ASSIGNMENT 1
·
SOLUTIONS
MAT 472
·
FALL 2011
Problem 1
(Exercise 1.2.2)
.
True or False? If false, provide a counterexample.
(a)
If
A
1
⊇
A
2
⊇
A
3
· · ·
are infinite sets, then
∩
∞
n
=1
A
n
is infinite.
(b)
If
A
1
⊇
A
2
⊇
A
3
· · ·
are finite nonempty sets, then
∩
∞
n
=1
A
n
is finite and nonempty.
(c)
A
∩
(
B
∪
C
) = (
A
∩
B
)
∪
C
.
(d)
A
∩
(
B
∩
C
) = (
A
∩
B
)
∩
C
.
(e)
A
∩
(
B
∪
C
) = (
A
∩
B
)
∪
(
A
∩
C
).
Partial Solution.
(a) is false: consider
A
n
=
{
n, n
+ 1
, n
+ 2
, . . .
} ⊆
N
for each
n
.
Then
certainly the
A
n
are infinite and nested as desired, but
∩
∞
n
=1
A
n
=
∅
.
(b) is true, but surprisingly delicate to prove. (We were not asked to prove it.) Compare
with Theorems 1.4.1 and 3.3.5.
(c) is false; (d) and (e) are true, with elementary proofs.
Problem 2
(Exercise 1.2.7)
.
Recall that for any function
f
:
D
→
R
and
B
⊆
R
, by
definition
f

1
(
B
) =
{
x
∈
D

f
(
x
)
∈
B
}
.
(a)
Let
f
(
x
) =
x
2
,
A
= [0
,
4], and
B
= [

1
,
1]. Find
f

1
(
A
) and
f

1
(
B
). Does
f

1
(
A
∩
B
) =
f

1
(
A
)
∩
f

1
(
B
)? Does
f

1
(
A
∪
B
) =
f

1
(
A
)
∪
f

1
(
B
)?
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 Spring '06
 Spielberg
 Sets, Zagreb, S. Kaliszewski, School of Mathematical and Statistical Sciences

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