Ordering/Examples/Integer Difference on Reals
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Example of Ordering
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 $\R$.
Proof
Reflexivity
We have that:
- $\forall a \in \R: a - a = 0 \in \Z_{\ge 0}$
Thus:
- $\forall a \in \R: a \preccurlyeq a$
So $\preccurlyeq$ has been shown to be reflexive.
$\Box$
Transitivity
Let $a, b, c \in \R$ such that:
| \(\ds a\) | \(\preccurlyeq\) | \(\ds b\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds b\) | \(\preccurlyeq\) | \(\ds c\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists m, n \in \Z_{\ge 0}: \, \) | \(\ds b - a\) | \(=\) | \(\ds m\) | Definition of $\preccurlyeq$ | |||||||||
| \(\, \ds \land \, \) | \(\ds c - b\) | \(=\) | \(\ds n\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists m, n \in \Z_{\ge 0}: \, \) | \(\ds \paren {b - a} + \paren {c - b}\) | \(=\) | \(\ds m + n\) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists m + n \in \Z_{\ge 0}: \, \) | \(\ds c - a\) | \(=\) | \(\ds m + n\) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(\preccurlyeq\) | \(\ds c\) | Definition of $\preccurlyeq$ |
So $\preccurlyeq$ has been shown to be transitive.
$\Box$
Antisymmetry
Let $a, b \in \R$ such that:
| \(\ds a\) | \(\preccurlyeq\) | \(\ds b\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds b\) | \(\preccurlyeq\) | \(\ds a\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists m, n \in \Z_{\ge 0}: \, \) | \(\ds b - a\) | \(=\) | \(\ds m\) | Definition of $\preccurlyeq$ | |||||||||
| \(\, \ds \land \, \) | \(\ds a - b\) | \(=\) | \(\ds n\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {a - b} + \paren {b - a}\) | \(=\) | \(\ds m + n\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds m + n\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds m = n\) | \(=\) | \(\ds 0\) | Definition of $m$ and $n$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds b\) | Definition of $\preccurlyeq$ |
So $\preccurlyeq$ has been shown to be antisymmetric.
$\Box$
$\preccurlyeq$ has been shown to be reflexive, transitive and antisymmetric.
Hence by definition it is an ordering.
$\blacksquare$
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: $(3)$