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.

axiomatic sets 21 (arithmetics in integers)

The addition of two integers is defined as follow.
Given two integers $[(a,b)],[(c,d)]\quad (a,b,c,d\in\mathbb{N})$ ,
\[  [(a,b)]+[(c,d)]=[(a+c,b+d)] .  \]
You will not have any questions.

For example,
\[ [(2,1)]+[(3,1)]=[(5,2)] . \]
This means $1+2=3$ .

\[ [(4,1)]+[(1,4)]=[(5,5)] .  \]
This means $3+(-3)=0$

\[ [(1,3)]+[(1,4)]=[(2,7)] .  \]
This means $(-2)+(-3)=-5$ and equal to
\[  [(21,23)]+[(41,44)]=[(62,67)] . \]

The multiplication of integers is defined same as addition.
\[ [(a,b)]\times [(c,d)]=[(ac+bd,ad+bc)] . \]
Intuitively, as $(a-b)\times(c-d)=(ac+bd)-(ad+bc)$ is true, it will be also true.

You will be able to see the integers are closed under addition (subtraction) and multiplication.

The integers have been constructed on natural numbers $\mathbb{N}$  and some axioms.

(On this definition, unfortunately, $\mathbb{N}\subset\mathbb{Z}$  is not true as a matter of form.  )