axiomatic sets 22 (rational numbers)

We saw the two kinds of numbers would be created.

Natural numbers $\mathbb{N}$  are constructed by $\phi$ and successor sets.

Integers $\mathbb{Z}$  are constructed by the equivalent classes $[(a,b)]$ of an ordered pair of two natural numbers.

In this post, we shall define rational numbers $\mathbb{Q}$ .

Rational numbers will be defined by the same way of integers.
These are constructed by the equivalent classes $[(p,q)]$ of an ordered pair of two integers.

Suppose an arbitrary ordered pair $(p,q)$ of two integers, where $q\ne 0$ .
(you can see it  $\frac{p}{q}$ . )

The equivalent relation "$\sim$" means, on two ordered pairs $(p,q)$  and $(r,s)$ (where $q\ne 0$ and $s\ne 0$),
\[ (p,q)\sim (r,s)\quad \leftrightarrow\quad ps=qr . \]

We make $\mathbb{Q}$  denote the set of all equivalent classes with respect to "$\sim$" .

The elements of $\mathbb{Q}$  will be called rational numbers $[(p,q)]$.

It is very simple.