ĐĎॹá>ţ˙ ňôţ˙˙˙đń{˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙ěĽÁ@ řż…bjbjîFîF 2ÄŒ,Œ,†y-˙˙˙˙˙˙ˆžžžžžžž˛šĺšĺšĺ8ŇĺT&çô˛)ę&éĘđé(ęęęęęęüüţüţüţüţüţüţü$ReŞ"ýÁžęęęęę"ýžžęęă˙¤ř¤ř¤řęđžęžęüü¤řęüü¤ř¤řžž¤řęé °Ö[>Çšĺď¤řĐř,ů˙0)¤ř÷ô¤ř˛˛žžžžž¤ř,¤řęęę"ý"ý˛˛rśsäqř ˛˛śsUse the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE 1. 1 +đ 1 =đ 3 if and only if 2 +đ 2 =đ 3. Ans:  True 2. If it is raining, then it is raining. Ans:  True 3. If 1 <đ 0, then 3 =đ 4. Ans:  True 4. If 2 +đ 1 =đ 3, then 2 =đ 3 -đ 1. Ans:  True 5. If 1 +đ 1 =đ 2 or 1 +đ 1 =đ 3, then 2 +đ 2 =đ 3 and 2 +đ 2 =đ 4. Ans:  False 6. Write the truth table for the proposition Řđ(r Žđ Řđq) Úđ (p Ůđ Řđr). Ans: p q r Řđ(r Žđ Řđq) Úđ (p Ůđ Řđr)TTTTTTFTTFTFTFFTFTTTFTFFFFTFFFFF 7. (a) Find a proposition with the given truth table. pq?TTFTFFFTTFFF(b) Find a proposition using only p,đq,đŘđ, and the connective Úđ that has this truth table. Ans: (a) Řđp Ůđ q, (b) Řđ(p Úđ Řđq). 8. Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false otherwise Ans: (a) p Ůđ Řđq Ůđ r. 9. Find a proposition with three variables p, q, and r that is true when exactly one of the three variables is true, and false otherwise Ans: (p Ůđ Řđq Ůđ Řđr) Úđ (Řđp Ůđ q Ůđ Řđr) Úđ (Řđp Ůđ Řđq Ůđ r). 10. Find a proposition with three variables p, q, and r that is never true Ans: (p Ůđ Řđp) Úđ (q Ůđ Řđq) Úđ (r Ůđ Řđr). 11. Find a proposition using only p,đq,đŘđ and the connective Úđ with the given truth table.   ?TTFTFTFTTFFF  Ans: Řđ(Řđp Úđ q) Úđ Řđ(p Úđ Řđq). 12. Determine whether p Žđ (q Žđ r) and p Žđ (q Ůđ r) are equivalent. Ans: Not equivalent. Let q be false and p and r be true. 13. Determine whether p Žđ (q Žđ r) is equivalent to (p Žđ q) Žđ r. Ans: Not equivalent. Let p, q, and r be false. 14. Determine whether (p Žđ q) Ůđ (Řđp Žđ q) şđ q. Ans: Both truth tables are identical: p q (p Žđ q) Ůđ (Řđp Žđ q) qTTTTTFFFFTTTFFFF 15. Write a proposition equivalent to p Úđ Řđq that uses only p,đq,đŘđ and the connective Ůđ. Ans: Řđ(Řđp Ůđ q). 16. Write a proposition equivalent to Řđp Ůđ Řđq using only p,đq,đŘđ and the connective Úđ. Ans: Řđ(p Úđ q). 17. Prove that the proposition  if it is not hot, then it is hot is equivalent to  it is hot . Ans: Both propositions are true when  it is hot is true and both are false when  it is hot is false. 18. Write a proposition equivalent to p Žđ q using only p,đq,đŘđ and the connective: Úđ. Ans: Řđp Úđ q. 19. Write a proposition equivalent to p Žđ q using only p,đq,đŘđ and the connective Ůđ. Ans: Řđ(p Ůđ Řđq). 20. Prove that p Žđ q and its converse are not logically equivalent. Ans: Truth values differ when p is true and q is false. 21. Prove that Řđp Žđ Řđq and its inverse are not logically equivalent. Ans: Truth values differ when p is false and q is true. 22. Determine whether the following two propositions are logically equivalent: p Úđ (q Ůđ r),đ(p Ůđ q) Úđ (p Ůđ r). Ans: No. 23. Determine whether the following two propositions are logically equivalent: p Žđ (Řđq Ůđ r),đŘđp Úđ Řđ(r Žđ q). Ans: Yes. 24. Prove that (q Ůđ (p Žđ Řđq)) Žđ Řđp is a tautology using propositional equivalence and the laws of logic. Ans: EMBED Equation.BREE4 , which is always true. 25. Determine whether this proposition is a tautology: ((p Žđ q) Ůđ Řđp) Žđ Řđq. Ans: No. 26. Determine whether this proposition is a tautology: ((p Žđ Řđq) Ůđ q) Žđ Řđp. Ans: Yes. Use the following to answer questions 27-33: In the questions below write the statement in the form “If …, then ….” 27. x is even only if y is odd. Ans: If x is even, then y is odd. 28. A implies B. Ans: If A, then B. 29. It is hot whenever it is sunny. Ans: If it is sunny, then it is hot. 30. To get a good grade it is necessary that you study. Ans: If you don't study, then you don't get a good grade (equivalently, if you get a good grade, then you study). 