Join is Dual to Meet

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b, c \in S$.

The following are dual statements:


 * $c = a \vee b$, the join of $a$ and $b$
 * $c = a \wedge b$ the meet of $a$ and $b$

Proof
By definition of join, $c = a \vee b$ iff:


 * $c = \sup \left\{{a, b}\right\}$

where $\sup$ denotes supremum.

The dual of this statement is:


 * $c = \inf \left\{{a, b}\right\}$

where $\inf$ denotes infimum, by Dual Pairs (Order Theory).

By definition of meet, this means $c = a \wedge b$.

Also see

 * Duality Principle (Order Theory)