38
Formal Logic
35. (P S Q) ` (P′ S Q) S Q
36. (A′ S B′) ` (A S C ) S (B S C )
37. (A′ S B) ` (B S C ) ` (C S D) S (A′ S D)
38. (A ~ B) ` (A S C ) ` (B S C ) S C
39. (Y S Z′) ` (X′ S Y) ` [Y S (X S W)] ` (Y S Z) S (Y S W)
40. (A ` B) ` (B S A′) S (C ` B′)
41. (A ` B)′ ` (C′ ` A)′ ` (C ` B′)′ S A′
42. (P ~ (Q ` R)) ` (R′ ~ S) ` (S S T′) S (T S P)
In Exercises 43–54, write the argument using propositional wffs (use the statement letters shown). Then, using
propositional logic, including the rules in Table 1.14, prove that the argument is valid.
43. If the program is efficient, it executes quickly. Either the program is efficient, or it has a bug. However, the
program does not execute quickly. Therefore, it has a bug. E, Q, B
44. If Jane is more popular, then she will be elected. If Jane is more popular, then Craig will resign. Therefore,
if Jane is more popular, she will be elected and Craig will resign. J, E, C
45. If chicken is on the menu, then don’t order fish, but you should have either fish or salad. So, if chicken is
on the menu, have salad. C, F, S
46. The crop is good, but there is not enough water. If there is a lot of rain or not a lot of sun, then there is
enough water. Therefore, the crop is good and there is a lot of sun. C, W, R, S
47. If the ad is successful, then the sales volume will go up. Either the ad is successful or the store will close.
The sales volume will not go up. Therefore, the store will close. A, S, C
48. If DeWayne is not tall then Jayden is not DeWayne’s brother. If DeWayne is tall then Trevor is DeWayne’s
brother. Therefore, if Jayden is DeWayne’s brother, then Trevor is DeWayne’s brother. D, J, T
49. Russia was a superior power, and either France was not strong or Napoleon made an error. Napoleon did
not make an error, but if the army did not fail, then France was strong. Hence, the army failed and Russia
was a superior power. R, F, N, A
50. It is not the case that if electric rates go up, then usage will go down, nor is it true that either new power plants
will be built or bills will not be late. Therefore, usage will not go down and bills will be late. R, U, P, B
51. If Jose took the jewelry or Mrs. Krasov lied, then a crime was committed. Mr. Krasov was not in town. If
a crime was committed, then Mr. Krasov was in town. Therefore, Jose did not take the jewelry. J, L, C, T
52. If the birds are flying south and the leaves are turning, then it must be fall. Fall brings cold weather. The
leaves are turning but the weather is not cold. Therefore, the birds are not flying south. B, L, F, C
53. If a Democrat is elected then taxes will go up. Either a Democrat will be elected or the bill will pass.
Therefore, if taxes do not go up, then the bill will pass. D, T, B
54. Either Emily was not home or if Pat did not leave the tomatoes, then Sophie was ill. Also, if Emily was not
home, then Olivia left the peppers. But it is not true that either Sophie was ill or Olivia left the peppers.
Therefore, Pat left the tomatoes and Olivia did not leave the peppers. E, P, S, O
55. a. Use a truth table to verify that A S (B S C ) 4 (A ` B) S C is a tautology.
b. Prove that A S (B S C ) 3 (A ` B) S C by using a series of equivalences.
c. Explain how this equivalence justifies the deduction method that says: to prove
P 1 ` P 2 ` … ` P n S (R S S ), deduce S from P 1 , P 2 , … , P n , and R.
56. The argument of the defense attorney at the beginning of this chapter was
If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason
Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard
Formal Logic
35. (P S Q) ` (P′ S Q) S Q
36. (A′ S B′) ` (A S C ) S (B S C )
37. (A′ S B) ` (B S C ) ` (C S D) S (A′ S D)
38. (A ~ B) ` (A S C ) ` (B S C ) S C
39. (Y S Z′) ` (X′ S Y) ` [Y S (X S W)] ` (Y S Z) S (Y S W)
40. (A ` B) ` (B S A′) S (C ` B′)
41. (A ` B)′ ` (C′ ` A)′ ` (C ` B′)′ S A′
42. (P ~ (Q ` R)) ` (R′ ~ S) ` (S S T′) S (T S P)
In Exercises 43–54, write the argument using propositional wffs (use the statement letters shown). Then, using
propositional logic, including the rules in Table 1.14, prove that the argument is valid.
43. If the program is efficient, it executes quickly. Either the program is efficient, or it has a bug. However, the
program does not execute quickly. Therefore, it has a bug. E, Q, B
44. If Jane is more popular, then she will be elected. If Jane is more popular, then Craig will resign. Therefore,
if Jane is more popular, she will be elected and Craig will resign. J, E, C
45. If chicken is on the menu, then don’t order fish, but you should have either fish or salad. So, if chicken is
on the menu, have salad. C, F, S
46. The crop is good, but there is not enough water. If there is a lot of rain or not a lot of sun, then there is
enough water. Therefore, the crop is good and there is a lot of sun. C, W, R, S
47. If the ad is successful, then the sales volume will go up. Either the ad is successful or the store will close.
The sales volume will not go up. Therefore, the store will close. A, S, C
48. If DeWayne is not tall then Jayden is not DeWayne’s brother. If DeWayne is tall then Trevor is DeWayne’s
brother. Therefore, if Jayden is DeWayne’s brother, then Trevor is DeWayne’s brother. D, J, T
49. Russia was a superior power, and either France was not strong or Napoleon made an error. Napoleon did
not make an error, but if the army did not fail, then France was strong. Hence, the army failed and Russia
was a superior power. R, F, N, A
50. It is not the case that if electric rates go up, then usage will go down, nor is it true that either new power plants
will be built or bills will not be late. Therefore, usage will not go down and bills will be late. R, U, P, B
51. If Jose took the jewelry or Mrs. Krasov lied, then a crime was committed. Mr. Krasov was not in town. If
a crime was committed, then Mr. Krasov was in town. Therefore, Jose did not take the jewelry. J, L, C, T
52. If the birds are flying south and the leaves are turning, then it must be fall. Fall brings cold weather. The
leaves are turning but the weather is not cold. Therefore, the birds are not flying south. B, L, F, C
53. If a Democrat is elected then taxes will go up. Either a Democrat will be elected or the bill will pass.
Therefore, if taxes do not go up, then the bill will pass. D, T, B
54. Either Emily was not home or if Pat did not leave the tomatoes, then Sophie was ill. Also, if Emily was not
home, then Olivia left the peppers. But it is not true that either Sophie was ill or Olivia left the peppers.
Therefore, Pat left the tomatoes and Olivia did not leave the peppers. E, P, S, O
55. a. Use a truth table to verify that A S (B S C ) 4 (A ` B) S C is a tautology.
b. Prove that A S (B S C ) 3 (A ` B) S C by using a series of equivalences.
c. Explain how this equivalence justifies the deduction method that says: to prove
P 1 ` P 2 ` … ` P n S (R S S ), deduce S from P 1 , P 2 , … , P n , and R.
56. The argument of the defense attorney at the beginning of this chapter was
If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason
Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard