Definition:Strict Ordering on Integers
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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 $a \ne b$.