# 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, $c = a \vee b$ if and only if:

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

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 $c = a \wedge b$.

$\blacksquare$