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4.26
Suppose
f
(
x
) =
f
(
y
). Then

f
(
x
)

f
(
y
)

= 0, so
c

x

y

α
≤
0. Since
c >
0 and
α >
0, this implies that

x

y

= 0 and hence
x
=
y
. Thus
f
is injective.
4.32
We have
f
(

x
) =

(

x
) =
x
and
g
(
y

1
) = (
y

1
)

1
=
y
for every
x
∈
F
and
y
∈
F
 {
0
}
, so
f
and
g
are surjective. If

x
1
=
f
(
x
1
) =
f
(
x
2
) =

x
2
and if
y

1
1
=
g
(
y
1
) =
g
(
y
2
) =
y

1
2
, then addition of
x
1
+
x
2
to both sides of the ﬁrst equation
and multiplication of both sides of the second equation by
y
1
y
2
yields
x
2
=
x
1
and
y
2
=
y
1
. Therefore
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This document was uploaded on 03/25/2011.
 Spring '10
 Math

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