Math 122 F01 and F02 2007
Assignment 2 Solutions
1. Let
A
and
B
be sets (
i.e.
subsets of some universe
U
).
(a) [10] Suppose
A
contains at least two elements. Prove that if every proper subset of
A
is
a subset of
B
, then
A
⊆
B
. (Hint: what does it mean for a subset of
A
with size one to be a
subset of
B
?)
Since
A
has at least two elements, for every
x
∈
A
the set
{
x
}
is a proper subset of
A
. By the condition,
each of these is a subset of
B
. From the deﬁnitions,
{
x
} ⊆
B
⇔
x
∈
B
. Therefore, for every
x
∈ U
, if
x
∈
A
then
x
∈
B
. That is,
A
⊆
B
.
(b) [4] Give an example and an explanation to demonstrate that the implication in part (a)
can be false if
A
contains only one element.
Let
A
=
{
1
}
and
B
=
{
2
}
. The only proper subset of
A
is the empty set, which is a subset of
B
. Thus,
every proper subset of
A
is a subset of
B
, but
A
6⊆
B
.
2. [10] Prove that
A
=
B
if and only if
P
(
A
) =
P
(
B
)
.
(
⇒
) Suppose
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 Spring '11
 stephenlang
 Calculus, Sets, Empty set

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