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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~",
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 Summer '10
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 Graph Theory

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