Meet with Complement is Bottom

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


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

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

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


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


\(\ds a\) \(=\) \(\ds r \wedge \neg r\) by hypothesis
\(\ds \) \(=\) \(\ds \paren {s \wedge \neg s} \vee \paren {r \wedge \neg r}\) Boolean Algebra: Axiom $(\text {BA}_2 5)$
\(\ds \) \(=\) \(\ds \paren {r \wedge \neg r} \vee \paren {s \wedge \neg s}\) Boolean Algebra: Axiom $(\text {BA}_2 1)$
\(\ds \) \(=\) \(\ds s \wedge \neg s\) Boolean Algebra: Axiom $(\text {BA}_2 5)$
\(\ds \) \(=\) \(\ds b\) by hypothesis

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.