axiomatic sets 20 (integers)

In preceding post, we have defined negative numbers using ordered pairs of $0$ and a natural number $n$.

\[ \mathbb{N}^-=\left\{(0,n) : n\in\mathbb{N},n\ne 0 \right\}  \]

Therefore, integers $\mathbb{Z}$ will be simply constructed by $\mathbb{N}\cup\mathbb{N}^-$ .

However, such integers are complicated as we always have to concern the difference of positive numbers and negative numbers. Because integers are made by two kinds of sets (0, positive numbers $1,2,3,\cdots$ and negative numbers $(0,1),(0,2),(0,3),\cdots$) .

We shall introduce an equivalent relation "$\sim$" to any ordered pair of two natural numbers.

For any natural numbers $k,h,n,m\in\mathbb{N}$ , $(k,h)\sim (n,m)$ if and only if $k+m=h+n$ .

That is,
\[ (1,0)\sim (2,1)\sim (3,2)\sim (4,3)\sim\cdots  , \]
\[ (2,0)\sim (3,1)\sim (4,2)\sim (5,3)\sim\cdots  , \]
\[ (0,0)\sim (1,1)\sim (2,2)\sim (3,3)\sim\cdots  , \]
\[ (0,1)\sim (1,2)\sim (2,3)\sim (3,4)\sim\cdots  , \]
$\cdots,\cdots$ .

We will write $[(k,h)]$  as meaning
\[ [(k,h)]=\left\{(n,m) : n,m\in\mathbb{N}, (k,h)\sim(n,m)  \right\} .  \]
For given $k,h$ , $[(k,h)]$  becomes a class of sets.

For example,
\[ [(1,0)]=\left\{(1,0),(2,1),(3,2),(4,3),\cdots  \right\} , \]
\[ [(2,0)]=\left\{(2,0),(3,1),(4,2),(5,3),\cdots  \right\} , \]
\[ [(0,0)]=\left\{(0,0),(1,1),(2,2),(3,3),\cdots  \right\} , \]
\[ [(0,1)]=\left\{(0,1),(1,2),(2,3),(3,4),\cdots  \right\} , \]
$\cdots,\cdots$ .

As a matter of course, for example,
\[ [(1,0)]=[(2,1)]=[(3,2)]=[(4,3)]=\cdots  .    \]

You will have to find $[(1,0)]$ as the intuitive natural number $1$ and
$[(2,0)]$  as $2$, $[(0,0)]$  as $0$ , $[(0,1)]$  as $-1$ , $\cdots\cdots$ .

We make $\mathbb{Z}$  denote the set of all equivalent classes with respect to $\sim$ .
The elements of $\mathbb{Z}$  will be called integers.

\[  \mathbb{Z}=\left\{ [(0,0)],[(1,0)],[(0,1)],[(2,0)],[(0,2)],\cdots  \right\} \]

You will see intuitively it means
\[   \mathbb{Z}=\left\{  0,1,-1,2,-2,\cdots \right\} . \]

Also, you can do define
\[  \mathbb{Z}=\left\{ [(0,0)],[(2,1)],[(2,3)],[(4,2)],[(3,5)],\cdots  \right\}  . \]