Definition:Usual Ordering

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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:

$6 \ge 4$


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.

Also known as

The usual ordering is also known as the natural ordering.

Also see