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Prof. Bjorn Poonen
October 16, 2001
MATH 55 MIDTERM SOLUTIONS (white)
(1)
(5 pts. each) For each of (a)(g) below: If the proposition is true, write TRUE.
If the proposition is false, write FALSE. (Please do not use the abbreviations T and
F, since in handwriting they are sometimes indistiguishable.) No explanations are
required in this problem.
(a) The value of
(

141)
mod
8
is

5
.
FALSE. The value of (

141)
mod
8 is an integer between 0 and 7.
(b) The function
3
n
2
log
n
+ 5
n
(log
n
)
4
is
O
(
n
3
)
.
TRUE. If
c >
0, then log
n
is
O
(
n
c
), so 3
n
2
log
n
is
O
(
n
2+
c
) and 5
n
(log
n
)
4
is
O
(
n
1+4
c
). Thus 3
n
2
log
n
+ 5
n
(log
n
)
4
is
O
(max
{
n
2+
c
, n
1+4
c
}
) for any
c >
0. In
particular, this holds for
c
= 1
/
2, in which case we ﬁnd that 3
n
2
log
n
+ 5
n
(log
n
)
4
is
O
(
n
3
).
(c) The ceiling function
f
(
x
) =
d
x
e
, considered as a function from
R
to
R
, has
an inverse function.
FALSE. The ceiling function is not injective since
b
1
c
=
b
1
/
2
c
, so it is deﬁnitely
not bijective. Hence it has no inverse function.
(d) The set
{
1
,
2
,
3
} ×
Z
is countable.
TRUE. One can list its elements in a sequence as follows:
(1
,
0)
,
(2
,
0)
,
(3
,
0)
,
(1
,
1)
,
(2
,
1)
,
(3
,
1)
,
(1
,

1)
,
(2
,

1)
,
(3
,

1)
,
(1
,
2)
,
(2
,
2)
,
(3
,
2)
,
(1
,

2)
, . . .
(e) The proposition
∀
x
∃
y
(
x
≤
y
)
is true, when the universe of discourse is the
set of natural numbers.
TRUE. In words, this says “For all natural numbers
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This note was uploaded on 03/08/2010 for the course MATH 55 taught by Professor Strain during the Spring '08 term at University of California, Berkeley.
 Spring '08
 STRAIN
 Math

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