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 rationalsREMARK:
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.

 

[blue_raver22]

Hello,

Thank you for your question. Yes, your explanation is correct. The set of rational numbers, Q, is countable because it can be put into a 1:1 correspondence with the set of positive integers, Z. This can be shown by listing all the rational numbers in a matrix, as you described, or by using the diagonalization method.

To show that Q is countable using a 1:1 function, we can define a function f: Q → Z, where f(p/q) = p + q. This function maps each rational number to a unique positive integer, thus showing that Q is a subset of a countable set (Z) and is therefore countable itself.

I hope this helps clarify your understanding. Let me know if you have any further questions.

Best,

Scientist