Lower Set is Dual to Upper Set

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

Let $T \subseteq S$.

The following are dual statements:


 * $T$ is a lower set in $S$
 * $T$ is an upper set in $S$

Proof
By definition, $T$ is a lower set in $S$ :


 * $\forall t \in T: \forall s \in S: s \preceq t \implies s \in T$

The dual of this statement is:


 * $\forall t \in T: \forall s \in S: t \preceq s \implies s \in T$

by Dual Pairs (Order Theory).

By definition, this means $T$ is an upper set in $S$.

The converse follows from Dual of Dual Statement (Order Theory).

Also see

 * Duality Principle (Order Theory)