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Homework 3
Instructions:
1. [A Formula for the Fibonacci Numbers]
Recall the defnition oF the ±ibonacci numbers
F
(
n
)
±ibonacci number is given by
F
(
n
)=
φ
n

(1

φ
)
n
√
5
, where
φ
=
1+
√
5
2
is the Golden Ratio, and is a solution
oF the equation
φ
2

φ

1 = 0
.
2. [Strengthening the Induction Hypothesis]
Prove that, For all
n
≥
1
, all entries oF the matrix
±
10
11
²
n
are bounded above by
n
.
[HINT: Look at the title oF this problem!]
3.
[
WellOrdering Principle]
This problem concerns the WellOrdering Principle. ±irst, we will see that the set oF all pairs oF natural
numbers
N
×
N
=
{
(
a,b
):
a
∈
N
,b
∈
N
}
is a wellordered set, provided we defne the ordering correctly.
Then we will use this Fact to do an inductive prooF over pairs oF natural numbers.
(a) Suppose we defne the ordering
≺
1
on
N
×
N
by
(
)
≺
1
(
c,d
)
iF
a
b
<
c
d
, or iF
a
b
=
c
d
and
a < c
this defnition, we assume that
(
)
≺
1
(
c,
0)
whenever
b>
0
, and that
(
a,
0)
≺
1
(
c,
0)
iF and only
a < c
.) Show that, with this ordering,
N
×
N
is
not
wellordered.
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 Fall '08
 HAULLGREN

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