CS70 Discrete Mathematics and Probability, Spring 2012
Homework 3
Out: 2 Feb. Due: 5pm, 9 Feb.
Instructions:
Start each problem on a new sheet. Write your name,
section number
and “CS70” on every sheet.
If you use more than one sheet for a problem, staple them together (but do
not
staple different problems together).
Put your solutions in the boxes on Soda level 2 by 5pm on Thursday: your solution to Q1 goes in box CS70–1, your
solution to Q2 goes in box CS70–2, and so on (
ﬁve
boxes total). You are encouraged to form small groups (two to four
people) to work through the homework, but you
must
write up all your solutions on your own.
1. [Strengthening the Induction Hypothesis]
Prove that, for all
n
≥
1
, all entries of the matrix
±
1
0
1
1
²
n
are bounded above by
n
.
[HINT: Look at the title of this problem! To come up with a stronger hypothesis, try evaluating the matrix
for the ﬁrst few values of
n
.]
2.
[
WellOrdering Principle]
This problem concerns the WellOrdering Principle. First, 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 deﬁne the ordering correctly.
Then we will use this fact to do an inductive proof over pairs of natural numbers.
(a) Suppose we deﬁne the ordering
≺
on
N
×
N
by
(
a,b
)
≺
(
c,d
)
if and only if either
a < c
, or
a
=
c
and
b < d
. (This is usually called the “lexicographic ordering.”) Show, using the WellOrdering Principle
for the natural numbers
N
, that
N
×
N
is wellordered with this ordering.
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 Spring '08
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