Meet with Complement is Bottom

Theorem

Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 2.

Then:

$\exists \bot \in S: \forall a \in S: a \wedge \neg a = \bot$

where $\wedge$ denotes the meet operation in $S$.

This element $\bot$ is unique for any given $S$, and is named bottom.

Let $\exists r, s \in S: r \wedge \neg r = a, \ s \wedge \neg s = b$

Then:

$\ds a$

$=$

$\ds r \wedge \neg r$

$\ds $

$=$

$\ds \paren {s \wedge \neg s} \vee \paren {r \wedge \neg r}$

$\ds $

$=$

$\ds \paren {r \wedge \neg r} \vee \paren {s \wedge \neg s}$

$\ds $

$=$

$\ds s \wedge \neg s$

$\ds $

$=$

$\ds b$

Thus, whatever $r$ and $s$ may be:

$r \wedge \neg r = s \wedge \neg s$

This unique element can be assigned the symbol $\bot$ and named bottom as required.

$\blacksquare$