16. I am happy.
18. (a) The President is not both a Democrat and a republican.
(b) Either the President is not a Democrat, or he is not a Republican, or
he is neither.
19.
p
¬ p
p
p
∨ ¬
(
)
T
F
T
F
T
T
Since there are only T’s in the p
p
∨ ¬
(
) column, we conclude that
p
p
∨ ¬
(
) is a tautology.
21. Hint: Show that all the entries in the last column are all F’s.
22. (
)
( )
¬ ∨ ∨ ¬
p q
r
23. You will get an A if you are clever and either the sun shines or it rains.
26. Either 0 1
≠ , or I am the Queen of Texas.
27. Yes
28. No (verify using Truth Table construction).
36. Valid.
38. (a) Valid
(b) Valid
(c) Valid
39. (a) There is a student who spends more than 6 hours every weekday in
class.
(b) Every student spends more than 6 hours every weekday in class.
(c) There is student who does not spend more than 6 hours every
weekday in class.
(d) No student spends more than 6 hours every weekday in class.
40. (a) ℑ
∧
x P x Q x
( ( )
( ))
(b) ℑ
∧ ¬
x P x
Q x
( ( )
( ))
41. ∨ ∨ ∨
⋅ ⋅ = ⋅ ⋅
x
y
z x y z x y z
((
)
(
))
43. (a) True
(b) False, unless the Universe of discourse has just one element
(c) True
46. PDNF P Q
P
Q
PCNF P
Q
P Q
:(
) (
)
:(
) (
)
∧ ∨ ¬ ∧ ¬
∨ ¬ ∨ ¬ ∨
47. (a) (
) (
) (
)
P Q
P Q
P
Q
∧ ∨ ¬ ∧ ∨ ¬ ∧ ¬
(b) (
) (
) (
) (
)
P Q R
P Q
R
P Q R
P Q
R
∧ ∧ ∨ ∧ ∧ ¬ ∨ ¬ ∧ ∧ ∨ ¬ ∧ ∧ ¬
48. P Q R
∨ ∨
49. (
) (
) (
)
P Q
P Q
P
Q
∧ ∨ ¬ ∧ ∨ ¬ ∧ ¬
Answers to Exercises
319