Math 3124
Tuesday, August 23
August 23 Ungraded Homework
Problem 1.2 on page 14
Let
α
,
β
and
γ
be mappings from
Z
to
Z
deﬁned by
α
(
n
) =
2
n
,
β
(
n
) =
n
+
1, and
γ
(
n
) =
n
3
for each
n
∈
Z
.
(a) Which of the three mappings are onto?
(b) Which of the three mappings are onetoone?
(c) Determine
α
(
N
)
,
β
(
N
)
, and
γ
(
N
)
.
(a)
β
is onto, the others are not.
(b)
α
,
β
,
γ
are all onetoone.
(c)
α
(
N
)
= positive even integers
{
2
,
4
,...
}
.
β
(
N
)
= integers
≥
2
{
2
,
3
,...
}
.
γ
(
N
)
= all
cubes
{
1
,
8
,
27
,...
}
.
Problem 1.14 on page 14
Deﬁne
f
:
R
→
R
by
f
(
x
) =
x
2
+
x
. Determine whether
f
is
onto, and whether
f
is onetoone. Also describe
f
(
P
)
where
P
is the positive real numbers.
We are given
f
:
R
→
R
deﬁned by
f
(
x
) =
x
2
+
x
. This is not onto; indeed by calculus the
minimum of
f
occurs when
f
0
(
x
) =
0, that is when 2
x
+
1
=
0, which is when
x
=

1
/
2 and
then
f
(
x
) =
1
/
4

1
/
2
=

1
/
4. Thus
f
(
x
)
≥ 
1
/
4 for all
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 Fall '08
 PARRY
 Math, Algebra, Addition, positive real numbers, positive square root

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