Antisymmetric Preordering is Ordering

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Theorem

Let $\mathcal R \subseteq S \times S$ be a preordering on a set $S$.

Let $\mathcal R$ also be antisymmetric.


Then $\mathcal R$ is an ordering on $S$.


Proof

By definition, a preordering on $S$ is a relation on $S$ which is:

$(1): \quad$ reflexive

and:

$(2): \quad$ transitive.


Thus $\mathcal R$ is a relation on $S$ which is reflexive, transitive and antisymmetric.

Thus by definition $\mathcal R$ is an ordering on $S$.

$\blacksquare$


Sources