# Lower Set is Dual to Upper Set

Jump to navigation
Jump to search

## Theorem

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

Let $T \subseteq S$.

The following are dual statements:

## Proof

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

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

The dual of this statement is:

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

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

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

$\blacksquare$