1. (15 points, 3 points each) Let
A
=
{
4
,
5
,
6
}
,
B
=
{
3
,
4
}
,
C
=
{
7
,
8
,
9
}
. Find
(a)
A
∪
C
=
{
4
,
5
,
6
,
7
,
8
,
9
}
(b)
A

B
=
{
5
,
6
}
(c)
A
×
B
=
{
(4
,
3)
,
(4
,
4)
,
(5
,
3)
,
(5
,
4)
,
(6
,
3)
,
(6
,
4)
}
(d)
A
∩
B
=
{
4
}
(e)
A
∩
C
=
∅
2. (15 points, 5 points each part)
(a) Find the Power Set of
X
=
{
a,b
}
{∅
,
{
a
}
,
{
b
}
,
{
a,b
}}
(b) Transform the following by making a change of variable
i
=
k
+ 1:
n
X
k
=0
k
2
k
+
n
n
+1
X
i
=1
(
i

1)
2
i

1 +
n
(c) Use summation notation to rewrite the following: 1

1
2
+
1
3

1
4
+
1
5

1
6
6
X
k
=1
(

1)
k
+1
k
3. (15 points, 3 points each part) Give concise deﬁnitions of the following terms
(a) Well ordering principle: Let
S
be a set containing one or more integers all of
which are greater than some ﬁxed integer. Then
S
has a least element.
(b) Cartesian Product: The Cartesian Product
A
×
B
of two sets is the set of all
ordered pairs (
a,b
) such that
a
∈
A
and
b
∈
B
,
A
×
B
=
{
(
a,b
)

a
∈
A,b
∈
B
}
(c) Ordered ntuple: An ordered
n

tuple (
x
1
,x
2
,...,x
n
) is a collection of not
necessarily distinct elements of some set
X
together with an ordering such that
x
1
comes before
x
2
, etc., in the tuple.
(d) Disjoint sets: