CMSC250 Spring 2004 Homework 2 Answers Due Wednesday February 11 at the beginning of your discussion section 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 For each of the following statements give its converse inverse and contrapositive in English sentences be sure to label the three parts of each answer You may change verb tenses to make your answers sound better a If one thinks one must reach conclusions 1 Answer Converse If one must reach conclusions then one thinks Inverse If one does not think then one must not reach conclusions Contrapositive If one must not reach conclusions then one does not think b If you look good and dress well you don t need a purpose in life 2 Answer Converse If you don t need a purpose in life then you look good and dress well Inverse If you do not look good or do not dress well then you need a purpose in life Contrapositive If you need a purpose in life then you do not look good or do not dress well c California is a fine place to live if you happen to be an orange 3 Answer Converse If California is a fine place to live then you happen to be an orange Inverse If you don t happen to be an orange then California is not a fine place to live Contrapositive If California is not a fine place to live then you don t happen to be an orange d You can be free only if I am free 4 Answer Converse If I am free then you can be free Inverse If you cannot be free then I am not free Contrapositive If I am not free then you cannot be free 2 Construct a complete truth table to help you determine if the following argument is valid or not State whether it is valid or not indicate the entries in the truth table that led you to your answer and explain why those entries support your answer p q r q p p r 1 Helen Keller 1880 1968 Robert Pante 3 Fred Allen 1894 1956 4 Clarence Darrow 1857 1938 2 1 Answer 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 0 0 0 0 1 1 1 1 q r 1 1 1 0 1 1 1 0 Premise p q r 1 1 1 0 1 1 1 1 Premise q p 0 0 1 1 1 1 1 1 Conclusion p r 1 0 1 0 1 1 1 1 Critical row Critical Critical Critical Critical row row row row The critical rows are the rows where all the premises are true Since in all five critical rows the conclusion is also true this argument is valid 3 Use the rules of inference you were given to complete the two proofs below Use the same format as was shown in class for these proofs each line of your proof must be justfied with the rule and line numbers you used to obtain that line a P1 x w P2 x y s z P3 z s w Answer Line 1 2 3 4 5 6 7 8 9 Statement x y s z x y s z x y s z x y s s x y x w s w Rule Definition of Definition of Associativity Disjunctive syllogism Assume Disjunctive syllogism Conjunctive simplification Disjunctive syllogism Closing conditional world Here s another way 2 Lines Used P2 1 2 3 P3 4 5 6 P1 7 5 8 Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Statement x y s z s w w x x y x y x y s z s z z z s w s w s w s w Rule Definition of Assume Conjunctive simplification Disjunctive syllogism Disjunctive addition Definition of Double negation Disjunctive syllogism Conjunctive simplification Modus ponens Conjunctive addition Closing cond world w contra DeMorgan s law Double negative Definition of Lines Used P2 2 P1 3 4 5 6 1 7 2 8 9 10 P3 2 11 12 13 14 And yet another way Line 1 2 3 4 5 6 7 8 9 10 11 b Statement s x y s z z z z x y x y x x w s w Rule Assume Assume Modus ponens Modus ponens Conjunctive addition Closing cond world w contra Definition of Conjunctive simplification Double negation Disjunctive syllogism Closing conditional world Lines Used 2 P2 3 1 P3 4 2 5 6 7 8 P1 9 1 10 P1 a b c a P2 a d f P3 c f a b Answer Line 1 2 3 4 5 6 7 8 9 10 Statement a b a c a b c a b c a d f f a f a c b Rule Commutativity Distributive law Conjunctive simplification Conjunctive simplification Disjunctive addition Modus ponens Conjunctive addition Double negative DeMorgan s law Modus tollens Disjunctive syllogism 3 Lines Used P1 1 2 2 3 5 P2 6 3 7 P3 8 9 4 Here s another way using proof by contradiction Line 1 2 3 4 5 6 7 8 9 10 11 12 13 Statement b a b a b c a a a d f c f a a f f f b Rule Assume Disjunctive addition DeMorgan s and Double neg Disjunctive syllogism Conjunctive simplification Disjunctive addition Modus ponens Conjunctive simplification Modus ponens Double negation Disjunctive syllogism Conjunctive addition Closing cond world w contra Lines Used 1 2 3 P1 4 5 P2 6 4 P3 8 4 9 10 7 11 1 12 4 Indiana Jones the famous archeologist is off on another adventure Indy knows that in all his adventures he always has three tasks to accomplish he must get the treasure save the girl and defeat the bad guy However being a college professor Dr Jones is also a very logical person In fact he has developed a set of rules to determine which of his three tasks he should complete first The rules are P1 If Indy is in Europe or South America he gets the treasure first P2 If Indy is in Asia or Africa he saves the girl or defeats the bad guy first P3 Indy was almost squashed by a rolling boulder if and only if he is in South America or Africa P4 Indy is in neither Europe nor South America if he falls into a pit of snakes P5 Indy never defeats the bad guy first if he is in Africa Indy can t remember which continent he is currently on but he does remember that earlier in this adventure he fell into a pit of snakes and was almost squashed by a rolling boulder Help him figure out which task he should do first You may use the …
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