10
(c)
The
converse is certainly true since
Ou
+
Ov
=
O.
(d)
The
negation is true: Take
a
=
u
=
b
= 1
and
v
=
1.
The contrapositive
of
A
is false since
A
is false.
5.
(a) There exists a countable set which is infinite.
(b) For all positive integers
n,
1
~
n.
Solutions
to'
Review
Exercises
6. (a) This is true.
If
x
is positive,
x
+
2 is positive.
In
addition,
if
x
is odd,
x
+
2 is odd.
(b) This is false. When
x
=
1,
x
+
2
=
+1
is a positive odd integer, while
x
is not.
7.
(a) This statement expresses a wellknown property
of
the real numbers.
It
is true.
(b) This is false.
The
conclusion would have us believe that every two real numbers are equal.
8.
The
desired formula is
ab
=
{a
+
b)2
~
{a

b)2
which holds because
{a
+
b)2

{a
_
b)2
=
(a
2
+
2ab
+
b
2
)

(a
2

2ab
+
b
2
)
=
4ab.
9.
(+)
Assume
n
3
is odd and suppose, to the contrary, that
n
is even. Thus
n
=
2x
for some integer
x.
But then
n
3
=
8x
3
=
2(
4x
3
)
is even, a contradiction. This means that
n
must
be
odd.
(t)
Assume
n
is odd. This means that
n
=
2x
+ 1
for some integer
x.
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 Summer '10
 any
 Real Numbers, Graph Theory, positive odd integer

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