Question on proving that the rationals are countably infinite
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I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $Bbb Q$ to $Bbb N$
$$f(x) = begin{cases}
0, & text{if $x$ = 0} \
2^m 3^n, & text{if $x$ > 0} \
2^m 3^n cdot 5 &text{if $x$ < 0}
end{cases} where x = frac mn and gcd(m,n) = 1 $$
One can see that it is injective by the uniqueness of prime factorization but how is it surjective?
functions rational-numbers
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
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|
show 3 more comments
$begingroup$
I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $Bbb Q$ to $Bbb N$
$$f(x) = begin{cases}
0, & text{if $x$ = 0} \
2^m 3^n, & text{if $x$ > 0} \
2^m 3^n cdot 5 &text{if $x$ < 0}
end{cases} where x = frac mn and gcd(m,n) = 1 $$
One can see that it is injective by the uniqueness of prime factorization but how is it surjective?
functions rational-numbers
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
$endgroup$
$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
1
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
|
show 3 more comments
$begingroup$
I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $Bbb Q$ to $Bbb N$
$$f(x) = begin{cases}
0, & text{if $x$ = 0} \
2^m 3^n, & text{if $x$ > 0} \
2^m 3^n cdot 5 &text{if $x$ < 0}
end{cases} where x = frac mn and gcd(m,n) = 1 $$
One can see that it is injective by the uniqueness of prime factorization but how is it surjective?
functions rational-numbers
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
$endgroup$
I just a question on a proof for the rationals being countably infinite from a textbook. We consider the following function, a mapping from $Bbb Q$ to $Bbb N$
$$f(x) = begin{cases}
0, & text{if $x$ = 0} \
2^m 3^n, & text{if $x$ > 0} \
2^m 3^n cdot 5 &text{if $x$ < 0}
end{cases} where x = frac mn and gcd(m,n) = 1 $$
One can see that it is injective by the uniqueness of prime factorization but how is it surjective?
functions rational-numbers
functions rational-numbers
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
edited Dec 25 '18 at 11:45
forward_behind1
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
asked Dec 25 '18 at 11:44
forward_behind1forward_behind1
32
32
$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
1
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
|
show 3 more comments
$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
1
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
1
1
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
|
show 3 more comments
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$begingroup$
surjective onto what?
$endgroup$
– mathworker21
Dec 25 '18 at 11:44
$begingroup$
Onto the natural numbers. Added it now to the question
$endgroup$
– forward_behind1
Dec 25 '18 at 11:47
1
$begingroup$
wait what? how could it possibly be surjective. how would you get 7
$endgroup$
– mathworker21
Dec 25 '18 at 11:48
$begingroup$
@mathworker21 What I was wondering. If we are trying to show that it is countably infinite then we wish to establish a bijection from Q to N and that what was given in the book
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50
$begingroup$
Or it is only necessary for there to be an injection between Q and N?
$endgroup$
– forward_behind1
Dec 25 '18 at 11:50