Lower Bound is Upper Bound for Inverse Ordering
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Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$.
Let $m$ be a lower bound for $\struct {T, \preceq}$.
Let $\succeq$ be the dual ordering of $\preceq$.
Then $m$ is an upper bound for $\struct {T, \succeq}$.
Proof
Let $m$ be a lower bound for $\struct {T, \preceq}$.
That is:
- $\forall a \in T: m \preceq a$
By definition of dual ordering, it follows that:
- $\forall a \in T: a \succeq m$
That is, $M$ is an upper bound for $\struct {T, \succeq}$.
$\blacksquare$