31. Studying is sufficient for passing. Ans: If you study, then you pass. 32. The team wins if the quarterback can pass. Ans: If the quarterback can pass, then the team wins. 33. You need to be registered in order to check out library books. Ans: If you are not registered, then you cannot check out library books (equivalently, if you check out library books, then you are registered). 34. Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win. Ans: Contrapositive: If you will not win, then you do not try hard. Converse: If you will win, then you try hard. Inverse: If you do not try hard, then you will not win. 35. Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday. Ans: Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If you sleep late, then it is Saturday. Inverse: If it is not Saturday, then you do not sleep late. Use the following to answer questions 36-38: In the questions below write the negation of the statement. (Don't write “It is not true that ….”) 36. It is Thursday and it is cold. Ans: It is not Thursday or it is not cold. 37. I will go to the play or read a book, but not both. Ans: I will go to the play and read a book, or I will not go to the play and not read a book. 38. If it is rainy, then we go to the movies. Ans: It is rainy and we do not go to the movies. 39. Explain why the negation of “Al and Bill are absent” is not “Al and Bill are present”. Ans: Both propositions can be false at the same time. For example, Al could be present and Bill absent. 40. Using c for  it is cold and d for  it is dry , write  It is neither cold nor dry in symbols. Ans: Řđc Ůđ Řđd. 41. Using c for  it is cold and r for  it is rainy , write  It is rainy if it is not cold in symbols. Ans: Řđc Žđ r. 42. Using c for  it is cold and w for  it is windy , write  To be windy it is necessary that it be cold in symbols. Ans: w Žđ c. 43. Using c for  it is cold , r for  it is rainy , and w for  it is windy , write  It is rainy only if it is windy and cold in symbols. Ans: r Žđ (w Ůđ c). 44. A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent? The system is in multiuser state if and only if it is operating normally. If the system is operating normally, the kernel is functioning. The kernel is not functioning or the system is in interrupt mode. If the system is not in multiuser state, then it is in interrupt mode. The system is in interrupt mode. Ans: Using m, n, k, and i, there are three rows of the truth table that have all five propositions true: the rows TTTT, FFTT, FFFT for m,đn,đk,đi. 45. On the island of knights and knaves you encounter two people, A and B. Person A says, "B is a knave." Person B says, "We are both knights." Determine whether each person is a knight or a knave. Ans: A is a knight, B is a knave. 46. On the island of knights and knaves you encounter two people. A and B. Person A says, "B is a knave." Person B says, "At least one of us is a knight." Determine whether each person is a knight or a knave. Ans: A is a knave, B is a knight. Use the following to answer questions 47-49: In the questions below suppose that Q(x) is “x +đ 1 =đ 2x , where x is a real number. Find the truth value of the statement. 47. Q(2). Ans:  False 48. "đxQ(x). Ans:  False 49. $đxQ(x). Ans:  True Use the following to answer questions 50-57: In the questions below P(x,đy) means  x +đ 2y =đ xy , where x and y are integers. Determine the truth value of the statement. 50. P(1,đ-đ1). Ans:  True 51. P(0,đ0). Ans:  True 52. $đyP(3,đy). Ans:  True 53. "đx$đyP(x,đy). Ans:  False 54. $đx"đyP(x,đy). Ans:  False 55. "đy$đxP(x,đy). Ans:  False 56. $đy"đxP(x,đy). Ans:  False 57. Řđ"đx$đyŘđP(x,đy). Ans:  False Use the following to answer questions 58-59: In the questions below P(x,đy) means  x and y are real numbers such that x +đ 2y =đ 5 . Determine whether the statement is true. 58. "đx$đyP(x,đy). Ans: True, since for every real number x we can find a real number y such that x +đ 2y =đ 5, namely y =đ (5 -đ x)/đ2. 