Section 4.3
89
(c)
We
must prove that
f
is onetoone and onto.
Onetoone:
Supposef(ml/n1)
=
f(m2/n2).
Assumefirstthatm1 = m2 =
1,n1
=p~l
...
p~t
and
n2
=
q{l
...
q£",
where P1,
...
,Pi
are distinct primes,
q1,
...
,qk
are distinct primes,
€i
> 0
and
fi
> 0 for all
i.
Then
p~el1
...
p~et1
=
q~h1
...
q~j,.l.
By the Fundamental Theorem
of
Arithmetic, we conclude that
n1
=
n2,
so
ml/n1
=
m2/n2.
The same approach can be applied
to the other eight possible cases for the pair (m1 /
n1
, m2 /
n2),
showing that
f
is onetoone.
Onto: Given
YEN,
if
y
= 1, then
y
=
f(1).
Otherwise,
y
is a natural
number>
1 and so,
by the Fundamental Theorem
of
Arithmetic, is the unique product
of
powers
of
distinct primes.
Separating the powers into even and odd powers, we can write
y
=
p~el
...
p%e"
if
there are no odd
powers,
y
=
q~h
1
...
q~ft1
if
there are no even powers, and
y
=
p~el
...
p%e"q~h
1
...
q~!t1
if
there are both odd and even powers.
In
the first case,
Y
=
f(m)
where m =
p~l
...
P~",
in the second case,
y
=
f(
~)
where
n
=
q{l
...
qft
and, in the third case,
y
=
f(
~)
where
m =
p~l
...
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '10
 any
 Graph Theory, Prime number, Divisor

Click to edit the document details