# Definition:Strict Ordering on Integers

## Definition

### Definition 1

The integers are strictly ordered on the relation $<$ as follows:

$\forall x, y \in \Z: x < y \iff y - x \in \Z_{>0}$

That is, $x$ is less than $y$ if and only if $y - x$ is (strictly) positive.

### Definition 2

The integers are strictly ordered on the relation $<$ 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:

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

where:

$+$ denotes natural number addition
$a < b$ denotes natural number ordering $a \le b$ such that $a \ne b$.