59. $đx"đyP(x,đy). Ans: False, if it were true for some number x0, then x0 = 5 -2y for every y, which is not possible. Use the following to answer questions 60-62: In the questions below P(m,đn) means  m d" n , where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? 60. "đnP(0,đn). Ans:  True 61. $đn"đmP(m,đn). Ans:  False 62. "đm$đnP(m,đn). Ans:  True Use the following to answer questions 63-68: In the questions below suppose P(x,y) is a predicate and the universe for the variables x and y is {1,2,3}. Suppose P(1,3), P(2,1), P(2,2), P(2,3), P(2,3), P(3,1), P(3,2) are true, and P(x,y) is false otherwise. Determine whether the following statements are true. 63. "đx$đyP(x,đy). Ans:  True 64. $đx"đyP(x,đy). Ans:  True 65. Řđ$đx$đy(P(x,đy) Ůđ ŘđP(y,x)). Ans:  False 66. "đy$đx(P(x,đy) ’! P(y,x)). Ans:  True 67. "đx"đy (x `" y ’! (P(x,y) Úđ đP(y,x)). Ans:  False 68. "đy$đx (x d" y Ůđ (P(x,y)). Ans:  False Use the following to answer questions 69-72: In the questions below suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(x,đy): student x is taking course y. Write the statement using these predicates and any needed quantifiers. 69. Eric is taking MTH 281. Ans: T(Eric, MTH 281). 70. All students are freshmen. Ans: "đxF(x). 71. Every freshman is a full-time student. Ans: "đx(F(x) Žđ B(x)). 72. No math course is upper-level. Ans: "đy(M(y) Žđ ŘđU(y)). Use the following to answer questions 73-75: In the questions below suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman A(x): x is a part-time student T(x,đy): student x is taking course y. Write the statement using these predicates and any needed quantifiers. 73. Every student is taking at least one course. Ans: "đx$đy T(x,đy). 74. There is a part-time student who is not taking any math course. Ans: $đx"đy[A(x) Ůđ (M(y) Žđ ŘđT(x,đy))]. 75. Every part-time freshman is taking some upper-level course. Ans: "đx$đy[(F(x) Ůđ A(x)) Žđ (U(y) Ůđ T(x,đy))]. Use the following to answer questions 76-78: In the questions below suppose the variable x represents students and y represents courses, and: F(x): x is a freshman A(x): x is a part-time student T(x,đy): x is taking y. Write the statement in good English without using variables in your answers. 76. F(Mikko). Ans: Mikko is a freshman. 77. Řđ$đyT(Joe,đy). Ans: Joe is not taking any course. 78. $đx(A(x) Ůđ ŘđF(x)). Ans: Some part-time students are not freshmen. Use the following to answer questions 79-81: In the questions below suppose the variable x represents students and y represents courses, and: M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(x,đy): x is taking y. Write the statement in good English without using variables in your answers. 79. "đx$đyT(x,đy). Ans: Every student is taking a course. 80. $đx"đyT(x,đy). Ans: Some student is taking every course. 81. "đx$đy[(B(x) Ůđ F(x)) Žđ (M(y) Ůđ T(x,đy))]. Ans: Every full-time freshman is taking a math course. Use the following to answer questions 82-84: In the questions below suppose the variables x and y represent real numbers, and L(x,đy) :đ x <đ y G(x) :đ x >đ 0 P(x) :đ x is a prime number. Write the statement in good English without using any variables in your answer. 82. L(7,đ3). Ans: 7 <đ 3. 83. "đx$đyL(x,đy). Ans: There is no largest number. 84. "đx$đy[G(x) Žđ (P(y) Ůđ L(x,đy))]. Ans: No matter what positive number is chosen, there is a larger prime. Use the following to answer questions 85-87: In the questions below suppose the variables x and y represent real numbers, and L(x,đy) :đ x <đ y Q(x,đy) :đ x =đ y E(x) :đ x is even I(x) :đ x is an integer. Write the statement using these predicates and any needed quantifiers. 85. Every integer is even. Ans: "đx(I(x) Žđ E(x)). 86. If x <đ y, then x is not equal to y. Ans: "đx"đy(L(x,đy) Žđ ŘđQ(x,đy)). 87. There is no largest real number. Ans: "đx$đyL(x,đy). Use the following to answer questions 88-89: In the questions below suppose the variables x and y represent real numbers, and E(x) :đ x is even G(x) :đ x >đ 0 I(x) :đ x is an integer. Write the statement using these predicates and any needed quantifiers. 88. Some real numbers are not positive. Ans: $đxŘđG(x). 89. No even integers are odd. Ans: Řđ$đx(I(x) Ůđ E(x) Ůđ ŘđE(x)]). Use the following to answer questions 90-92: In the questions below suppose the variable x represents people, and F(x): x is friendly T(x): x is tall A(x): x is angry. Write the statement using these predicates and any needed quantifiers. 90. Some people are not angry. Ans: $đxŘđA(x). 91. All tall people are friendly. Ans: "đx(T(x) Žđ F(x)). 92. No friendly people are angry. Ans: "đx(F(x) Žđ ŘđA(x)). Use the following to answer questions 93-94: In the questions below suppose the variable x represents people, and F(x): x is friendly T(x): x is tall A(x): x is angry. Write the statement using these predicates and any needed quantifiers. 93. Some tall angry people are friendly. Ans: $đx(T(x) Ůđ A(x) Ůđ F(x)). 94. If a person is friendly, then that person is not angry. Ans: "đx(F(x) Žđ ŘđA(x)). Use the following to answer questions 95-97: In the questions below suppose the variable x represents people, and F(x): x is friendly T(x): x is tall A(x): x is angry. Write the statement in good English. Do not use variables in your answer. 95. A(Bill). Ans: Bill is angry. 96. Řđ$đx(A(x) Ůđ T(x)). Ans: No one is tall and angry. 97. Řđ"đx(F(x) Žđ A(x)). Ans: Some friendly people are not angry. Use the following to answer questions 98-100: In the questions below suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course S(x): x is a sophomore F(x): x is a freshman T(x,đy): x is taking y. Write the statement using these predicates and any needed quantifiers. 98. There is a course that every freshman is taking. Ans: $đy"đx(F(x) Žđ T(x,đy)). 99. No freshman is a sophomore. Ans: Řđ$đx(F(x) Ůđ S(x)]. 100. Some freshman is taking an advanced course. Ans: $đx$đy(F(x) Ůđ A(y) Ůđ T(x,đy)). Use the following to answer questions 101-102: In the questions below suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course F(x): x is a freshman T(x,đy): x is taking y P(x,đy): x passed y. Write the statement using the above predicates and any needed quantifiers. 101. No one is taking every advanced course. Ans: Řđ$đx"đy (A(y) Žđ T(x,đy)). 102. Every freshman passed calculus. Ans: "đx(F(x) Žđ P(x,đcalculus)). Use the following to answer questions 103-105: In the questions below suppose the variable x represents students and the variable y represents courses, and T(x,đy): x is taking y P(x,đy): x passed y. Write the statement in good English. Do not use variables in your answers. 103. ŘđP(Wisteria,đMAT100). Ans: Wisteria did not pass MAT 100. 104. $đy"đxT(x,đy). Ans: There is a course that all students are taking. 105. "đx$đyT(x,đy). Ans: Every student is taking at least one course. Use the following to answer questions 106-110: In the questions below assume that the universe for x is all people and the universe for y is the set of all movies. Write the English statement using the following predicates and any needed quantifiers: S(x,đy): x saw y L(x,đy): x liked y A(y): y won an award C(y): y is a comedy. 106. No comedy won an award. Ans: "đy[C(y) Žđ ŘđA(y)]. 107. Lois saw Casablanca, but didn't like it. Ans: S(Lois,đCasablanca) Ůđ ŘđL(Lois,đCasablanca). 