CMSC250, Spring 2004 Homework 2 AnswersDue 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 oryou will lose points.1. For each of the following statements, give its converse, inverse, and contrapositive in Englishsentences; be sure to label the three parts of each answer. You may change verb tenses tomake your answers sound better.(a) “If one thinks, one must reach conclusions.”1Answer:• 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.”2Answer:• Converse: If you don’t need a purpose in life, then you look good and dress well.• Inverse: If you do not look go od or do not dress well, then you need a purpose inlife.• Contrapositive: If you need a purpose in life, then you do not look good or do notdress well.(c) “California is a fine place to live if you happen to be an orange.”3Answer:• 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 tolive.• Contrapositive: If California is not a fine place to live, then you don’t happen tobe an orange.(d) “You can be free only if I am free.”4Answer:• 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 ornot. State whether it is valid or not, indicate the entries in the truth table that led you toyour answer, and explain why those entries support your answer.p → (q ∨ r)q →∼ p∴ p → r1Helen Keller (1880–1968)2Rob ert Pante3Fred Allen (1894–1956)4Clarence Darrow (1857–1938)1Answer:Premise Premise Conclusionp q r ∼ p q ∨ r p → (q ∨ r) q →∼ p p → r1 1 1 0 1 1 0 11 1 0 0 1 1 0 01 0 1 0 1 1 1 1 ← Critical row1 0 0 0 0 0 1 00 1 1 1 1 1 1 1 ← Critical row0 1 0 1 1 1 1 1 ← Critical row0 0 1 1 1 1 1 1 ← Critical row0 0 0 1 0 1 1 1 ← Critical rowThe critical rows are the rows where all the premises are true. Since in all five critical rowsthe 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 sameformat as was shown in class for these proofs — each line of your proof must be justfied withthe rule and line numbers you used to obtain that line.(a)P1 ∼ x ∨ wP2 (x → y) → (s → z)P3 ∼ z∴ s → wAnswer:Line Statement Rule Lines Used1 ∼ (x → y) ∨ (s → z) Definition of → P22 (x∧ ∼ y) ∨ (∼ s ∨ z) Definition of → 13 (x∧ ∼ y)∨ ∼ s ∨ z Associativity 24 (x∧ ∼ y)∨ ∼ s Disjunctive syllogism 3, P35 s Assume —6 x∧ ∼ y Disjunctive syllogism 4,57 x Conjunctive simplification 68 w Disjunctive syllogism P1, 79 s → w Closing conditional world 5–8Here’s another way.2Line Statement Rule Lines Used1 ∼ (x → y) ∨ (s → z) Definition of → P22 s∧ ∼ w Assume —3 ∼ w Conjunctive simplification 24 ∼ x Disjunctive syllogism P1, 35 ∼ x ∨ y Disjunctive addition 46 x → y Definition of → 57 ∼∼ (x → y) Double negation 68 s → z Disjunctive syllogism 1, 79 s Conjunctive simplification 210 z Modus ponens 8, 911 z∧ ∼ z Conjunctive addition 10, P312 ∼ (s∧ ∼ w) Closing cond world w/ contra 2–1113 ∼ s∨ ∼∼ w DeMorgan’s law 1214 ∼ s ∨ w Double negative 1315 s → w Definition of → 14And yet another way.Line Statement Rule Lines Used1 s Assume —2 x → y Assume —3 s → z Modus ponens 2, P24 z Modus ponens 3, 15 z ∧ ∼ z Conjunctive addition P3, 46 ∼ (x → y) Closing cond world w/ contra 2–57 x∧ ∼ y Definition of → 68 x Conjunctive simplification 79 ∼∼ x Double negation 810 w Disjunctive syllogism P1, 911 s → w Closing c onditional world 1–10(b)P1 (a∧ ∼ b) ∨ (c ∧ a)P2 (a ∨ d) →∼ fP3 c → (f∨ ∼ a)∴ ∼ bAnswer:Line Statement Rule Lines Used1 (a∧ ∼ b) ∨ (a ∧ c) Commutativity P12 a ∧ (∼ b ∨ c) Distributive law 13 a Conjunctive simplification 24 ∼ b ∨ c Conjunctive s implification 25 a ∨ d Disjunctive addition 36 ∼ f Modus ponens 5, P27 ∼ f ∧ a Conjunctive addition 6, 38 ∼ (f ∨ ∼ a) Double negative, DeMorgan’s law 79 ∼ c Modus tollens P3, 810 ∼ b Disjunctive syllogism 9, 43Here’s another way using proof by contradiction.Line Statement Rule Lines Used1 b Assume —2 ∼ a ∧ b Disjunctive addition 13 ∼ (a∨ ∼ b) DeMorgan’s and Double neg 24 c ∧ a Disjunctive syllogism 3, P15 a Conjunctive simplification 46 a ∨ d Disjunctive addition 57 ∼ f Modus ponens P2, 68 c Conjunctive simplification 49 f∨ ∼ a Modus ponens P3, 810 ∼∼ a Double negation 411 f Disjunctive syllogism 9, 1012 f∧ ∼ f Conjunctive addition 7, 1113 ∼ b Closing cond world w/ contra 1–124. Indiana Jones, the famous archeologist, is off on another adventure. Indy knows that in allhis adventures he always has three tasks to accomplish: he must get the treasure, save thegirl, and defeat the bad guy. However, being a college professor, Dr. Jones is also a verylogical person. In fact, he has developed a set of rules to determine which of his three taskshe 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 orAfrica.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 e arlierin 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 following propositions:e = “Indy is in Europe.” t = “Indy gets the treasure first.”f = “Indy is in Africa.” g = “Indy saves the girl first.”s = “Indy is in South America.” b = “Indy defeats the bad
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