This preview shows page 1. Sign up to view the full content.
Section 0.2
5
(b) Converse:
If
n~l
is not an integer, then
n
is an integer. This is false:
n
=
!
is a counterexample
(
n
_
1)
n+1

3"
•
Contrapositive:
If
n~l
is an integer, then
n
is not an integer. This is false:
n
=
0 is a counterex
ample.
Negation: There exists an integer
n
such that
n~l
is an integer. This is true: Take
n
=
o.
9. (a)
n
prime
+
2
n

1 prime.
(b)
n
prime is sufficient for
2
n

1 to be prime.
(c)
A
is false. For example,
n
=
11 is prime, but 211 
1
=
2047
=
23(89) is not. The integer
n
=
11 is a counterexample to
A.
(d)
2
n

1 prime
+
n
prime.
(e) The converse of
A
is true. To show this, we establish the contrapositive. Thus, we assume
n
is not
prime. Then there exists a pair of integers
a
and
b
such that
a
>
1,
b
>
1, and
n
=
abo
Using the
hint, we can factor
2
n

1 as
2
n
1
=
(2
a
)b
1
=
(2
a
_1)[(2
a
)b1
+
(2
a
)b2
+ .
.. +
2
a
+
1].
Since
a>
1 and
b
>
1, we have
2
a
1
>
1 and
(2
a
)b1
+
(2
a
)b2
+ .
.. +
2
a
+
1
>
1, so
2
n
1
is the product of two integers both of which exceed one. Hence,
This is the end of the preview. Sign up
to
access the rest of the document.
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

Click to edit the document details