Ordering/Examples/Example Ordering on Integers

Example of Ordering
Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:
 * $a \preccurlyeq b$ $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

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

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

Reflexivity
Let $0 \le a$.

Then $0 \le a \le a$ and so $a \preccurlyeq a$.

Let $a < 0$.

Then $a \le a < 0$ and so $a \preccurlyeq a$.

Thus in both cases $a \preccurlyeq a$.

So $\preccurlyeq$ has been shown to be reflexive.

Transitivity
Let $a, b, c \in \R$ such that:


 * $a \preccurlyeq b \text { and } b \preccurlyeq c$

Let $0 \le a \le b$.

Then:
 * $0 \le b \le c$

and so:
 * $0 \le a \le c$

That is: $a \preccurlyeq c$

Let $b \le a < 0$.

Then there are two cases for $c$:


 * $(1): \quad c \le b < 0$

Then we have:
 * $c \le a < 0$

That is: $a \preccurlyeq c$


 * $(2): \quad b < 0 \le c$

Then we have:
 * $a < 0 \le c$

That is: $a \preccurlyeq c$

In both cases:

$a \preccurlyeq c$

Let $a < 0 \le b$.

Then:
 * $0 \le b \le c$

and so:
 * $0 < a \le c$

That is: $a \preccurlyeq c$

In all cases: $a \preccurlyeq c$

So $\preccurlyeq$ has been shown to be transitive.

Antisymmetry
Let $a, b \in \R$ such that:


 * $a \preccurlyeq b \text { and } b \preccurlyeq a$

Let $0 \le a \le b$.

Then as $b \ge 0$:


 * $0 \le b \le a$

Thus:
 * $a \le b \text { and } b \le a$

and as $\le$ is the usual ordering, and so antisymmetric:


 * $a = b$

Let $b \le a < 0$.

Then as $b < 0$:


 * $a \le b < 0$

Thus:
 * $a \le b \text { and } b \le a$

and as $\le$ is the usual ordering, and so antisymmetric:


 * $a = b$

Let $a < 0 \le b$.

But then because $b \le a$:


 * $0 \le b \le a$

which contradicts $a < 0$.

Hence if both $a \le b$ and $b \le a$, it cannot happen that $0 \le b \le a$.

All cases are accounted for, and it has been shown that:


 * $a \preccurlyeq b \text { and } b \preccurlyeq a \implies a = b$

So $\preccurlyeq$ has been shown to be antisymmetric.

$\preccurlyeq$ has been shown to be reflexive, transitive and antisymmetric.

Hence by definition it is an ordering.

It is seen that $\preccurlyeq$ looks like:


 * $-1 \preccurlyeq -2 \preccurlyeq -3 \preccurlyeq \cdots \preccurlyeq 0 \preccurlyeq 1 \preccurlyeq 2 \preccurlyeq \cdots$