<p>PHIL 012 - Study Sheet for Exam 2 Part I</p><p>1. Which of the following statements are logically equivalent to this sentence: Home(Max)  (Home(Claire)  Happy(Carl))</p><p>I. (Home(Max)  Home(Claire))  Happy(Carl) II. (Home(Max)  (Home(Claire)  Happy(Carl) III. (Home(Max)  Home(Claire))  (Home(Max)  Happy(Carl)) a. I b. I & II only c. II & III only d. 1, II, and III e. None of the above</p><p>2. Which of the following statements are logically equivalent to this sentence: Home(Max)  Home(Claire)  Home(Max)</p><p>I. Home(Max)  Home(Claire) II. Home(Max) ^ Home(Claire) ^ (Happy(Carl)  Happy(Carl)) III. Home(Max)  Home(Claire) a. I b. I & II only c. II & III only d. I, II, and III e. None of the above.</p><p>3. Which of the following is a negation normal form of the following sentence: ((A  B)  C a. (A  B)  C b. (A  B)  C c. (A  B)  C d. All of the above e. None of the above</p><p>4. Which of the following is a negation normal form of the following sentence: (A  B)  C  ((B  A)  B) a. (A  B)  C  (B  A) b. (A  B)  C c. (A  B)  C  (B  A d. All of the above e. None of the above Translate the following sentences into FOL: Note that being unhappy is the opposite of being happy.</p><p>5. Either a is not large or it is in back of b.</p><p>6. Neither e nor a is to the right of c and to the left of b.</p><p>7. Either Carl is happy and Max is unhappy or it is not the case that either Claire is home and Carl is home or Max is happy.</p><p>8. It is not the case that Claire is happy, but Max is at home and Carl is unhappy.</p><p>Choose the best statements to describe the following sentences: Assume that all sentences like A, B, and C are atomic sentences. All sentences such as Cube(a) follow the rules of Tarski's world.</p><p>9. (A  B)  C a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.</p><p>10. (Cube(a)  Cube(b))  Cube(c) a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.</p><p>11. a = a a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true. 12. Larger(a, b)  Larger(b, c)  Larger(c, a) a. The sentence is unsatisfiable. b. The sentence is satisfiable. c. The sentence is a tautology. d. The sentence is logically true. e. The sentence is tautological and logically true.</p><p>For the following questions, use these rules: Conjunction Elimination ( Elim) Disjunction Elimination ( Elim) Conjunction Introduction ( Intro) Disjunction Introduction ( Intro) Negation Elimination ( Elim) Negation Introduction ( Intro) Distribution of  over  Distribution of  over  DeMorgan's Theorems (DeM) Reiteration (Reit)</p><p>Consider the following proof:</p><p>1. (P  R) 2. (P  R) 3. P 4.P   R 5.(P  R)  (P  R) 6. P 7. R 8. P  R 9. (P  R)  (P  R) 10.R 11.P  R 12.(P  R) 13.(P  R)  (P  R) 14. P  R</p><p>13. Lines 3 and 7 are: a. Invalid b. the assumptions for a  Elim c. the assumptions for a  Intro d. the assumptions for two  Intro's e. None of the above.</p><p>14. The rule for line 8 is a. Invalid b.  Elim, 2 c. DeM, 1 d.  Elim, 7 e. None of the above. 15. Line 12 is a. Invalid b. Valid c. Unnecessary d.  Intro e. None of the above.</p><p>15. This is a proof of a. DeMorgan's Theorem b. Distribution of  over  c.  Intro d. Nothing because it is invalid e. None of the above</p>