MATH 135, Fall 2010
Solution of Assignment #2
Problem 1
.
Let
S
=
{
5
,

4
,

3
,

2
,

1
,
0
,
1
,
2
,
3
,
4
,
5
}
. List all of the elements in each of the
following sets.
(a)
A
=
±
x
∈
S
²
²
x
is even and
x
is a multiple of 3
³
.
(b)
B
=
±
x
∈
S
²
²
if
x
is odd, then
x
is a multiple of 5
³
.
(c)
A
∪
B
.
(d)
A
∩
B
.
Solution.
(a) We have
A
=
±
x
∈
S
²
²
x
is even
³
∪
±
x
∈
S
²
²
x
is a multiple of 3
³
=
{
4
,

2
,
0
,
2
,
4
} ∩ {
3
,
0
,
3
}
=
{
0
}
(b) Since
P
=
⇒
Q
is equivalent to (NOT
P
) OR
Q
, the statement “if
x
is odd, then
x
is a
multiple of 5” is equivalent to the statement “
x
is odd or
x
is a multiple of 5”, so we have
B
=
±
x
∈
S
²
²
x
is even or
x
is a multiple of 5
³
=
±
x
∈
S
²
²
x
is even
³
∪
±
x
∈
S
²
²
x
is a multiple of 5
³
=
{
4
,

2
,
0
,
2
,
4
} ∪ {
5
,
0
,
5
}
=
{
5
,

4
,

2
,
0
,
2
,
4
,
5
}
(c) Since
A
⊆
B
,
A
∪
B
=
B
=
{
5
,

4
,

2
,
0
,
2
,
4
,
5
}
.
(d) Since
A
⊆
B
,
A
∩
B
=
A
=
{
0
}
.
Problem 2