108. Some people have seen every comedy. Ans: $đx"đy [C(y) Žđ S(x,đy)]. 109. No one liked every movie he has seen. Ans: Řđ$đx"đy[S(x,đy) Žđ L(x,đy)]. 110. Ben has never seen a movie that won an award. Ans: Řđ$đy[A(y) Ůđ S(Ben,đy)]. Use the following to answer questions 111-113: In the questions below assume that the universe for x is all people and the universe for y is the set of all movies. Write the statement in good English, using the predicates S(x,đy): x saw y L(x,đy): x liked y. Do not use variables in your answer. 111. $đyŘđS(Margaret,đy). Ans: There is a movie that Margaret did not see. 112. $đy"đxL(x,đy). Ans: There is a movie that everyone liked. 113. "đx$đyL(x,đy). Ans: Everyone liked at least one movie. Use the following to answer questions 114-123: In the questions below suppose the variable x represents students, y represents courses, and T(x,đy) means  x is taking y . 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OJQJ^JYLz¤zŕzâzJ{‚{„{â{ä{F}Ž}Ú}Ü} ~p~r~–~î~đ~fčŃÉčŃÉžžž°žÉčŃÉčŃÉčŃ $$$1$7$8$H$a$ $$1$7$8$H$1$7$8$H$$$ ĆĐ„„çú1$7$8$H$^„`„çú$$ Ćv„Đ„0ý1$7$8$H$^„Đ`„0ýđ}~~~~~€~‚~„~ˆ~Š~Œ~Ž~~ü~ţ~  $€&€R€T€†€ˆ€Š€Œ€Ž€€˘€¤€ş€ź€vxz|€‚„†ˆ”–˜šž ˘¤Ś˛´ś¸źžŔÂÄĐÔÖŘÜŢŕâäđňôöřúüţ‚ "üňëüňëňëüëňëüňëňëüëňëüëüëüëüëňëüëüëüňëňëüëňëüňëňëüëňëüňëňëüëňëüňëňëüëňëüňëňëňëüëéňëüňëňëüëňëU h! 6]h! OJQJ^Jh! \fhĆČnŒŞČč&Ff†ŞĐňôj|~Ôî÷ěěěěěěěěěěěěěěě÷Őž÷Őž$$ ĆĐ„„çú1$7$8$H$^„`„çú$$ Ćv„Đ„0ý1$7$8$H$^„Đ`„0ý $$1$7$8$H$1$7$8$H$,đy) 6. "đy$đxT(x,đy) 7. $đy"đxŘđT(x,đy) 8. Řđ"đx$đyT(x,đy) 9. Řđ$đy"đxT(x,đy) 10. Řđ"đx$đyŘđT(x,đy) 11. Řđ"đxŘđ"đyŘđT(x,đy) 12. "đx$đyŘđT(x,đy) 114. Every course is being taken by at least one student. Ans: 6. 115. Some student is taking every course. Ans: 1, 10. 116. No student is taking all courses. Ans: 12. 117. There is a course that all students are taking. Ans: 2. 118. Every student is taking at least one course. Ans: 3. 119. There is a course that no students are taking. Ans: 7. 120. Some students are taking no courses. Ans: 5, 8, 11. 121. No course is being taken by all students. Ans: 9. 122. Some courses are being taken by no students. Ans: 7. 123. No student is taking any course. Ans: 4. Use the following to answer questions 124-134: In the questions below suppose the variable x represents students, F(x) means  x is a freshman , and M(x) means  x is a math major . Match the statement in symbols with one of the English statements in this list: 1. Some freshmen are math majors. 2. Every math major is a freshman. 3. No math major is a freshman. 124. "đx(M(x) Žđ ŘđF(x)). Ans: 3. 125. Řđ$đx(M(x) Ůđ ŘđF(x)). Ans: 2. 126. "đx(F(x) Žđ ŘđM(x)). Ans: 3. 127. "đx(M(x) Žđ F(x)). Ans: 2. 128. $đx(F(x) Ůđ M(x)). Ans: 1. 129. Řđ"đx(ŘđF(x) Úđ ŘđM(x)). Ans: 1. 130. "đx(Řđ(M(x) Ůđ ŘđF(x))). Ans: 2. 131. "đx(ŘđM(x) Úđ ŘđF(x)). Ans: 3. 132. Řđ$đx(M(x) Ůđ ŘđF(x)). Ans: 2. 133. Řđ$đx(M(x) Ůđ F(x)). Ans: 3. 134. Řđ"đx(F(x) Žđ ŘđM(x)). Ans: 1. Use the following to answer questions 135-138: In the questions below let F(A) be the predicate  A is a finite set and S(A,đB) be the predicate  A is contained in B . Suppose the universe of discourse consists of all sets. Translate the statement into symbols. 135. Not all sets are finite. Ans: $đAŘđF(A). 136. Every subset of a finite set is finite. Ans: "đA"đB[(F(B) Ůđ S(A,đB)) Žđ F(A)]. 137. No infinite set is contained in a finite set. Ans: Řđ$đA$đB(ŘđF(A) Ůđ F(B) Ůđ S(A,đB)). 138. The empty set is a subset of every finite set. Ans: "đA[F(A) Žđ S(Ćđ,đA)]. Use the following to answer questions 139-143: In the questions below write the negation of the statement in good English. Don't write  It is not true that & .đ 139. Some bananas are yellow. Ans: No bananas are yellow. 