# Dual Ordering is Ordering

## Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $\succeq$ denote the dual ordering of $\preceq$.

Then $\succeq$ is an ordering on $S$.

## Proof

By definition of ordering, $\preceq$ is reflexive, transitive and antisymmetric.

By definition, $\succeq$ is the inverse relation to $\preceq$.

By Inverse of Reflexive Relation is Reflexive, $\succeq$ is reflexive.

By Inverse of Antisymmetric Relation is Antisymmetric, $\succeq$ is antisymmetric.

By Inverse of Transitive Relation is Transitive, $\succeq$ is transitive.

Thus by definition $\succeq$ is an ordering on $S$.

$\blacksquare$