Math 347 C1
HOUR EXAM I
29 June 2010
SOLUTIONS
1. Let
S
=
{
x
∈
R

x
2
> x
+ 6
}
and let
T
=
{
x
∈
R

x >
3
}
. Determine whether the
following statements are true. Justify your answers.
a)
T
⊆
S
.
b)
S
⊆
T
.
SOLUTION
Clearly
T
= (3
,
∞
)
.
S
consists of those real numbers satisfying:
x
2

x

6
>
0
,
and
x
2

x

6
>
0 if
and only if (
x

3)(
x
+ 2)
>
0.
This is equivalent to either
x

3
>
0 and
x
+ 2
>
0 OR
x

3
<
0 and
x
+ 2
<
0.
In the ﬁrst case,
x >
3 and
x >

2, whence
x >
3
.
In the second case
x <
3 and
x <

2 whence
x <

2
.
Therefore
S
= (3
,
∞
)
∪
(
∞
,

2)
.
a) Clearly
T
⊆
S
.
b)
S
6⊆
T.
2. Suppose there are
n
married couples, each consisting of a man and a woman. How
many pairs of a man and a woman are there in which the man and the woman in
the same pair are not married?
SOLUTION
The total number of pairs of men and women is
n
2
.
Among these are
the pairs of a man and his wife; there are exactly
n
of these. Therefore there are
exactly
n
2

n
pairs of men and women where no one is paired with their spouse.
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 Spring '11
 Ashby
 Logic, Logical connective, Truth value

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