# Dual of Preordered Set is Preordered Set

## 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

$\succeq$ is reflexive.
$\succeq$ is transitive.

Hence $\succeq$ is a preordering.

$\blacksquare$