Definition:Usual Ordering

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Definition

Let $X$ be one of the sets of numbers: $\N$, $\Z$, $\Q$, $\R$.

The usual ordering on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers.


Natural Numbers

Let $\N$ denote the natural numbers.

The ordering on $\N$ is the relation $\le$ everyone is familiar with.

For example, we use it when we say:

James has $6$ apples, which is more than Mary, who has $4$.

which can be symbolised as:

$6 \ge 4$


Integers

Definition 1

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

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

if and only if:

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

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 known as

The usual ordering is also known as the natural ordering.


Also see


Sources