Dual of Preordered Set is Preordered Set

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Theorem

Let $P = \left({S, \preceq}\right)$ be a preordered set.

Then dual of $P$, $P^{-1} = \left({S, \succeq}\right)$ is also a preordered set.


Proof

By Inverse of Reflexive Relation is Reflexive:

$\succeq$ is reflexive.

By Inverse of Transitive Relation is Transitive:

$\succeq$ is transitive.

Hence $\succeq$ is a preordering.

$\blacksquare$


Sources