Show that Q, that set of all rational numbers, is countable.

To show:

ℚ the set of all rational numbers, is countable.

Given information:

ℚ: The set of rational numbers.

Concept used:

The set of rational number is defined as follows:

ℚ={pq|p,q∈ℤ and q≠0}

The set rational number can also be written as ℚ=ℚ−∪{0}∪ℚ+.

Proof:

From the definition of the rational numbers the set of positive rational numbers can be formed in a sequence as follows.

11  21  31  41  51  61   ...12  22  32  42  52  62  ...13  23  33  43  53  63   ...14  24  34  44  54  64