140. All integers ending in the digit 7 are odd. Ans: Some integers ending in the digit 7 are not odd. 141. No tests are easy. Ans: Some tests are easy. 142. Roses are red and violets are blue. Ans: Roses are not red or violets are not blue. 143. Some skiers do not speak Swedish. Ans: All skiers speak Swedish. 144. A student is asked to give the negation of  all bananas are ripe . (a) The student responds  all bananas are not ripe . Explain why the English in the student's response is ambiguous. (b) Another student says that the negation of the statement is  no bananas are ripe . Explain why this is not correct. (c) Another student says that the negation of the statement is  some bananas are ripe . Explain why this is not correct. (d) Give the correct negation. Ans: (a) Depending on which word is emphasized, the sentence can be interpreted as  all bananas are non-ripe fruit (i.e., no bananas are ripe) or as  not all bananas are ripe (i.e., some bananas are not ripe). (b) Both statements can be false at the same time. (c) Both statements can be true at the same time. (d) Some bananas are not ripe. 145. Explain why the negation of  Some students in my class use e-mail is not  Some students in my class do not use e-mail . Ans: Both statements can be true at the same time. 146. What is the rule of inference used in the following: If it snows today, the university will be closed. The university will not be closed today. Therefore, it did not snow today. Ans: Modus tollens. 147. What is the rule of inference used in the following: If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material. Ans: Hypothetical syllogism. 148. Explain why an argument of the following form is not valid: p Žđ q Řđp ________ \đ Řđq Ans: p false and q true yield true hypotheses but a false conclusion. 149. Determine whether the following argument is valid: p Žđ r q Žđ r Řđ(p Úđ q) ________ \đ Řđr Ans: Not valid: p false, q false, r true 150. Determine whether the following argument is valid: p Žđ r q Žđ r q Úđ Řđr ________ \đ Řđp Ans: Not valid: p true, q true, r true 151. Show that the hypotheses  I left my notes in the library or I finished the rough draft of the paper and  I did not leave my notes in the library or I revised the bibliography imply that  I finished the rough draft of the paper or I revised the bibliography . Ans: Use resolution on l Úđ f and Řđl Úđ r to conclude f Úđ r. 152. Determine whether the following argument is valid. Name the rule of inference or the fallacy. If n is a real number such that n >đ 1, then n2 >đ 1. Suppose that n2 >đ 1. Then n >đ 1. Ans: Not valid: fallacy of affirming the conclusion. 153. Determine whether the following argument is valid. Name the rule of inference or the fallacy. If n is a real number such that n >đ 2, then n2 >đ 4. Suppose that n Łđ 2. Then n2 Łđ 4. Ans: Not valid: fallacy of denying the hypothesis. 154. Determine whether the following argument is valid: She is a Math Major or a Computer Science Major. If she does not know discrete math, she is not a Math Major. If she knows discrete math, she is smart. She is not a Computer Science Major. Therefore, she is smart. Ans: Valid. 155. Determine whether the following argument is valid. Rainy days make gardens grow. Gardens don't grow if it is not hot. It always rains on a day that is not hot. Therefore, if it is not hot, then it is hot. Ans: Valid. 156. Determine whether the following argument is valid. If you are not in the tennis tournament, you will not meet Ed. If you aren't in the tennis tournament or if you aren't in the play, you won't meet Kelly. You meet Kelly or you don't meet Ed. It is false that you are in the tennis tournament and in the play. Therefore, you are in the tennis tournament. Ans: Not valid. 157. Show that the premises  Every student in this class passed the first exam and  Alvina is a student in this class imply the conclusion  Alvina passed the first exam . Ans: Universal instantiation. 