CMPSC 360
Fall 2008
Homework 5: PRACTICE EXAM 1 Solutions
1.
(25 points, 5 each) Logic, Quantifiers, Proofs
(a)
i. True. For each
x
, we can choose
y
= 0, and then it is true that
x
+ 0 =
x
=
x

0.
ii. True. We can choose
y
= 0, and then the statement says for all
x
,
x
+ 0 =
x
=
x

0, which is true.
iii. False. A counterexample would be
y
= 1, in which case there does not exist
and
x
such that
x
+ 1 =
x

1.
(b) Note that
Q
(
x, y
) is the same as
¬
P
(
x, y
).
¬
(
∀
y
∃
x P
(
x, y
))
≡
∃
y
¬∃
x P
(
x, y
)
≡
∃
y
∀
x
¬
P
(
x, y
)
≡
∃
y
∀
x Q
(
x, y
)
(c) Assume
a
and
b
are rational. Then
a
=
a
1
/a
2
and
b
=
b
1
/b
2
where
a
1
, a
2
, b
1
, b
2
∈
Z
,
and
a
2
, b
2
= 0.
Multiplying, we have
ab
=
a
1
a
2
·
b
1
b
2
=
a
1
b
1
a
2
b
2
.
Since
a
1
b
1
∈
Z
and
a
2
b
2
∈
Z
(and is not zero),
ab
is a rational number.
(d) Proof by contraposition. Assume
x
1
/
3
is rational. Then
x
1
/
3
=
a/b
, for
a, b
∈
Z
,
b
= 0. Raising both side to the 3rd power, we get
x
=
a
3
/b
3
. Since
a
3
, b
3
∈
Z
,
and
b
3
= 0,
x
is rational.
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 Fall '08
 HAULLGREN
 Prime number

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