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 $1$.
$\blacksquare$