Upper Bound is Dual to Lower Bound
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $a \in S$ and $T \subseteq S$.
The following are dual statements:
- $a$ is an upper bound for $T$
- $a$ is a lower bound for $T$
Proof
By definition, $a$ is an upper bound for $T$ if and only if:
- $\forall t \in T: t \preceq a$
The dual of this statement is:
- $\forall t \in T: a \preceq t$
By definition, this means $a$ is a lower bound for $T$.
The converse follows from Dual of Dual Statement (Order Theory).
$\blacksquare$