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CMPS 101
Midterm 1
Some solutions to review problems and one additional problem
Problem 2 from the Midterm 1 review sheet
State whether the following assertions are true or false.
If any statements are false, give a related
statement which is true.
a.
))
(
(
)
(
n
g
O
n
f
=
implies
))
(
(
)
(
n
g
o
n
f
=
.
False
))
(
(
)
(
n
g
o
n
f
=
implies
))
(
(
)
(
n
g
O
n
f
=
b.
))
(
(
)
(
n
g
O
n
f
=
if and only if
))
(
(
)
(
n
f
n
g
Ω
=
.
True
c.
))
(
(
)
(
n
g
n
f
Θ
=
if and only if
L
n
g
n
f
n
=
∞
→
))
(
/
)
(
(
lim
, where
∞
<
<
L
0
.
False
∞
<
<
L
0
and
L
n
g
n
f
n
=
∞
→
))
(
/
)
(
(
lim
implies
))
(
(
)
(
n
g
n
f
Θ
=
Problem 3 from the Midterm 1 review sheet
Prove that
))
(
)
(
(
))
(
(
))
(
(
n
g
n
f
n
g
n
f
⋅
Θ
=
Θ
⋅
Θ
.
In other words, if
))
(
(
)
(
1
n
f
n
h
Θ
=
and
))
(
(
)
(
2
n
g
n
h
Θ
=
,
then
))
(
)
(
(
)
(
)
(
2
1
n
g
n
f
n
h
n
h
⋅
Θ
=
⋅
.
Proof:
By hypothesis there exist positive constants
1
n
,
2
n
,
1
a
,
1
b
,
2
a
, and
2
b
such that
)
(
)
(
)
(
0
:
1
1
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 Spring '09
 AgoreBack

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