Definition:Ordering on Integers
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Definition
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.
Also see
- Results about orderings on integers can be found here.