Ordering/Examples
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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 $\Z$.
Monarchy
Let $K$ denote the set of British monarchs.
Let $\MM$ denote the relation on $K$ defined as:
- $a \mathrel \MM b$ if and only if $a$ was monarch after or at the same time as $b$.
Its dual $\MM^{-1}$ is defined as:
- $a \mathrel {\MM^{-1} } b$ if and only if $a$ was monarch before or at the same time as $b$.
Then $\MM$ and $\MM^{-1}$ are orderings on $K$.
American Presidency
Let $S$ denote the set of American presidents.
Let $\PP$ denote the relation on $S$ defined as:
- $a \mathrel \PP b$ if and only if $a$ was president after or at the same time as $b$.
Because Grover Cleveland was president both before and after Benjamin Harrison, $\PP$ is not an antisymmetric relation.
Thus $\PP$ is not an ordering on $S$.