158. Show that the premises  Jean is a student in my class and  No student in my class is from England imply the conclusion  Jean is not from England . Ans: Universal instantiation. 159. Determine whether the premises  Some math majors left the campus for the weekend and  All seniors left the campus for the weekend imply the conclusion  Some seniors are math majors . Ans: The two premises do not imply the conclusion. 160. Show the premises "Everyone who read the textbook passed the exam", and "Ed read the textbook" imply the conclusion "Ed passed the exam". Ans: Let R(x) be the predicate "x has read the textbook" and P(x) be the predicate "x passed the exam". The following is the proof: 1. EMBED Equation.BREE4 hypothesis 2. R(Ed) ’! P(Ed) universal instantiation on 1 3. R(Ed) hypothesis 4. P(Ed) modus ponens on 2 and 3 161. Determine whether the premises "No juniors left campus for the weekend" and "Some math majors are not juniors" imply the conclusion "Some math majors left campus for the weekend." Ans: The two premises do not imply the conclusion. 162. Show that the premise  My daughter visited Europe last week implies the conclusion  Someone visited Europe last week . Ans: Existential generalization. 163. Suppose you wish to prove a theorem of the form  if p then q . (a) If you give a direct proof, what do you assume and what do you prove? (b) If you give an indirect proof, what do you assume and what do you prove? (c) If you give a proof by contradiction, what do you assume and what do you prove? Ans: (a) Assume p, prove q. (b) Assume Řđq, prove Řđp. (c) Assume p Ůđ Řđq, show that this leads to a contradiction. 164. Suppose that you had to prove a theorem of the form  if p then q . Explain the difference between a direct proof and a proof by contraposition. Ans: Direct proof: Assume p, show q. Indirect proof: Assume Řđq, show Řđp. 165. Give a direct proof of the following:  If x is an odd integer and y is an even integer, then x +đ y is odd . Ans: Suppose x =đ 2k +đ 1, y =đ 2l. Therefore x +đ y =đ 2k +đ 1 +đ 2l =đ 2(k +đ l) +đ 1, which is odd. 166. Give a proof by contradiction of the following:  If n is an odd integer, then n2 is odd . Ans: Suppose n =đ 2k +đ 1 but n2 =đ 2l. Therefore (2k +đ 1)2 =đ 2l, or 4k2 +đ 4k +đ 1 =đ 2l. Hence 2(2k2 +đ 2k -đ l) =đ -đ1 (even = odd), a contradiction. Therefore n2 is odd. 167. Consider the following theorem:  if x and y are odd integers, then x +đ y is even . Give a direct proof of this theorem. Ans: Let x =đ 2k +đ 1, y =đ 2l +đ 1. Therefore x +đ y =đ 2k +đ 1 +đ 2l +đ 1 =đ 2(k +đ l +đ 1) , which is even. 168. Consider the following theorem:  if x and y are odd integers, then x +đ y is even . Give a proof by contradiction of this theorem. Ans: Suppose x =đ 2k +đ 1 and y =đ 2l +đ 1, but x +đ y =đ 2m +đ 1. Therefore (2k +đ 1) +đ (2l +đ 1) =đ 2m +đ 1. Hence 2(k +đ l -đ m +đ 1) =đ 1 (even = odd), which is a contradiction. Therefore x +đ y is even. 169. Give a proof by contradiction of the following: If x and y are even integers, then xy is even. Ans: Suppose x =đ 2k and y =đ 2l, but xy =đ 2m +đ 1. Therefore 2k ×đ 2l =đ 2m +đ 1. Hence 2(2kl -đ m) =đ 1 (even = odd), which is a contradiction. Therefore xy is even. 170. Consider the following theorem: If x is an odd integer, then x +đ 2 is odd. Give a direct proof of this theorem Ans: Let x =đ 2k +đ 1. Therefore x +đ 2 =đ 2k +đ 1 +đ 2 =đ 2(k +đ 1) +đ 1, which is odd. 171. Consider the following theorem: If x is an odd integer, then x +đ 2 is odd. Give a proof by contraposition of this theorem. Ans: Suppose x +đ 2 =đ 2k. Therefore x =đ 2k -đ 2 =đ 2(k -đ 1), which is even. 172. Consider the following theorem: If x is an odd integer, then x +đ 2 is odd. Give a proof by contradiction of this theorem. Ans: Suppose x is odd but x +đ 2 is even. Therefore x =đ 2k +đ 1 and x +đ 2 =đ 2l. Hence (2k +đ 1) +đ 2 =đ 2l. Therefore 2(k +đ 1 -đ l) =đ -đ1 (even = odd), a contradiction. 