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MODEL ANSWERS TO THE SECOND HOMEWORK
1. (a)
σ
is surjective. Indeed, let
t
∈
R
be a nonnegative real
number. Then
t
has a square root
s
∈
R
. In this case
f
(
s
)=
s
2
=
t
, by
defnition oF the square root.
σ
is not injective. ±or example,
σ
(1) =
σ
(

1) = 1.
(b) This is surjective, For the same reason as (a). It is now also
injective, since every positive real number has a unique positive root.
(c) Not surjective, since there is no positive integer whose square is
two. Not injective, For the same reason as (a).
(d) This is injective. Indeed suppose that
f
(
a
)=
f
(
b
). Then 2
a
=2
b
,
so that
a
=
b
. This is not surjective. Indeed there is not integer
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This note was uploaded on 07/15/2008 for the course MATH 111 taught by Professor Long during the Spring '08 term at UCSB.
 Spring '08
 LONG
 Algebra

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