Solutions to Midterm 1
Question 1
(a) Someone has both been to every state and doesn’t know everyone.
(b) It is not the case that every book that has been made into a movie is liked
by everyone. (OR, there is a book that has been made into a movie and someone
doesn’t like the book.)
Question 2
(a)
∃
x
¬∀
yD
(
x, y, Klondike bar
)
or
∃
x
∃
y
¬
D
(
x, y, Klondike bar
)
.
(b) Let
B
(
s
) =
student
s
goes to beach,
H
(
s
) =
student
s
ﬁnishes their home
work.
∀
xH
(
x
)
→
B
(
x
)
Question 3
¬∃
x
∀
y
(
P
(
x
)
∧ ∃
z
(
Q
(
x
)
→ ¬
R
(
y, z
)))
≡∀
x
∃
y
¬
(
P
(
x
)
∧ ∃
z
(
Q
(
x
)
→ ¬
R
(
y, z
)))
≡∀
x
∃
y
(
¬
P
(
x
)
∨ ¬∃
z
(
Q
(
x
)
→ ¬
R
(
y, z
)))
≡∀
x
∃
y
(
¬
P
(
x
)
∨ ∀
z
¬
(
Q
(
x
)
→ ¬
R
(
y, z
)))
≡∀
x
∃
y
(
¬
P
(
x
)
∨ ∀
z
¬
(
¬
Q
(
x
)
∨ ¬
R
(
y, z
)))
≡∀
x
∃
y
(
¬
P
(
x
)
∨ ∀
z
(
Q
(
x
)
∧
R
(
y, z
)))
Question 4
1
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¬
p
∧
q
)
∨
(
¬
q
∧
r
)
≡
[(
¬
p
∧
q
)
∨ ¬
q
]
∧
[(
¬
p
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 Summer '08
 Staff
 codomain, Klondike bar

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