173. Consider the following theorem: If n is an even integer, then n +đ 1 is odd. Give a direct proof of this theorem. Ans: Let n =đ 2k. Therefore n +đ 1 =đ 2k +đ 1, which is odd. 174. Consider the following theorem: If n is an even integer, then n +đ 1 is odd. Give a proof by contraposition of this theorem. Ans: Suppose n +đ 1 is even. Therefore n +đ 1 =đ 2k. Therefore n =đ 2k -đ 1 =đ 2(k -đ 1) +đ 1, which is odd. 175. Consider the following theorem: If n is an even integer, then n +đ 1 is odd. Give a proof by contradiction of this theorem. Ans: Suppose n =đ 2k but n +đ 1 =đ 2l. Therefore 2k +đ 1 =đ 2l (even = odd), which is a contradiction. 176. Prove that the following is true for all positive integers n: n is even if and only if 3n2 +đ 8 is even. Ans: If n is even, then n =đ 2k. Therefore 3n2 +đ 8 =đ 3(2k)2 +đ 8 =đ 12k2 +đ 8 =đ 2(6k2 +đ 4), which is even. If n is odd, then n =đ 2k +đ 1. Therefore 3n2 +đ 8 =đ 3(2k +đ 1)2 +đ 8 =đ 12k2 +đ 12k +đ 11 =đ 2(6k2 +đ 6k +đ 5) +đ 1, which is odd. 177. Prove the following theorem: n is even if and only if n2 is even. Ans: If n is even, then n2 =đ (2k)2 =đ 2(2k2), which is even. If n is odd, then n2 =đ (2k +đ 1)2 =đ 2(2k2 +đ 2k) +đ 1, which is odd. 178. Prove: if m and n are even integers, then mn is a multiple of 4. Ans: If m =đ 2k and n =đ 2l, then mn =đ 4kl. Hence mn is a multiple of 4. 179. Prove or disprove: For all real numbers x and y, ëđx -đ yűđ =đ ëđxűđ -đ ëđyűđ. Ans: False: x =đ 2 y =đ 1/đ2. 180. Prove or disprove: For all real numbers x and y, ëđx +đ ëđxűđűđ =đ ëđ2xűđ. Ans: False: x =đ 1/đ2. 181. Prove or disprove: For all real numbers x and y, ëđxyűđ =đ ëđxűđ ×đ ëđyűđ. Ans: False: x =đ 3/đ2, y =đ 3/đ2. 182. Give a proof by cases that EMBED Equation.BREE4for all real numbers x. Ans: Case 1, EMBED Equation.BREE4: then EMBED Equation.BREE4, so EMBED Equation.BREE4. Case 2, EMBED Equation.BREE4: here EMBED Equation.BREE4 and EMBED Equation.BREE4, so EMBED Equation.BREE4. 183. Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if 3n + 5 is even, then n is odd. Which type of proof would be easier to give? Explain why. Ans: It is easier to give a contraposition proof; it is usually easier to proceed from a simle expression (such as n) to a more complex expression (such as 3n + 5 is even). Begin by supposing that n is not odd. Therefore n is even and hence n = 2k for some integer k. Therefore 3n + 5 = 3(2k) + 5 = 6k + 5 = 2(3k + 2) + 1, which is not even. If we try a direct proof, we assume that 3n + 5 is even; that is, 3n + 5 = 2k for some integer k. From this we obtain n = (2k - 5)/3, and it is not obvious from this form that n is even. 184. Prove that the following three statements about positive integers n are equivalent: (a) n is even; (b) n3 + 1 is odd; (c) n2 - 1 is odd. Ans: Prove that (a) and (b) are equivalent and that (a) and (c) are equivalent. 185. Given any 40 people, prove that at least four of them were born in the same month of the year. Ans: If at most three people were born in each of the 12 months of the year, there would be at most 36 people. 186. Prove that the equation EMBED Equation.BREE4has no positive integer solutions. Ans: Give a proof by cases. There are only six cases that need to be considered: x = y = 1; x = 1, y = 2; x = 1, y = 3; x = 2, y = 1; x = y = 2; x = 2, y = 3. 187. What is wrong with the following "proof" that -3 = 3, using backward reasoning? Assume that -3 = 3. Squaring both sides yields (-3)2 = 32, or 9 = 9. Therefore -3 = 3. Ans: The steps in the "proof" cannot be reversed. 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