Totally Ordered Set/Examples/Example Ordering on Integers
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Examples of Totally Ordered Sets
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 $\struct {\Z, \preccurlyeq}$ is a totally ordered set.
Proof
From Example Ordering on Integers, $\preccurlyeq$ is an ordering on $\Z$.
Let $a, b \in \Z$.
Let $a, b \ge 0$.
Then either $a \le b$ or $b \le a$.
Hence either $a \preccurlyeq b$ or $b \preccurlyeq a$.
Let $a, b < 0$.
Then either $a \le b$ or $b \le a$.
Hence either $a \preccurlyeq b$ or $b \preccurlyeq a$.
Let $a < 0$ but $b \ge 0$.
Then $a \preccurlyeq b$.
Let $b < 0$ but $a \ge 0$.
Then $b \preccurlyeq a$.
In all cases, either $a \preccurlyeq b$ or $b \preccurlyeq a$.
Hence the result.
$\blacksquare$
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $4$