# Equivalence of Definitions of Ordering on Integers

## Theorem

The following definitions of the concept of **Ordering on Integers** are equivalent:

### Definition 1

The integers are ordered on the relation $\le$ as follows:

- $\forall x, y \in \Z: x \le y$

- $\exists c \in P: x + c = y$

where $P$ is the set of positive integers.

That is, $x$ is **less than or equal** to $y$ if and only if $y - x$ is non-negative.

### Definition 2

The integers are ordered on the relation $\le$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

Then:

- $x < y \iff x_1 + y_2 \le x_2 + y_1$

where:

- $+$ denotes natural number addition
- $\le$ denotes natural number ordering.

## Proof

Let $x, y \in \Z$ such that $x \le y$.

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

### $(1)$ implies $(2)$

Let $\le$ be an ordering on integers by definition $1$.

Then by definition:

- $y - x$ is non-negative

That is:

- $\map \PP {y - x}$

where $\PP$ is the positivity property.

Thus:

- $\eqclass {y_1, y_2} {} - \eqclass {x_1, x_2} {}$ is non-negative

By definition of integer subtraction:

- $\eqclass {y_1, y_2} {} + \eqclass {x_2, x_1} {}$ is non-negative

and by the formal definition of integers:

- $\eqclass {y_1 + x_2, y_2 + x_1} {}$ is non-negative

We have that $y_1 + x_2$ and $y_2 + x_1$ are natural numbers.

Thus by definition of natural number ordering:

- $y_1 + x_2 \ge y_2 + x_1$

Thus $\le$ is an ordering on integers by definition $2$.

$\Box$

### $(2)$ implies $(1)$

Let $\le$ be an ordering on integers by definition $2$.

Then by definition:

- $x_1 + y_2 \le x_2 + y_1$

That is:

- $x_2 + y_1 \ge x_1 + y_2$

Hence by the formal definition of integers:

- $\eqclass {y_1 + x_2, y_2 + x_1} {}$ is non-negative

By definition of integer addition:

- $\eqclass {y_1, y_2} {} + \eqclass {x_2, x_1} {}$ is non-negative

By definition of integer subtraction:

- $\eqclass {y_1, y_2} {} - \eqclass {x_1, x_2} {}$ is non-negative

That is:

- $\map \PP {y - x}$

where $\PP$ is the positivity property.

Thus $\le$ is an ordering on integers by definition $1$.

$\blacksquare$