Dual of Dual Ordering

Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\left({S, \succeq}\right)$ be its dual.

Then the dual of $\left({S, \succeq}\right)$ is again $\left({S, \preceq}\right)$.

Proof

Denote with $\preceq'$ the dual of $\succeq$.

By definition of dual ordering, we thus have for all $a, b \in S$:

$a \preceq b$ iff $b \succeq a$
$b \succeq a$ iff $a \preceq' b$

Hence $a \preceq b$ iff $a \preceq' b$.

The result follows from Equality of Relations.

$\blacksquare$