MAT 300
, Mathematical Structures
Name: ANSWERS
Exam 1
February 25, 2010
1. (30 points, 5 points each) Shortanswer questions.
(a) Complete the following definition, making sure to introduce notation and using
the appropriate quantifier:
For integers
a
and
b
,
a
divides
b
if
there is an integer
k
such that
b
=
ak
.
(b) Complete the following definition, making sure to introduce notation and using
the appropriate quantifier:
For sets
A
and
B
,
A
is a
subset
of
B
if
for all
x
∈
A
,
x
∈
B
.
(c) Negate the following statement: “For every real number
x
satisfying
x
2

3
x

18 =
0, then
x
is negative or
x
is greater than 5.”
Logical form
:
∀
x
∈
R
,
(
P
⇒
Q
∨
R
) where
P
is
x
2

3
x

18 = 0,
Q
is
x <
0,
and
R
is
x >
5.
Logical negation
:
∃
x
∈
R
(
¬
(
P
⇒
Q
∨
R
)), or
∃
x
∈
R
(
P
∧ ¬
(
Q
∨
R
)) (see
part (f)) or
∃
x
∈
R
(
P
∧ ¬
Q
∧ ¬
R
)
Negation
: There exists a real number
x
satisfying
x
2

3
x

18 = 0 and
x
6
<
0
and
x
6
>
5, or
There exists a real number
x
satisfying
x
2

3
x

18 = 0 and 0
≤
x
≤
5.
Notes:
(1) The negation of “everyone wearing a blue shirt is wearing black pants”
is “someone wearing a blue shirt is
NOT
wearing black pants.” In other words,
we are looking for an
x
that
IS
a root to the given equation and that does
NOT
satisfy the given property.
(2) Be very careful in negating order relations for real numbers. The negation of
“
x
is negative” is “
x
is not negative” or “
x
is nonnegative” or “
x
≥
0”. Similarly,
the negation of “
x
is greater than 5” is “
x
is less than or equal to 5” or “
x
≤
5”.
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 Spring '07
 thieme
 Math, Integers, Negative and nonnegative numbers, 5m, 30 pts

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