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 Dual Pairs (Order Theory).


By definition, this means $a$ is the smallest element of $S$.


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

$\blacksquare$


Also see