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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 COURTNEY
 Math, Natural number, Prime number

Click to edit the document details