MATH 347: SOLUTIONS FOR MIDTERM 1
NOTE: 100% = 40 points.
1
(10 points)
:
Suppose
C
and
D
are subsets of the domain of a function
f
.
(a)
Prove that
f
(
C
∩
D
)
⊂
f
(
C
)
∩
f
(
D
).
By the definition of a subset, we have to show that any element of
f
(
C
∩
D
)
is also an element of
f
(
C
)
∩
f
(
D
). To this end, suppose
x
∈
f
(
C
∩
D
). Then
there exists
a
∈
C
∩
D
s.t.
f
(
a
) =
x
. As
a
∈
C
, we have
x
=
f
(
a
)
∈
f
(
C
).
Similarly,
x
∈
f
(
D
). Thus,
x
∈
f
(
C
)
∩
f
(
D
).
(b)
Give an example showing that the inlusion in part (a) may be strict.
It may happen that
f
(
C
∩
D
) is a
strict
subset of
f
(
C
)
∩
f
(
D
). To provide
an example, consider
U
=
{
1
,
2
,
3
}
,
C
=
{
1
,
3
}
, and
D
=
{
2
,
3
}
. Define
the function
f
:
U
→
N
by setting
f
(1) =
f
(2) = 5 and
f
(3) = 10. Then
f
(
C
) =
f
(
D
) =
{
5
,
10
}
, hence
f
(
C
)
∩
f
(
D
) =
{
5
,
10
}
. On the other hand,
C
∩
D
=
{
3
}
, hence
f
(
C
∩
D
) =
{
f
(3)
}
=
{
10
}
.
2
(10 points)
:
For
x
∈
R
, consider the following statements: (i)
P
: 0
<
x
2
<
3; (ii)
Q
:

x

= 1; (iii)
R
:
x
2
/
∈
Z
; (iv)
S
:
x
2
= 2. Which of the
following statements are true for any
x
∈
R
?
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 Spring '09
 ?
 Math, Addition, Multiplication, Sets, Natural number, conditional statement

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