MAD 2104
May 13, 2010
Quiz II and Key
Prof. S. Hudson
1) [45 pts] Find a counterexample to each of these, where the domain is
Z
, the set of
integers:
a)
∀
x,
∀
y,
(
x
2
=
y
2
→
x
3
=
y
3
)
b)
∀
x,
∃
y,
(
y
2
=
x
)
c)
∀
x,
∀
y,
(
y
2
=
x
3
)
2) [30pts] Let
A
i
=
{
i, i
+ 1
, i
+ 2
, . . .
}
for every
i
∈
Z
. Find these sets (we did one of these
in class, but I changed the other slightly):
a)
S
∞
i
=1
A
i
b)
T
5
i
=1
A
i
3) [25 pts] Prove or disprove that the product of a nonzero rational number and an irrational
number is irrational. If you need to use the back, leave a note here.
Remarks and Answers:
The average was about 57 / 100. The unofficial scale for this
quiz is
A’s = 70 to 79
B’s = 60 to 69
C’s = 50 to 59
D’s = 40 to 49
1a) Let
x
= 1 and
y
=

1. Then
x
2
=
y
2
, but
x
3
6
=
y
3
. Notice that the counterexample
includes values for both
x
and
y
because both are quantified by
∀
.
1b) Let
x
=

1. There is no
y
∈
R
such that
y
2
<
0, so
y
2
=
x
is always false. Notice
that the counterexample includes a value only for
x
, because only
x
is quantified by
∀
.
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 Fall '08
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 Rational number, Irrational number, nonzero rational number

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