modus ponens Modus Tolens Double negative law Disjunctive addition Conjunctive Simplification Disjunctive addition Modus ponens Disjunctive addition Conjunctive addition Lines Used P1 1 P3 and 2 P4 and 3 4 5 1 7 P2 and 8 9 6 and 10 x y y q r q g g x r Statement Reason Lines Used g Assume r q Disj Syll P3 and 1 r Conj Simp 2 q Conj Simp 2 y Disj Syll P2 and 4 x Modus Tollens P1 and 5 x r Conj Addition 3 and 6 x r De Morgan s Law 7 g x r Close Condit Wld 1 8 4
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