Question on proving that the rationals are countably infinite
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
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forward_behind1
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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 yesterday
forward_behind1
11
New contributor
forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
surjective onto what?
– mathworker21
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
1
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
@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
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday
|
show 3 more comments
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 yesterday
forward_behind1
11
New contributor
forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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 yesterday
forward_behind1
11
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forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
forward_behind1
11
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forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
asked yesterday
forward_behind1
11
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forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked yesterday
forward_behind1
11
asked yesterday
forward_behind1
11
11
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New contributor
forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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forward_behind1 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
surjective onto what?
– mathworker21
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
1
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
@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
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday
|
show 3 more comments
surjective onto what?
– mathworker21
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
1
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
@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
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday
surjective onto what?
– mathworker21
yesterday
surjective onto what?
– mathworker21
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
1
1
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
@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
– forward_behind1
yesterday
@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
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday
|
show 3 more comments
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surjective onto what?
– mathworker21
yesterday
Onto the natural numbers. Added it now to the question
– forward_behind1
yesterday
1
wait what? how could it possibly be surjective. how would you get 7
– mathworker21
yesterday
@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
– forward_behind1
yesterday
Or it is only necessary for there to be an injection between Q and N?
– forward_behind1
yesterday