CS 173: Discrete Structures, Spring 2010
Homework 7 Solutions
This homework contains 4 problems worth a total of 40 regular points. It is due on Friday,
March 19th at 4pm. Put your homework in the appropriate dropbox in the Siebel basement.
1.
BigO proofs [10 points]
For both parts of this problem, make sure your proof is presented clearly, in forward order,
and spells out the (maybe 45 steps of) algebra clearly.
(a) Prove that 3
x
is
O
(
x
2
x
+2
).
Solution:
We need to prove that for some
c
and
k
that

3
x

< c

x
2
x
+2

for all
x > k
.
Let
k
= 9
and
c
= 4
.

3
x

<

3
x
+ (
x

8)

= 4

x

2

= 4

x

2
x
+4
x
+2

<
4

x

2
x
x
+2

= 4

x
2
x
+2

This inequality proves that

3
x

<
4

x
2
x
+2

for all
x >
9
.
Alternate Solution:
Without loss of generality we may assume
k >
0. Since we always take
x > k
this
allows us to ignore absolute values and avoid multiplication and division by negative
numbers and zero. Observe that for all
x
≥
k
the following inequalities are equivalent
(but we are NOT assuming them to be true yet):
3
x
≤
c
·
x
2
x
+ 2
3
x
(
x
+ 2)
≤
c
·
x
2
3(
x
+ 2)
≤
c
·
x
3
x
+ 6
≤
c
·
x
So to prove the ﬁrst inequality holds for all
x
≥
k
, it suﬃces to choose
c
and
k
so
that the last inequality is satisﬁed for all
x
≥
k
. Choosing
c
= 6 and
k
= 2 obviously
works.
(b) Use a proof by contradiction to show that
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 Spring '08
 Erickson
 Mathematical Induction, Inductive Reasoning, Negative and nonnegative numbers, Mathematical logic, Proof by contradiction

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