§
1.4 Problem 1 (
a
): “For every number
x
, there is a bigger number
y
.”
(
b
): “For any two numbers
x
and
y
, if
x
and
y
are nonnegative then so is their product.”
(
c
): “For any two numbers
x
and
y
, there is another number
z
that is equal to their product.”
§
1.4 Problem 9 (
a
):
∀
xL
(
x,
Jerry)
(
b
):
∀
x
∃
yL
(
x,y
)
(
c
):
∃
x
∀
yL
(
y,x
)
(
d
):
¬∃
x
∀
yL
(
x,y
)
(
e
):
∃
x
¬
L
(Lydia
,x
)
(
f
):
∃
x
∀
y
¬
L
(
y,x
)
(
g
):
∃
x
∀
y
(
x
=
y
)
↔
(
∀
zL
(
z,y
))
(
h
):
∃
x
∃
y
(
(
x
negationslash
=
y
)
∧ ∀
z
(
x
=
z
∨
x
=
y
)
↔
L
(Lynn
,z
)
)
(
i
):
∀
xL
(
x,x
)
(
j
):
∃
x
∀
y
(
x
negationslash
=
y
)
→ ¬
L
(
x,y
)
The book gives the inequivalent solution
∃
x
∀
y
(
L
(
x,y
)
↔
x
=
y
)
. This says that everyone loves them-
selves and nobody else, whereas my solution allows for the possibility that people might not love themselves.
It is unclear from the English which is intended.
§
1.4 Problem 12 (
e
):
∀
(
x
(
x
negationslash
= Joseph)
↔
C
(Sanjay
,x
)
)
(
f
):
∃
x
¬
I
(
x
)
(
g
):
¬∀
xI
(
x
)
(
h
):
∃
x
∀
y
(
(
x
=
y
)
↔
I
(
y
)
)
(
i
):
∃
x
∀
y
(
(
x
negationslash
=
y
)
↔
I
(
y
)
)
(
j
):
∀
x
(
I
(
x
)
→ ∃
y
(
(
y
negationslash
=
x
)
∧
C
(
x,y
)
))
(
k
):
∃
x
(
I
(
x
)
∧ ∀
y
(
(
y
negationslash
=
x
)
→ ¬
C
(
x,y
)
))
(
l
):
∃
x
∃
y
(
(
x
negationslash
=
y
)
∧ ¬
C
(
x,y
)
)
(
m
):
∃
x
∀
yC
(
x,y
)
(
n
):
∃
x
∃
y
∃
z
(
(
x
negationslash
=
y
)
∧ ¬
C
(
x,z
)
∧ ¬
(
y,z
)
)
(
o
):
∃
x
∃
y
bracketleftbig
(
x
negationslash
=
y
)
∧ ∀
z
(
C
(
x,z
)
∨
C
(
y,z
)
)bracketrightbig
Some of these are open to interpretation, due to the vagueness of the English language. For instance,
take part (
n
), for which the solutions manual gives
∃
x
∃
y
(
x
negationslash
=
y
∧ ∀
z
¬
(
C
(
x,z
)
∧
C
(
y,z
)
))
My solution says, “There are two people
x
and
y
and a person
z
with whom
x
and
y
have not chatted,” so
that
x
and
y
have not chatted with the same person. The solutions manual’s solution says, “There are two