MATH 74 HOMEWORK 8 (DUE WEDNESDAY OCTOBER 31)
1(a).
#18 on p. 118 in Eccles.
1(b).
If
f
:
X
→
Y
and
g
:
Y
→
Z
are functions, and
g
◦
f
is surjective, is
g
necessarily surjective? Prove that it is, or give a counterexample.
1(c).
If
f
:
X
→
Y
and
g
:
Y
→
Z
are functions, and
g
◦
f
is surjective, is
f
necessarily surjective? Prove that it is, or give a counterexample.
2(a).
#9.4 on p. 114 of Eccles.
2(b).
If
f
:
X
→
Y
and
g
:
Y
→
Z
are functions, and
g
◦
f
is injective, is
f
necessarily injective? Prove that it is, or give a counterexample.
2(c).
If
f
:
X
→
Y
and
g
:
Y
→
Z
are functions, and
g
◦
f
is injective, is
g
necessarily injective? Prove that it is, or give a counterexample.
3.
Let
f
:
R
2
→
R
2
be the function given by
f
(
x, y
) = (
x
+
y, x

y
)
,
(
x, y
)
∈
R
2
.
3(a).
Is
f
surjective? Prove that it is, or prove that it is not. (This would involve
exhibiting a specific point (
a, b
)
∈
R
2
and proving that (
a, b
)
6∈
Im
f
.)
3(b).
Is
f
injective? Prove that it is, or prove that it is not.
4.
Let
g
:
Z
2
→
Z
2
be the function given by
g
(
x, y
) = (
x
+
y, x

y
)
,
(
x, y
)
∈
R
2
.
Note that
g
is given by the same formula as
f
of the previous problem. You may
use any relevant results from the previous problem.
4(a).
Is
g
injective? (Prove that it is, or prove that it is not.)
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 Fall '07
 COURTNEY
 Math, Inverse function, codomain, Prove, previous problem

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