Math 361, Problem Set 1 Solutions
September 10, 2010
1. (1.2.9) If
C
1
,C
2
3
, . . .
are sets such that
C
k
⊇
C
k
+1
,
k
=1
,
2
,
3
, . . . ,
∞
,
we deFne lim
k
→∞
C
k
as the intersection
±
∞
k
=1
C
k
=
C
1
∩
C
2
∩
. . .
. ±ind
lim
k
→∞
C
k
for the following, and draw a picture of a typical ’
C
k
’ on the
line or plane, as appropriate:
a.
C
k
=
{
x
:2

1
/k < x
≤
2
}
,
k
,
2
,
3
, . . .
.
b.
C
k
=
{
x
<x
≤
2+
1
k
}
,
k
,
2
,
3
, . . .
.
c.
C
k
=
{
(
x, y
) : 0
≤
x
2
+
y
2
≤
1
k
}
,
k
,
2
,
3
, . . .
.
Note:
In addition to the book problem, I ask for a picture of the set.
Answer:
±or (
a
), note that 2
∈
C
k
for every
k
, but 2

±
is not in every
C
k
for any
± >
0. This is because if
k>
1
±
,2

±
±∈
C
k
. Therefore
lim
k
→∞
C
k
=
±
∞
k
=1
C
k
=
{
2
}
.
±or (
b
) note that 2
±∈
C
k
for any
k
. As before, 2 +
±
is not in
C
k
for
1
/±
. Therefore lim
k
→∞
C
k
=
∅
.
2. (1.2.4) Let Ω denote the set of points interior to or on the boundary of a
cube with edge of length 1. Moreover, say the cube is in the Frst octant
with one vertex at the point (0
,
0
,
0) and an opposite vertex at the point
(1
,
1
,
1). Let
Q
(
C
)=
² ² ²
C
dxdydz
.
(a.) If
C
⊆
Ω is the set
{
(
x, y, z
):0
< x < y < z <
1
}
compute
Q
(
C
).
Describe the set
C
in words (or picture).
(b.) If
C
⊆
Ω is the set
{
(
x, y, z
=
y
=
z<
1
}
compute
Q
(
C
).
Describe the set
C
in words (or picture).