Solutions to Homework 6  Math 2000
The solutions to 3.7,3.15,3.21 may be found in the book.
Exercises 3.4,3.15 were done during the tutorial.
(# 3.2) Let
n
∈
N
. Prove that if

n

1

+

n
+ 1
 ≤
1, then

n
2

1
 ≤
4.
Solution
. Let
S
=
Z
and
P
(
n
)
, Q
(
n
) be the open statements:
P
(
n
) :

n

1

+

n
+ 1
 ≤
1
Q
(
n
) :

n
2

1
 ≤
4
.
We now give a vacuous proof. Let
n
∈
N
. Thus
n
≥
1. It follows that
n
+ 1
≥
2
and so that

n
+ 1
 ≥
2
.
Since

n

1
 ≥
0 we deduce that

n

1

+

n
+ 1
 ≥
2
>
1
Therefore
P
(
n
) is false for all
n
∈
N
and hence it follows that if

n

1

+

n
+ 1
 ≤
1,
then

n
2

1
 ≤
4.
(# 3.8) Prove that if
a
and
c
are odd integers and
b
is an integer, then
ab
+
bc
is even.
Solution
. Since
a
and
c
are odd integers, there exist integers
k, l
∈
Z
such that
a
= 2
k
+ 1 and
c
= 2
l
+ 1
.
It follows that
ab
+
bc
=
b
(
a
+
c
) =
b
(2
k
+ 1 + 2
l
+ 1) =
b
(2
k
+ 2
l
+ 2) = 2
b
(
k
+
l
+ 1)
is even since
b
(
k
+
l
+ 1)
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 Spring '09
 dd
 Math, Odd, Z.

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