(
m
1
,m
2
)
±
= (
n
1
,n
2
)
, but
f
(
m
1
,m
2
) =
f
(
n
1
,n
2
)
. There are many possibili
ties, such as
(1
,
0)
and
(0
,
1)
. In fact, since
+
is commutative,
(
m,n
)
,
(
n,m
)
is a counterexample for any
m,n
.
E
XAMPLE
4.10
The function
f
on natural numbers deFned by
f
(
x
) =
x
2
is onetoone, but
the similar function
f
on integers is not. The function
f
on integers deFned
by
f
(
x
) =
x
+ 1
is surjective, but the similar function on natural numbers is
not.
E
XAMPLE
4.11
The function
f
on the real numbers given by
f
(
x
) = 4
x
+ 3
is a bijective
function. To prove that
f
is onetoone, suppose that
f
(
n
1
) =
f
(
n
2
)
, which
means that
4
n
1
+ 3 = 4
n
2
+ 3
. It follows that
4
n
1
= 4
n
2
, and hence
n
1
=
n
2
.
To prove that
f
is onto, let
n
be an arbitrary real number.
We have
f
((
n

3)
/
4) =
n
by deFnition of
f
, and hence
f
is onto. Since
f
is both
onetoone and onto, it is bijective. Notice that the function
f
on the
natural
numbers given by
f
(
x
) =
x
+ 3
is onetoone but
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 Spring '10
 Koskesh
 Math, Natural Numbers, Finite set, n pigeonholes

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