Legal proof that set of Rational numbers is countable?

[Liquid7800]

Hello,

Just wondering if this is a correct way to say the set of RATIONAL numbers is countable:

Rationals (Q) is countable , because for every Q = p/q , such that p & q are positive INTS (Z)

and since the set of positive INTs (Z) is countable ( a 1:1 correspondence) Q is countable because it is a SUBSET of a countable set.....

it can be listed by listing those Q's with
denominator q = 1 in the first row of a listing matrix
denominator q =2 in the second row and so on...
with p1 = 1 , p2 =2, etc... for each row...

will eventually cover ALL rationals


REMARK:
Ive seen the little N x N matrix and path listing for each rational...but I would like to know...
How would you show this as a mapped 1:1 function?? such that A ---> B showing f(a) = some b??


Thanks for any input...

[daveyinaz]

How would you show this as a mapped 1:1 function?? such that A ---> B showing f(a) = some b??
Thanks for any input...

You might be asking for an onto function. 1:1 means if f(a)=f(b), then a = b.

[Liquid7800]

Thanks for the reply...
However, when I said
How would you show this as a mapped 1:1 function?? such that A ---> B showing f(a) = some b??
I shouldve said:
A 1:1 correspondence function or a bijective function...

[mathman]

The standard proof (Cantor) is as follows: Let a(i,j)=i/j. Then the order to show countable is a(1,1), a(2,1), a(1,2), a(3,1), [a(2,2)], a(1,3), a(4,1), a(3,2), a(2,3), a(1,4),.... skipping terms in [] since these are not in lowest terms.