Greatest Element is Dual to Smallest Element
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $a \in S$.
The following are dual statements:
- $a$ is the greatest element of $S$
- $a$ is the smallest element of $S$
Proof
By definition, $a$ is the greatest element of $S$ if and only if:
- $\forall b \in S: b \preceq a$
The dual of this statement is:
- $\forall b \in S: a \preceq b$
By definition, this means $a$ is the smallest element of $S$.
The converse follows from Dual of Dual Statement (Order Theory).
$\blacksquare$