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Math 104  Lecture 1  Exam 1 Solutions
Monday, July 12, 2010
1. Recall that for any set
S
, the power set of
S
, denoted
P
(
S
), is the collection of all sub
sets of
S
. That is
P
(
S
) =
{
X
⊂
S
}
. Prove that if
A
and
B
have the same cardinality,
then
P
(
A
) and
P
(
B
) have the same cardinality.
Let
f
:
A
→
B
be a bijection. Deﬁne
g
:
P
(
A
)
→ P
(
B
)
by
g
(
X
) =
f
[
X
] =
{
f
(
x
) :
x
∈
X
}
. Since
f
is a bijection,
f
inverse exists. Then the map
h
(
Y
) =
f

1
[
Y
]
is an inverse
of
g
because
h
(
g
(
X
)) =
f

1
h
f
[
X
]
±
=
X
and
g
(
h
(
Y
)) =
f
h
f

1
[
Y
]
i
=
Y
.
2. Prove that if
a < b
are real numbers, then there is a rational number
q
so that
a < q < b
.
(Do not use the theorem in the book which includes this fact.)
Let
a
be the cut
A

A
0
and
b
be the cut
B

B
0
. Since
a < b
,
A
(
B
. Pick
q
0
∈
B
\
A
.
Since
B
is a the left side of a cut, pick some
q
∈
B q > q
0
. Then
a < q < b
as desired.
3. Prove or disprove:
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 Summer '08
 RIEMAN
 Sets

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