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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
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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