Answers To see an answer to any odd-numbered exercise, just click on the exercise number.
Determine the truth value (if it exists) of each of the following statements.
1. "If 1 = 1, then 2 = 2." |
2."If 1 = 1, then 2 = 3." |
3. "If 1 = 0, then 1 = 1." |
4."If 1 0, then 2 2." |
5. "If 1 = 1 and 1 = 2, then 1 = 2." |
6."If 1 = 3 or 1 = 2 then 1 = 1." |
7. "If everything I say is false, then everything I say is true." |
8."If everything I say is false, then 1 = 2." |
9. "A sufficient condition for 1 to equal 2 is 1 = 3." |
10."1 = 0 is a sufficient condition for 1 to equal 2." |
11. "1 = 0 is a necessary condition for 1 to equal 2." |
12."1 = 1 is a necessary condition for 1 to equal 2." |
13. "1 = 2 is a necessary condition for 1 to be unequal to 2." |
14."2 is a necessary condition for 1 to be unequal to 3." |
15. "If I pay homage to the great Den, then the sun will rise in the east." |
16."If I fail to pay homage to the great Den, then the sun will still rise in the east." |
17. "In order for the sun to rise in the east, it is necessary that it sets in the west." |
18."In order for the sun to rise in the east, it is sufficient that it sets in the west." |
19. "The sun rises in the west only if it sets in the west." |
20."The sun rises in the east only if it sets in the east." |
21. "The sun sets in the west only if it rises in the west" |
22."The Milky Way Galaxy will not fall into a great black hole only if 1 = 1." |
23. "1 = 2 is a necessary and sufficient condition for 1 to be unequal to 2." |
24."1 2 is a necessary and sufficient condition for 1 to be unequal to 3." |
25. "The sun will rise in the east if and only if it sets in the west." |
26."The sun will rise in the east if and only if it does not set in the west." |
27. "In order for the sun to rise in the west, it is necessary and sufficient that it sets in the east." |
28."In order for the sun to rise in the east, it is necessary and sufficient that it sets in the west." |
Construct the truth table for each of the following statements, and indicate which (if any) are tautologies or contradictions.
29. p(qp) | 30. p(pq)~p |
31. (pq)~p | 32.. (p~p)~p |
33. (p~p)p | 34. p(p~p) |
35. (p~p)q. | 36. ~((p~p)q) |
37. p(pq) | 38. (pq)~p |
39. (p~p)(q~q) | 40. (p~p)(q~q) |
Use truth tables to demonstrate the following equivalences.
Give the contrapositive and converse of each of the following statements, phrasing your answers in words:
48. "If I do not think, then I do not exist."
49. "If I do not think, then I am Buddha."
50. "If I am Buddha, then I think."
51."These birds are of a feather only if they flock together."
52. "These birds flock together only if they are of a feather."
53. "In order to worship Den, it is necessary to sacrifice beasts of burden."
54. "In order to read the Tarot, it is necessary to consult the Oracle."
Express each of the following statements in equivalent disjunctive form.
56. "I think if I am."
57. "Symphony orchestras will cease to exist without government subsidy."
58. "The education system will collapse without continued taxpayer support."
59. "Research in the pure sciences will continue if our society wishes it."
60. "Nuclear physicists would be out of work if their accomplishments were measured purely by the generation of profit."
Translate the following into compound statements utilizing either the conditional or the biconditional, and using "p" for the statement "I am Julius Caesar" and "q" for the statement "You are Brutus"
62. "It is not the case that if I am Julius Caesar then you are Brutus."
63."I am Julius Caesar only if you are not Brutus."
64. "You are Brutus only if I am not Julius Caesar."
65. "I am Julius Caesar if and only if you are not Brutus."
66. "You are not Brutus if and only if I am not Julius Caesar."
67. "Either you are Brutus, or I am Julius Caesar."
68. "Either I am not Julius Caesar, or you are Brutus."
69. "In order for you to be Brutus, it is necessary and sufficient that I am not Julius Caesar."
70. "In order for you to not be Brutus, it is necessary and sufficient that I am not Julius Caesar."
Communication and Reasoning Exercises
71. Give an example of an instance where pq means that q causes p.
72. Complete the following. If pq, then its convese,, is the statement thatand (is/is not) logically equivalent to pq.
73. Complete the following sentence. If both pq and its are true, then the biconditional, , is.
74. If B is a tautology, why is AB also a tautology, regardless of A?
75. If A is a contradiction, why is AB a tautology, regardless of B?
76. If A is a tautology and B is a contradiciton, what can you say about AB?
77. If A and B are both contradictions, what can you say about AB?
78. Give an instance of a biconditional pq where neither p nor q causes the other.