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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.
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This note was uploaded on 02/10/2010 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.
 Fall '07
 COURTNEY
 Math

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