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MATH 74 HOMEWORK 7 (DUE WEDNESDAY OCTOBER 17)
1(a).
There is
f
:
A
→
B
given by
f
(1) = 1 and
f
(2) = 1, and
g
:
A
→
B
given
by
g
(1) = 1 and
g
(2) = 2, and
h
:
A
→
B
given by
h
(1) = 2 and
h
(2) = 1, and
k
:
A
→
B
given by
k
(1) = 2 and
k
(2) = 2.
1(b).
Three of them do. (As I named them,
f
,
g
, and
k
;
h
doesn’t.)
2.
Consider e.g.
f
1
:
C
→
C
given by
f
1
(
x
) = 1 for all
x
, and
f
2
:
C
→
C
given by
f
2
(
x
) = 2 for all
x
, and
f
3
:
C
→
C
given by
f
3
(
x
) = 3 for all
x
, and
f
4
:
C
→
C
given by
f
4
(
x
) =
x
for all
x
.
[Other solutions possible.]
3.
It is not necessarily true that
f
(
x
) =
x
for all
x
∈
R
. Consider for example the
function
f
:
R
→
R
given by
x
7→ 
x
. Then
f
(
f
(
x
)) =

(

x
)) =
x
for all
x
∈
R
,
but
f
(2) =

2 is not 2. [Other solutions possible.]
4(a).
The range of
g
is the set
R
=
{
1
,
3
,
7
,
9
}
.
4(b).
We ﬁrst prove
R
⊆
Im
g
. To see this, note that 1
∈
Im
g
because
g
(4) = 1
(as 3
4
= 81), that 3
∈
Im
g
because
g
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 Fall '07
 COURTNEY
 Math

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