Maximal Element is Dual to Minimal Element
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$, and $a \in T$.
The following are dual statements:
- $a$ is a maximal element of $T$
- $a$ is a minimal element of $T$
Proof
By definition, $a$ is a maximal element of $T$ if and only if:
- $\forall t \in T: a \preceq t$ implies $a = t$
The dual of this statement is:
- $\forall t \in T: t \preceq a$ implies $a = t$
By definition, this means $a$ is a minimal element of $T$.
The converse follows from Dual of Dual Statement (Order Theory).
$\blacksquare$