# Ordering/Examples

## Examples of Orderings

### Integer Difference on Reals

Let $\preccurlyeq$ denote the relation on the set of real numbers $\R$ defined as:

$a \preccurlyeq b$ if and only if $b - a$ is a non-negative integer

Then $\preccurlyeq$ is an ordering on $\R$.

### Example Ordering on Integers

Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:

$a \preccurlyeq b$ if and only if $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

where $\le$ is the usual ordering on $\Z$.

Then $\preccurlyeq$ is an ordering on $\Z$.